Lesson PLanning CaLendar - iblog. · PDF fileMode Mean Median Measures of Variation ... Building Vocabulary: Crossword Puzzle offers students an opportunity to improve their understanding

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Lesson PLanning CaLendar

Use this Lesson Planning Calendar to determine how much time to allot for each topic.

Schedule Day One Day TwoTraditional Period (50 minutes) Frequency Distributions

Measures of Central Tendency

Measures of Variation

Normal Distribution

Comparative Statistics

Correlation Coefficient

Statistical Inference

Block schedule (90 minutes) Frequency Distributions



Normal Distribution




Psychology’s Statistics

B2E3e_book_ATE.indb 1 3/19/12 11:26 AM

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aCTiviTy PLanner From The TeaCher’s resourCe maTeriaLs

Use this Activity Planner to bring active learning to your daily lessons.

Topic ActivitiesFrequency distributions Building Vocabulary: Crossword Puzzle (10 min.)

Application Activity: Organizing and Interpreting Data (15 min.)

Application Activity: Describing Data (15 min.)

measures of Central Tendency Digital Connection: Technology Application Activity: PsychSim: “Descriptive Statistics” (20 min.)

measures of variation

normal distribution Cooperative Learning Activity: A Tasty Sample(r): Teaching About Sampling Using M&Ms® (20 min.)

Comparative statistics Application Activity: The Water Cup Toss Test (15 min.)

Correlation Coefficient Application Activity: Creating a Scatterplot (15 min.)

Application Activity: Correlation and the Challenger Disaster (15 min.)

statistical inference Application Activity: When Is a Difference Significant? (15 min.)

Portfolio Project: Applying Research Skills

41

Frequency Distributions

Measures of Central Tendency●Mode●Mean●Median

Measures of Variation●Range●StandardDeviation

Normal Distribution




Statistics are more a matter of attitude than numbers. Many people (not you, ofcourse)haveanegativeviewofstatisticsandconvincethemselvesthatthisisatopicbeyondcomprehension.Maybeit’stheword—statistics ishardtopronounce!Theconceptsthemselvesarereallynotsohard.Lotsofpeopleenjoythesubject,andthereisnodenyingitsusefulness.Thinkoftryingtoparticipateinafantasysportsleague,orarealsportsleagueforthatmatter,withoutthepresenceofstatisticaldatatoguideyourdecisionsortodeterminewinnersandlosers.Thinkofalltheadvertis-ersthattrytoswayconsumerswithstatisticstheysayprovetheirproductisbest.Alackofunderstandingofstatisticalprocessesputsyouataseriousdisadvantagebecausestatisticalinformationisallaroundus.Television,newspapers,magazines,andtheInternetareallfullofstatisticaldata.Somepeopledousestatisticsaccu-ratelyandappropriately,butothersusestatisticsinappropriatelybecauseoflackofknowledgeorinanefforttodeliberatelymislead.Tothinkcriticallyaboutalltheinfor-mationbombardingyou,you’llneedtosortthegoodstufffromthebad.Educatedpeopleneedstatisticalliteracyasmuchascomputerliteracy.

Statisticsmayseemdifficultatfirstsimplybecausethisisanewwayoflookingatthings.Ridingabikeordrivingacarseemshardthefirsttimeyoutry,too.Thinkofthefirsttimeyouwentbowling,thefirsttimeyoupickedupamusicalinstrument,or


MODULE 3

Psychologistsusestatisticstomakedatamoremeaningfulanduseful.


41

3inTroduCe The moduLe

Getting StartedHave students consider how math and psychology are related. What applica-tions could any kind of math have in understanding behavior? Emphasize to students that statistics provide mean-ing to data collected in psychological studies.

Building Vocabulary TRMBuilding Vocabulary: Crossword Puzzle offers students an opportunity to improve their understanding of the module’s terms.

TeaChing TiPHelp convince students of the impor-tance of statistics by showing them news reports that use statistics of any kind. Have the students look up the original studies showcased in the articles to see if the statistics are pre-sented correctly and clearly.

resource managerActivities TE Web/Multimedia TEApplication 42, 43, 48, 50, 51, 52, 53 Technology Application 45, 48, 51

Cooperative Learning 49

Critical Thinking 51

Enrichment 51

Portfolio Project 54

Vocabulary 41

Getting StartedHave students consider how math and psychology are related. What applications could any kind of math have in understanding behavior? Emphasize to students that statistics provide meaning to data collected in psychological studies.

Building Vocabulary BuildingPuzzleto improve their understanding of the module’s terms.

TeaHelp convince students of the importance of statistics by showing them news reports that use statistics of any kind. Have the students look up the original studies showcased in the articles to see if the statistics are presented correctly and clearly.

r m

41


Measures of Central Tendency●Mode●Mean●Median

Measures of Variation●Range●StandardDeviation

Normal Distribution




Statistics are more a matter of attitude than numbers. Many people (not you, ofcourse)haveanegativeviewofstatisticsandconvincethemselvesthatthisisatopicbeyondcomprehension.Maybeit’stheword—statistics ishardtopronounce!Theconceptsthemselvesarereallynotsohard.Lotsofpeopleenjoythesubject,andthereisnodenyingitsusefulness.Thinkoftryingtoparticipateinafantasysportsleague,orarealsportsleagueforthatmatter,withoutthepresenceofstatisticaldatatoguideyourdecisionsortodeterminewinnersandlosers.Thinkofalltheadvertis-ersthattrytoswayconsumerswithstatisticstheysayprovetheirproductisbest.Alackofunderstandingofstatisticalprocessesputsyouataseriousdisadvantagebecausestatisticalinformationisallaroundus.Television,newspapers,magazines,andtheInternetareallfullofstatisticaldata.Somepeopledousestatisticsaccu-ratelyandappropriately,butothersusestatisticsinappropriatelybecauseoflackofknowledgeorinanefforttodeliberatelymislead.Tothinkcriticallyaboutalltheinfor-mationbombardingyou,you’llneedtosortthegoodstufffromthebad.Educatedpeopleneedstatisticalliteracyasmuchascomputerliteracy.

Statisticsmayseemdifficultatfirstsimplybecausethisisanewwayoflookingatthings.Ridingabikeordrivingacarseemshardthefirsttimeyoutry,too.Thinkofthefirsttimeyouwentbowling,thefirsttimeyoupickedupamusicalinstrument,or


MODULE 3

Psychologistsusestatisticstomakedatamoremeaningfulanduseful.


42

3TeaCh

Beyond the ClassroomGuest Speaker Contact your athletic department and ask who keeps the sta-tistics for each team. Have that person come and discuss the importance of understanding statistics in athletics.

Beyond the Classroom TRMApply Have students compute statis-tics for a local sports team. Go to the team’s website and gather any data the students might be interested in (win- loss record, home- away win record, and so forth). Have students use the data to calculate any statis-tics from this module that apply. Ask students to discuss the meaning of the particular statistic they calculated.

You may want to use Application Activity: Organizing and Interpret-ing Data.

department and ask who keeps the sta-tistics for each team. Have that person

students to discuss the meaning of the

42 ■❚❙ S c i e n t i f i c i n q u i r y ■❚❙ ThinkingAboutPsychologicalScience

thefirsttimeyouattemptedanewcomputergame.Thesewereallsignificantchal-lengesthefirsttimearound.Ifyou’relikemostpeople,you’rewillingtoworkhardtolearnsomething,despitetheinitialdifficulty,ifyoubelieveinanactivity’svalue.Andstatisticsdoeshavevalue—youreffortstounderstandwillarmyouwithtoolsthatwillhelpyouthinkcriticallyaboutlotsofchoicesyou’llbemakinginlife.So,stickaroundandtakeourguidedtourofthisimportantarea.

Beforewebegin, takenoteof thiskeyconcept:Thepurposeofstatistics is tomakedatameaningful.Ifafriendtoldyoushegot27questionsrightonherhistorytest,howmeaningfulwould thatbe?Wouldn’t youalsoneed toknowhowmanyitemswereonthetest?Ifshegot27rightoutof30,that’sgreat,butifshegot27outof100....Well,yougetthepicture.Andisn’titalsoimportanttoknowhowotherstudentsdid?Even ifshedidscore27outof30,thatmayhavebeenthe lowestscoreintheclassonaveryeasytest.Or27outof100couldbeagoodscoreifitwasthehighestscoreintheclassonanexceptionallydifficulttest.Statisticsareanimportantextensionofcriticalthinking.Theyprovideamethodfororganizinginforma-tionsothatwecanunderstandwhatanumberreallymeans.

Toillustratethestatisticswediscuss,we’llrelyonanexampleweusedinMod-ule2 todemonstrate researchstrategies:Howwouldbanning theuseofmusicplayersaffectstudentlearninginstudyhalls?Tofindout,wedesignedanexperi-mentwithtwogroups(seeFigure 3.1).Theexperimentalgroupcontainedstudentsassigned to listen tomusicwhile instudyhall; thecontrolgroupcontainedstu-dentswhowerenotallowedtolistentomusicwhilethere.Wedecidedtomeasuretheeffectof listening tomusic (our independent variable)byexamining the twogroups’ averageend-of-quartergrades (ourdependent variable).Wesettledonahypothesisthattheexperimentalgroupstudents,wholistenedtomusic,wouldhavehighergradesattheendofthequarterthanthecontrolgroupstudents,whodidnotlistentomusic.

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Statistical Literacy

Peopleinmodernsocietyneedtounderstandstatis-ticswellenoughtomakeinformeddecisionsaboutdatapresentedinthemedia.Thenumberofrunsscoredbyabaseballplayercouldbeinfluencedbyhisbattingaverage,thenumberofbaseshesteals,orhispositioninthebattingorder.

Psychology’sStatistics ❙❚■ M o d u l e 3 ❙❚■ 43

Figure 3.1

All studyhall students

40 studentsrandomlyselected

20 students randomly

assigned toexperimental

group

Averagegrades at the end of

the quarter


assigned tocontrol group

Listen to music daily in

study hall

Music not allowed in study hall

Average grades at the

end ofthe quarter

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Frequency DistributionsWHAT’S THE POINT?

3-1 What can we learn from frequency distributions?

A frequency distribution is, quite simply, a list of scores ordered from high-est to lowest. Figure 3.2 shows some possible data from the hypothetical experiment, before and after being made into an ordered list. Do you notice

805897779369678993788473848489

648375729268877994748277687169

No music

Grades, in random order

Music979393898984848480787773696758

949287838279777574727169686864

No music

Frequency distributions

Music

frequencydistribution Alistofscoresorderedfromhighesttolowest.


Puttingscoresinordercre-atesafrequencydistributionandmakestherawdatamoremeaningful.

Design for Music Experiment

Thepurposeofthisexperi-ment,fullyexplainedinModule2,istodeterminewhetherornotstudentswholistentomusicduringstudyhallwillhavehighergrades.

Figure 3.2


43

3TeaChing TiPAn easy way to teach frequency distri-bution is to have students take some data and create a frequency table and chart. You can use any kind of data, from test scores to heights of students in the class to shoe sizes.

● To create a frequency table, have students list elements of one cate-gory. For example, the category shoe size would list sizes: 5, 5½, 6, 6½, and so forth. In a second column, students note the number of times each element occurs in a population. For shoe sizes in your class, students should determine how many stu-dents wear size 5, how many wear size 5½, and so forth.

● To create a frequency chart, have students plot the data for each ele-ment on a graph. Have students debate what type of graph would show a frequency distribution. (Typi-cally, a bar graph is best.)

Beyond the Classroom TRMApply Give students the test scores from a recent test (omit names to protect privacy) and have them create a frequency distribution of the data. Students should calculate the mode, median, and mean of the data. This will allow them to compare their own test scores to the distribution and mea-sures of central tendency.

You may want to use Application Activity: Describing Data.


thefirsttimeyouattemptedanewcomputergame.Thesewereallsignificantchal-lengesthefirsttimearound.Ifyou’relikemostpeople,you’rewillingtoworkhardtolearnsomething,despitetheinitialdifficulty,ifyoubelieveinanactivity’svalue.Andstatisticsdoeshavevalue—youreffortstounderstandwillarmyouwithtoolsthatwillhelpyouthinkcriticallyaboutlotsofchoicesyou’llbemakinginlife.So,stickaroundandtakeourguidedtourofthisimportantarea.

Beforewebegin, takenoteof thiskeyconcept:Thepurposeofstatistics is tomakedatameaningful.Ifafriendtoldyoushegot27questionsrightonherhistorytest,howmeaningfulwould thatbe?Wouldn’t youalsoneed toknowhowmanyitemswereonthetest?Ifshegot27rightoutof30,that’sgreat,butifshegot27outof100....Well,yougetthepicture.Andisn’titalsoimportanttoknowhowotherstudentsdid?Even ifshedidscore27outof30,thatmayhavebeenthe lowestscoreintheclassonaveryeasytest.Or27outof100couldbeagoodscoreifitwasthehighestscoreintheclassonanexceptionallydifficulttest.Statisticsareanimportantextensionofcriticalthinking.Theyprovideamethodfororganizinginforma-tionsothatwecanunderstandwhatanumberreallymeans.

Toillustratethestatisticswediscuss,we’llrelyonanexampleweusedinMod-ule2 todemonstrate researchstrategies:Howwouldbanning theuseofmusicplayersaffectstudentlearninginstudyhalls?Tofindout,wedesignedanexperi-mentwithtwogroups(seeFigure 3.1).Theexperimentalgroupcontainedstudentsassigned to listen tomusicwhile instudyhall; thecontrolgroupcontainedstu-dentswhowerenotallowedtolistentomusicwhilethere.Wedecidedtomeasuretheeffectof listening tomusic (our independent variable)byexamining the twogroups’ averageend-of-quartergrades (ourdependent variable).Wesettledonahypothesisthattheexperimentalgroupstudents,wholistenedtomusic,wouldhavehighergradesattheendofthequarterthanthecontrolgroupstudents,whodidnotlistentomusic.

© D

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iel

Be

nD

jy/i

sto

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ph

oto

Statistical Literacy

Peopleinmodernsocietyneedtounderstandstatis-ticswellenoughtomakeinformeddecisionsaboutdatapresentedinthemedia.Thenumberofrunsscoredbyabaseballplayercouldbeinfluencedbyhisbattingaverage,thenumberofbaseshesteals,orhispositioninthebattingorder.

TeaAn easy way to teach butiondata and create a frequency table and chart. You can use any kind of data, from test scores to heights of students in the class to shoe sizes.

●

●

Beyond the Classroom Applyfrom a recent test (omit names to protect privacy) and have them create a frequency distribution of the data. Students should calculate the mode, median, and mean of the data. This will allow them to compare their own test scores to the distribution and measures of central tendency.

Activity: Describing Data.


Figure 3.1

All studyhall students

40 studentsrandomlyselected


assigned toexperimental

group

Averagegrades at the end of

the quarter


assigned tocontrol group

Listen to music daily in

study hall

Music not allowed in study hall

Average grades at the

end ofthe quarter

© E

© S

UP

ER

STO

CK

/P

UR

ES

TOC

K/

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TOC

K

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CK

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UP

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CK

© IN

DE

XS

TOC

K/S

UP

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© L

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MY

Frequency DistributionsWHAT’S THE POINT?


A frequency distribution is, quite simply, a list of scores ordered from high-est to lowest. Figure 3.2 shows some possible data from the hypothetical experiment, before and after being made into an ordered list. Do you notice

805897779369678993788473848489

648375729268877994748277687169

No music

Grades, in random order

Music979393898984848480787773696758

949287838279777574727169686864

No music

Frequency distributions

Music

frequencydistribution Alistofscoresorderedfromhighesttolowest.


Puttingscoresinordercre-atesafrequencydistributionandmakestherawdatamoremeaningful.

Design for Music Experiment

Thepurposeofthisexperi-ment,fullyexplainedinModule2,istodeterminewhetherornotstudentswholistentomusicduringstudyhallwillhavehighergrades.

Figure 3.2

Cultural VariabilityHave students find variability among people from different cul-tures for the following data:

● Height● Weight● Income● Standardized test scores (for

example, IQ tests)● Health conditions (for example,

heart disease and cancer)

Discuss variables that would lead to different statistics in dif-ferent cultures. Have students discuss how these statistics might influence their views of other cultures.

Multicultural connections


44

3 44 ■❚❙ S c i e n t i f i c i n q u i r y ■❚❙ ThinkingAboutPsychologicalScience

Figure 3.3 how much more useful these numbers are after we arrange them in a fre-quency distribution from highest to lowest? At a glance, you can tell the high score for each group and the low score for each group.

The information from the frequency distribution can be easily presented as a graph, like the bar graphs in Figure 3.3. By viewing our scores this way, we can discover the meaning behind the numbers even more easily because we can see at a glance how the scores cluster and what the most common scores are.

Pause Now or Move oN

Turn to page 55 to review and apply what you’ve learned.

Measures of Central TendencyWHAT’S THE POINT?

3-2 What are the three measures of central tendency?

The next thing we need to know about a frequency distribution is where its center— the “normal” score— is. For the study hall research, we have two groups— one that listens to music and one that does not— and you just learned how to create a frequency distribution for each. If the center of one frequency distribution is higher than the center of the other, this may help us decide whether the hypothesis is correct. It seems as though there should be some easy and reliable way to determine the center. However, it isn’t quite this sim-ple. There are three methods— mode, mean, and median— and each is appropri-ate only in certain situations.

ModeThe mode is the most frequently occurring score or scores in a distribution. Using the sample data from our frequency distributions, we can see that the mode for the music group is 68 and the mode for the no- music group is

109876543210

109876543210

51–60 61–70 91–10071–80 81–90

Number ofstudents

Grades51–60 61–70 91–10071–80 81–90

Number ofstudents

Grades

No music Music

mode Themostfrequentlyoccurringscoreorscoresinadistribution.

Bar Graphs

Datafromafrequencydistributioncaneasilybeconvertedtoabargraph.


84

= 81121515

Mode(Most common)

Mean(Average)

Median(Middle score)

No music Music

84

68

= 77115515

75

84 (see Figure 3.4). Modes are not the most ideal source of information— at least for our purposes here. It’s possible for the most common score (the mode) to not be near the center of the distribution. For example, I once gave a test to my students on which almost everyone did either very well or very poorly; the most frequent scores did not represent the center of the distribution. The mode is most useful when the data can only be put into distinct groups. For example, if you were categorizing people as male or female— by assigning males the number 1 and females the number 2— it would not make any sense to say that the average sex for the group was 1.5. When the numbers represent distinct groups, the mode is the only appropriate way to establish central tendency. Here’s another example: If a high school had 100 sophomores, 200 juniors, and 150 seniors, the mode would be grade 11, the group with the largest number. In finding the cen-ter of this distribution, the best you can do is use the mode and say that the most common student is a junior.

MeanThe most familiar measure of central tendency is the mean, the mathemati-cal average of a distribution. As you know, we compute averages by adding all the scores and dividing by the total number of scores. The means for the two groups in the music study are presented in Figure 3.4— 81 for the no- music group and 77 for the music group. Under most circumstances, the mean is the statistic of choice for central tendency.

Sometimes, however, the mean can mislead us. This occurs when a few scores are either extremely high or extremely low. It’s not a good idea to use the mean, for example, to report the central tendency for housing costs in a community because most communities have a few very valuable homes. When you calculate a mean, these few expensive homes will affect the mean much more than each of the moderately priced homes will. As a result, hous-ing will appear to be more expensive than it really is.

MedianThe third measure of central tendency is the median, the middle score in a ranked distribution; half of the scores are above the median and half are below. To remember this, just think of the median of a rural interstate highway, usually a strip of grass running down the center of the highway between the two sets of lanes. An extreme score in the top or bottom half of

mean Themathematicalaver-ageofadistribution,obtainedbyaddingthescoresandthendividingbythenumberofscores.

median Themiddlescoreinarankeddistribution;halfthescoresareaboveit,andhalfarebelowit.

Figure 3.4

The Three Measures of Central Tendency

Thethreeprimarymethodsoffindingthecenterofadistributionofscoresarethemode,mean,andmedian.Eachhasitsownstrengthsandweaknesses.

● Nominal data simply identify categories. Examples include gender, “yes- or- no” answers on surveys, and class level in school (senior, junior, and so forth).

● Ordinal data identify the order in which data fall in a set. Ranking of items, from one- hit wonders to class rank to home- run sluggers, pro-duces ordinal data.

● Interval data include data that fall along a number line that has a zero point. Weight has a real zero point in that a weight of zero means no weight. Height is another example of interval data.

● Ratio data include data that fall in a number line where zero is just another number on the line. For instance, a tempera-ture of zero degrees does not mean there is no temperature. Test scores can also be ratio data in that a score of zero usu-ally doesn’t mean “absence of knowledge.”

FYi


45

3Beyond the Classroom TRMApply Have students explore what the modal, median, and mean home prices are in their community. Ask them to evaluate which statistic is most mean-ingful to determining the value of homes in the area.

You may want to use Technol-ogy Application Activity: PsychSim: “Descriptive Statistics.”


Figure 3.3 how much more useful these numbers are after we arrange them in a fre-quency distribution from highest to lowest? At a glance, you can tell the high score for each group and the low score for each group.

The information from the frequency distribution can be easily presented as a graph, like the bar graphs in Figure 3.3. By viewing our scores this way, we can discover the meaning behind the numbers even more easily because we can see at a glance how the scores cluster and what the most common scores are.



Measures of Central TendencyWHAT’S THE POINT?


The next thing we need to know about a frequency distribution is where its center— the “normal” score— is. For the study hall research, we have two groups— one that listens to music and one that does not— and you just learned how to create a frequency distribution for each. If the center of one frequency distribution is higher than the center of the other, this may help us decide whether the hypothesis is correct. It seems as though there should be some easy and reliable way to determine the center. However, it isn’t quite this sim-ple. There are three methods— mode, mean, and median— and each is appropri-ate only in certain situations.

ModeThe mode is the most frequently occurring score or scores in a distribution. Using the sample data from our frequency distributions, we can see that the mode for the music group is 68 and the mode for the no- music group is

109876543210

109876543210

51–60 61–70 91–10071–80 81–90

Number ofstudents

Grades51–60 61–70 91–10071–80 81–90

Number ofstudents

Grades

No music Music

mode Themostfrequentlyoccurringscoreorscoresinadistribution.

Bar Graphs

Datafromafrequencydistributioncaneasilybeconvertedtoabargraph.

Beyond the ClassroomApplymodal, median, and mean home prices are in their community. Ask them to evaluate which statistic is most meaningful to determining the value of homes in the area.

ogy“Descriptive Statistics.”


84

= 81121515

Mode(Most common)

Mean(Average)

Median(Middle score)

No music Music

84

68

= 77115515

75

84 (see Figure 3.4). Modes are not the most ideal source of information— at least for our purposes here. It’s possible for the most common score (the mode) to not be near the center of the distribution. For example, I once gave a test to my students on which almost everyone did either very well or very poorly; the most frequent scores did not represent the center of the distribution. The mode is most useful when the data can only be put into distinct groups. For example, if you were categorizing people as male or female— by assigning males the number 1 and females the number 2— it would not make any sense to say that the average sex for the group was 1.5. When the numbers represent distinct groups, the mode is the only appropriate way to establish central tendency. Here’s another example: If a high school had 100 sophomores, 200 juniors, and 150 seniors, the mode would be grade 11, the group with the largest number. In finding the cen-ter of this distribution, the best you can do is use the mode and say that the most common student is a junior.

MeanThe most familiar measure of central tendency is the mean, the mathemati-cal average of a distribution. As you know, we compute averages by adding all the scores and dividing by the total number of scores. The means for the two groups in the music study are presented in Figure 3.4— 81 for the no- music group and 77 for the music group. Under most circumstances, the mean is the statistic of choice for central tendency.

Sometimes, however, the mean can mislead us. This occurs when a few scores are either extremely high or extremely low. It’s not a good idea to use the mean, for example, to report the central tendency for housing costs in a community because most communities have a few very valuable homes. When you calculate a mean, these few expensive homes will affect the mean much more than each of the moderately priced homes will. As a result, hous-ing will appear to be more expensive than it really is.

MedianThe third measure of central tendency is the median, the middle score in a ranked distribution; half of the scores are above the median and half are below. To remember this, just think of the median of a rural interstate highway, usually a strip of grass running down the center of the highway between the two sets of lanes. An extreme score in the top or bottom half of

mean Themathematicalaver-ageofadistribution,obtainedbyaddingthescoresandthendividingbythenumberofscores.

median Themiddlescoreinarankeddistribution;halfthescoresareaboveit,andhalfarebelowit.

Figure 3.4

The Three Measures of Central Tendency

Thethreeprimarymethodsoffindingthecenterofadistributionofscoresarethemode,mean,andmedian.Eachhasitsownstrengthsandweaknesses.

ResearchHave students survey teachers in your building to determine the most common way of computing course averages. Do most teach-ers use a total points system or a weighted averages system? What are the advantages and disadvan-tages of each method?

active learning


46

3

TeaChing TiPRemind students about how important sampling is to the meaning of statis-tics. If data from a sample like the “Cs- Only Club” are used to calculate statistics, the data will be meaningful only for those who share qualities with that group. Groups that are chosen at random are more likely to produce data that represent a variety of people.

ReteachPositive and Negative Skew Point out that positive skew results from scores pulling the mean toward the tail end of the scores. Hence, the mean is more “positive,” or greater than the rest of the scores. Negative skew is the opposite, with the mean pulled down toward the lower end of the scores.

Remind students about how important

only for those who share qualities with

data that represent a variety of people.

rest of the scores. Negative skew is the


Figure 3.5

Central Tendency in a Skewed Distribution

Thisdiagramshowsthethreemeasuresofcentraltendencyforadistributionofincomesskewedbyafewfamilieswithveryhighincomes.Noticethat,underthesecircumstances,themean(which,astheaver-age,“balances”thedistri-bution)producesaresultthatisfarabovewhatmostpeoplewouldconsiderthecenterofthisdistribu-tion.Incaseslikethis,themedianisprobablyabetterrepresentationofcentraltendencybecauseitislessinfluencedbyskew.

the distribution will have no greater effect than any other score. Figure 3.4 also shows the median for the students who listen to music (75) and students who don’t (84).

When the mean, median, and mode are all the same, it’s easy to identify cen-tral tendency. The three measures of central tendency can, however, be vastly different. Figure 3.5 shows what can happen when a distribution is distorted, or skewed— not evenly distributed around the mean. In a skewed distribu-tion, there is an unusual number of high scores or low scores.



Measures of VariationWHAT’S THE POINT?

3-3 How do we determine the variation of a distribution of scores?

It is important to have a sense of where the center of a distribution falls. To truly understand the meaning of the numbers, however, we need to add another piece of the puzzle. Two distributions can have the same center and still be different. Consider a school “Cs- Only Club” that was open only to people who got a grade of C in every class they took. The mode for this club’s grades would be C. The mean and median grades would also be C. But can you now imagine a situation in which all students in a school were invited to join an “Everybody’s Welcome Club” and the mode, mean, and median for the grades would still be a C? The B students would balance the D students, and the A students would balance the fail-ing students. The students in the “Cs- Only Club” are packed together at the same point on the grade distribution. The students in the “Everybody’s Welcome Club” are spread throughout the distribution. These differences are represented in Fig-

ure 3.6. In this section, we examine ways to measure how spread out scores are.

30 40 50 60 70 80 90 100140

180 950 1420

MeanMedian

One family Income per family in thousands of dollars

Mode

skewed Distorted;notevenlydistributedaroundthemean.


Figure 3.6 RangeThe simplest measure of variation is the range, or the difference between the highest and the lowest scores in a distribution. The range is a simple, often helpful, indicator of how much variation there is in a distribution. It’s nice to know, for example, that the range of grades for the music group was 30 (from a high of 94 to a low of 64 points) and that the range for the no- music group was 39 (97 to 58). The only problem with this measure is that a range consid-ers only two scores: the highest and the lowest. Let’s assume for a minute that the student who got a 58 in the no- music group missed a lot of school because of illness. This already marginal student’s grade might easily have dropped to a 38 because of the absences. A change in this one student’s grade could add 20 points to the range! A better statistic would consider every score, not just the two extremes. That’s where the standard deviation comes in.

Standard DeviationStandard deviation is a statistic that tells us how much scores vary around the mean score of a distribution. The higher the standard deviation, the more spread out the scores are; the smaller the standard deviation, the more closely the scores are packed near the mean. In fact, if a distribution had a standard deviation of zero, it would signify that every score was the mean score— there would be no variation at all! This would happen in the “Cs- Only Club.” The standard deviation would be zero because every grade would be the average grade of C. The “Everybody’s Welcome Club” would have a higher standard deviation because its members have a variety of grades.

Figure 3.7 shows an example of how the standard deviation is calculated for a small set of scores (in this case, punting distances for a football player). It is a logical process, and you should take a careful look to see how it is done. However, the important thing to understand here is the meaning behind the statistic. When you compute the standard deviation, you are coming up with a number that represents how far the scores spread from the mean. If there were two punters on my team who both have a mean of 40 yards, the punter with the smaller standard deviation is the one who is most consistent.

Cs-Only Club

Number ofstudents

Number ofstudents

Everybody’s Welcome Club

A B C D F A B C D F

Grades GradesCentral tendency = C Variation: None Central tendency = C Variation: High

range Thedifferencebetweenthehighestandlowestscoresinadistribution.

standarddeviation Acom-putedmeasureofhowmuchscoresvaryaroundthemeanscoreofadistribution.

Variation Makes a Difference

Bothofthesebargraphsrepresentdistributionsofstudentsinwhichthemeasuresofcentralten-dencyareCgrades.It’sclear,however,thatthedistributionsaredifferent.Forthe“Cs-OnlyClub,”thereisnovariation;forthe“Everybody’sWelcomeClub,”thereishighvaria-tion.Measuresofvariationallowustounderstandsuchdifferences.

● Positive skew occurs when scores congregate toward the lower end of a distribution. In this case, the mode is less than the mean.

● Negative skew occurs when scores congregate toward the higher end of a distribution. In this case, the mode is greater than the mean.

FYi


47

3TeaChing TiPStandard deviation is an often-misunderstood but highly valuable statistic. Knowing the standard devia-tion for a set of data can reveal how similar the scores are. The higher the standard deviation, the less similar the scores are.

For example, if a set of test scores has a standard deviation of 5, then everyone’s scores were fairly similar. If the standard deviation is 50, then the test results were not similar at all!


Figure 3.5

Central Tendency in a Skewed Distribution

Thisdiagramshowsthethreemeasuresofcentraltendencyforadistributionofincomesskewedbyafewfamilieswithveryhighincomes.Noticethat,underthesecircumstances,themean(which,astheaver-age,“balances”thedistri-bution)producesaresultthatisfarabovewhatmostpeoplewouldconsiderthecenterofthisdistribu-tion.Incaseslikethis,themedianisprobablyabetterrepresentationofcentraltendencybecauseitislessinfluencedbyskew.

the distribution will have no greater effect than any other score. Figure 3.4 also shows the median for the students who listen to music (75) and students who don’t (84).

When the mean, median, and mode are all the same, it’s easy to identify cen-tral tendency. The three measures of central tendency can, however, be vastly different. Figure 3.5 shows what can happen when a distribution is distorted, or skewed— not evenly distributed around the mean. In a skewed distribu-tion, there is an unusual number of high scores or low scores.



Measures of VariationWHAT’S THE POINT?


It is important to have a sense of where the center of a distribution falls. To truly understand the meaning of the numbers, however, we need to add another piece of the puzzle. Two distributions can have the same center and still be different. Consider a school “Cs- Only Club” that was open only to people who got a grade of C in every class they took. The mode for this club’s grades would be C. The mean and median grades would also be C. But can you now imagine a situation in which all students in a school were invited to join an “Everybody’s Welcome Club” and the mode, mean, and median for the grades would still be a C? The B students would balance the D students, and the A students would balance the fail-ing students. The students in the “Cs- Only Club” are packed together at the same point on the grade distribution. The students in the “Everybody’s Welcome Club” are spread throughout the distribution. These differences are represented in Fig-

ure 3.6. In this section, we examine ways to measure how spread out scores are.

30 40 50 60 70 80 90 100140

180 950 1420

MeanMedian

One family Income per family in thousands of dollars

Mode

skewed Distorted;notevenlydistributedaroundthemean.

TeaStandard deviation misunderstood but highly valuable statistic. Knowing the standard deviation for a set of data can reveal how similar the scores are. The higher the standard deviation, the less similar the scores are.

has a standard deviation of 5, then everyone’s scores were fairly similar. If the standard deviation is 50, then the test results were not similar at all!


Figure 3.6 RangeThe simplest measure of variation is the range, or the difference between the highest and the lowest scores in a distribution. The range is a simple, often helpful, indicator of how much variation there is in a distribution. It’s nice to know, for example, that the range of grades for the music group was 30 (from a high of 94 to a low of 64 points) and that the range for the no- music group was 39 (97 to 58). The only problem with this measure is that a range consid-ers only two scores: the highest and the lowest. Let’s assume for a minute that the student who got a 58 in the no- music group missed a lot of school because of illness. This already marginal student’s grade might easily have dropped to a 38 because of the absences. A change in this one student’s grade could add 20 points to the range! A better statistic would consider every score, not just the two extremes. That’s where the standard deviation comes in.

Standard DeviationStandard deviation is a statistic that tells us how much scores vary around the mean score of a distribution. The higher the standard deviation, the more spread out the scores are; the smaller the standard deviation, the more closely the scores are packed near the mean. In fact, if a distribution had a standard deviation of zero, it would signify that every score was the mean score— there would be no variation at all! This would happen in the “Cs- Only Club.” The standard deviation would be zero because every grade would be the average grade of C. The “Everybody’s Welcome Club” would have a higher standard deviation because its members have a variety of grades.

Figure 3.7 shows an example of how the standard deviation is calculated for a small set of scores (in this case, punting distances for a football player). It is a logical process, and you should take a careful look to see how it is done. However, the important thing to understand here is the meaning behind the statistic. When you compute the standard deviation, you are coming up with a number that represents how far the scores spread from the mean. If there were two punters on my team who both have a mean of 40 yards, the punter with the smaller standard deviation is the one who is most consistent.

Cs-Only Club

Number ofstudents

Number ofstudents

Everybody’s Welcome Club

A B C D F A B C D F

Grades GradesCentral tendency = C Variation: None Central tendency = C Variation: High

range Thedifferencebetweenthehighestandlowestscoresinadistribution.

standarddeviation Acom-putedmeasureofhowmuchscoresvaryaroundthemeanscoreofadistribution.

Variation Makes a Difference

Bothofthesebargraphsrepresentdistributionsofstudentsinwhichthemeasuresofcentralten-dencyareCgrades.It’sclear,however,thatthedistributionsaredifferent.Forthe“Cs-OnlyClub,”thereisnovariation;forthe“Everybody’sWelcomeClub,”thereishighvaria-tion.Measuresofvariationallowustounderstandsuchdifferences.

cross- curricular connection

Math and StatisticsTeam teach these lessons with a math colleague:

● Because many statistical measures can be done by calculator, invite a math teacher to lead a tutorial on how to use the statistical functions on the students’ calculators.

● If your schedule permits (or an after- school session is feasible), team teach a lesson in basic statistics with a math colleague. While the math teacher offers calculation instructions, you can offer instruction in the practical use of statistics in psychological research.

Marilyn vos Savant has the world’s highest recorded IQ. Her score is 220. Have students calculate how many standard deviations her score falls from the mean IQ (100).

FYi


48

3

TeaChing TiPMost scientific calculators can com-pute the standard deviation with the click of a button. Work with your school’s math department to learn how to calculate standard deviation both by hand and by calculator.

school’s math department to learn how to calculate standard deviation both by


Figure 3.7

Calculating the standard deviation by hand would be tedious if you have more than just a few scores. Thank goodness for calculators and computer spreadsheets that will do the job with a few pushes of a button or clicks of a mouse. My computer tells me that the standard deviation for our music group is 8.7 points and the standard deviation for the control group is 10.5 points. This means that the scores for the music listeners are packed a little closer together. In other words, there is less variation among the grades of the music listeners than there is for those who weren’t allowed to listen to music in study hall.



Normal DistributionWHAT’S THE POINT?

3-4 What are the important characteristics of a normal distribution?

Much psychological data can be represented in a graph called a normal dis-tribution, a frequency distribution shaped like a symmetrical bell. In a normal distribution, most scores fall near the mean with fewer scores at the extremes. A normal distribution, like the one in Figure 3.8, is not skewed; its left and right sides are mirror images of each other. Furthermore, the high point of a normal

36 yards38 yards41 yards45 yards

1. Calculate the mean

4. Take the square root of the mean of column 3

= 40 yards

Standard deviation = = = 3.4 yards

Mean = 1604

16 yards2

4 yards2

1 yard2 25 yards2

3. Square the deviations

46 yards2 = Sum of (deviations)2

Sum of (deviations)2

Number of punts

–4 yards–2 yards+1 yard +5 yards

2. Determine deviation from the

mean (40 yards)

46 yards2

4

Steps in Calculating the Standard Deviation

1. Calculate the mean.2. Determine how far each score (punt distances, in this example) deviates (differs)

from the average.3. Square the deviation scores and add them together. Note that you cannot just

average the deviations without squaring them because the sum of the deviation scores will always be zero.

4. Take the square root of the average of the squared deviation scores. This step brings you back to the original units—yards rather than yards squared.

normaldistribution Afrequencydistributionthatisshapedlikeasymmetricalbell.

Calculating Standard Deviation

Hereishowtocalculatethestandarddeviationofasmallsetofscores.


distribution is in the center. This high point represents all three measures of central tendency: the mode, the mean, and the median. Many collections of data produce a normal distribution. In intelligence test scores, for example, most people “pile up” at or near the middle of the distribution. The further you move above or below the mean, the fewer people are represented. This distinctive pileup produces the bell shape of the normal distribution.

Data distributed in a bell- shaped curve illustrate some useful principles. Let’s take a look at scores on intelligence tests. Scores on the Wechsler intelligence tests (the most widely used family of tests) produce a mean score of 100 points and a standard deviation of 15 points. This means that a person with a score of 115 is one standard deviation above average on this test. A person with a score of 80 falls about 1.33 standard deviations below average. There are some remarkable consistencies about normally distributed data. The most important things to remember, illustrated in Figure 3.8, are these:

● Approximately 68 percent of the population will fall within one stan-dard deviation of the average. In the case of the Wechsler test scores, this means that 68 percent of the population scores between 85 (one standard deviation below average) and 115 (one standard deviation above average). In other words, roughly two- thirds of any normal population falls in this range.

● If you move one more standard deviation on both sides of the mean, you have now accounted for about 96 per-cent of any normal population. In other words, about 24 out of 25 people fall within two standard deviations of the mean. For the Wechsler test scores, this represents the spread from a score of 70 (two standard deviations below average) to a score of 130 (two standard devia-tions above average).

● By the time you’ve gone one more standard deviation, you’ve included just about everyone. Slightly more than 99.7 percent of any normal population falls within three standard deviations of the mean. For the Wechsler test

34%14% 34% 14%0.1% 0.1%

55 70 85 100Wechsler intelligence score

115 130 145

2%2%

68%

96%

Ninety-six percent of allpeople fall within twostandard deviationsfrom the mean

Sixty-eight percent of all people score within onestandard deviationfrom the mean

Number ofstudents

Marilyn vos Savant

MarilynvosSavant,whowritesaSundaypuzzlecolumnforParademagazine,wasoncelistedbytheGuin-nessBookofWorldRecordsasthepersonwiththehighestintelligencetestscore.Guin-nessnolongermaintainsthiscategorybecausethescoresarenotreliableenoughtodetermineasingle“winner.”Herscorewasreportedtobeover220.Iftrue,thiswouldputherovereightstandarddeviationsaboveaverage,trulyastonishingwhenyourealizethatonly0.3percentofthepopulationismorethanthreestandarddeviationsawayfromaverage!

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A Normal Distribution or Bell- Shaped Curve

Intelligencetestscoresformanormaldistributionwithameanof100pointsandastandarddeviationof15points.Thepercentagefiguresshownaretrueforanynormaldistribution,notjustforintelligencescores.

Figure 3.8

● A normal distribution is a distribution in which the mode, median, and mean are the same.

● Any large data set should have a normal distribution.

FYi


Math and StatisticsIf your schedule does not permit you to team teach a lesson in basic statistics, record a col-laborative lesson delivered by you and a math colleague. Each of you can use the video or video clips to enhance the lessons provided in your own classrooms.

At this point, you may wish to use Application Activity: Organizing and Interpret-ing Data, Application Activity: Describing Data, and Technology Application Activity: PsychSim: “Descriptive Statistics.”

TRM


49

3Beyond the Classroom TRMApply Use Cooperative Learning Activity: A Tasty Sample(r): Teach-ing About Sampling Using M&Ms® to teach about normal distribution. The individual packets of M&Ms will not typically reflect a normal distri-bution, but the more data that are pooled, the more likely a normal distri-bution will emerge.

Beyond the ClassroomBellringers Use the following prompts as discussion starters:

● Now that you know about the nor-mal distribution, would you like for your grades to be based on the “bell curve”? Why or why not?

● What does being “normal” mean to you? Do people who have always lived in the United States like being thought of as “normal”? Why or why not?

Beyond the ClassroomResearch Have students collect data from their classmates on one factor (height, weight, and so forth). Then have students gather data on this fac-tor from students in other classes. Ask students how many classes it takes before their data resemble a normal distribution.


Figure 3.7

Calculating the standard deviation by hand would be tedious if you have more than just a few scores. Thank goodness for calculators and computer spreadsheets that will do the job with a few pushes of a button or clicks of a mouse. My computer tells me that the standard deviation for our music group is 8.7 points and the standard deviation for the control group is 10.5 points. This means that the scores for the music listeners are packed a little closer together. In other words, there is less variation among the grades of the music listeners than there is for those who weren’t allowed to listen to music in study hall.



Normal DistributionWHAT’S THE POINT?


Much psychological data can be represented in a graph called a normal dis-tribution, a frequency distribution shaped like a symmetrical bell. In a normal distribution, most scores fall near the mean with fewer scores at the extremes. A normal distribution, like the one in Figure 3.8, is not skewed; its left and right sides are mirror images of each other. Furthermore, the high point of a normal

36 yards38 yards41 yards45 yards

1. Calculate the mean

4. Take the square root of the mean of column 3

= 40 yards

Standard deviation = = = 3.4 yards

Mean = 1604

16 yards2

4 yards2

1 yard2 25 yards2

3. Square the deviations

46 yards2 = Sum of (deviations)2

Sum of (deviations)2

Number of punts

–4 yards–2 yards+1 yard +5 yards

2. Determine deviation from the

mean (40 yards)

46 yards2

4

Steps in Calculating the Standard Deviation

1. Calculate the mean.2. Determine how far each score (punt distances, in this example) deviates (differs)

from the average.3. Square the deviation scores and add them together. Note that you cannot just

average the deviations without squaring them because the sum of the deviation scores will always be zero.

4. Take the square root of the average of the squared deviation scores. This step brings you back to the original units—yards rather than yards squared.

normaldistribution Afrequencydistributionthatisshapedlikeasymmetricalbell.

Calculating Standard Deviation

Hereishowtocalculatethestandarddeviationofasmallsetofscores.

Beyond the ClassroomApplyActivity: A Tasty Sample(r): Teaching About Sampling Using M&Msto teach about normal distribution. The individual packets of M&Ms will not typically reflect a normal distribution, but the more data that are pooled, the more likely a normal distribution will emerge.

Beyond the ClassroomBellringersas discussion starters:

●

●

Beyond the ClassroomResearchfrom their classmates on one factor (height, weight, and so forth). Then have students gather data on this factor from students in other classes. Ask students how many classes it takes before their data resemble a normal distribution.


distribution is in the center. This high point represents all three measures of central tendency: the mode, the mean, and the median. Many collections of data produce a normal distribution. In intelligence test scores, for example, most people “pile up” at or near the middle of the distribution. The further you move above or below the mean, the fewer people are represented. This distinctive pileup produces the bell shape of the normal distribution.

Data distributed in a bell- shaped curve illustrate some useful principles. Let’s take a look at scores on intelligence tests. Scores on the Wechsler intelligence tests (the most widely used family of tests) produce a mean score of 100 points and a standard deviation of 15 points. This means that a person with a score of 115 is one standard deviation above average on this test. A person with a score of 80 falls about 1.33 standard deviations below average. There are some remarkable consistencies about normally distributed data. The most important things to remember, illustrated in Figure 3.8, are these:

● Approximately 68 percent of the population will fall within one stan-dard deviation of the average. In the case of the Wechsler test scores, this means that 68 percent of the population scores between 85 (one standard deviation below average) and 115 (one standard deviation above average). In other words, roughly two- thirds of any normal population falls in this range.

● If you move one more standard deviation on both sides of the mean, you have now accounted for about 96 per-cent of any normal population. In other words, about 24 out of 25 people fall within two standard deviations of the mean. For the Wechsler test scores, this represents the spread from a score of 70 (two standard deviations below average) to a score of 130 (two standard devia-tions above average).

● By the time you’ve gone one more standard deviation, you’ve included just about everyone. Slightly more than 99.7 percent of any normal population falls within three standard deviations of the mean. For the Wechsler test

34%14% 34% 14%0.1% 0.1%

55 70 85 100Wechsler intelligence score

115 130 145

2%2%

68%

96%

Ninety-six percent of allpeople fall within twostandard deviationsfrom the mean

Sixty-eight percent of all people score within onestandard deviationfrom the mean

Number ofstudents

Marilyn vos Savant

MarilynvosSavant,whowritesaSundaypuzzlecolumnforParademagazine,wasoncelistedbytheGuin-nessBookofWorldRecordsasthepersonwiththehighestintelligencetestscore.Guin-nessnolongermaintainsthiscategorybecausethescoresarenotreliableenoughtodetermineasingle“winner.”Herscorewasreportedtobeover220.Iftrue,thiswouldputherovereightstandarddeviationsaboveaverage,trulyastonishingwhenyourealizethatonly0.3percentofthepopulationismorethanthreestandarddeviationsawayfromaverage!

er

ika

la

rs

en

/re

Du

x

A Normal Distribution or Bell- Shaped Curve

Intelligencetestscoresformanormaldistributionwithameanof100pointsandastandarddeviationof15points.Thepercentagefiguresshownaretrueforanynormaldistribution,notjustforintelligencescores.

Figure 3.8

Multicultural connection

Use this information about the normal distribution to have students consider how a sta-tistical definition of “normal behavior” may lead to different views of “normal” in different cultures. For example, people who live in the southern United States may view saying, “yes, ma’am,” and, “yes, sir,” as normal, while people who live in other parts of the United States may view this as unusual. Extend these comparisons to cultural definitions of men-tal illness highlighted in Modules 30 through 32.


50

3Beyond the Classroom TRMApply● Ask students what a score of 74

percent on a test means. (They most likely will make value judgments about getting a mid- C on a test.)

● Then ask students why a score of 74 percent is a desirable percentage to achieve.

● Finally, ask whether a score of 74 percent is considered average if 76 percent is the highest possible percentage for the test.

You may want to use Application Activity: The Water Cup Toss Test.

(They most

Then ask students why a score of 74


Figure 3.9

scores, this is the range from a score of 55 to a score of 145. Statisti-cally, then, it is very unusual for people to have Wechsler test scores below 55 or above 145. No matter what is being tested, in a normal distribution, it is unusual for any individual to exceed three standard deviations from the mean in either direction.



Comparative StatisticsWHAT’S THE POINT?

3-5 What is the difference between percentage and percentile rank?

The two major comparative statistics are percentage and percentile rank. Per-centage, as you probably know, compares a score to a perfect score of 100 points. If a student scores 83 percent on a test, for example, that student would have had 83 right on a test with 100 questions.

The percentile rank compares one score with other scores in an imagi-nary group of 100 individuals. Percentile rank tells you where a particu-lar score stands in that group and how many people had equal or lower scores. If our student scores at the 83rd percentile, it means that score would have equaled or exceeded the score of 83 of every 100 people who took the test. Figure 3.9 shows an example of how percentages and percen-tile ranks are calculated.

Assume Jack gets 160 points on a 200-point test. His score is good enough totop 27 students out of his class of 36 students.

Percentage

× 100 = 80%

100 points

Meaning: If the testhad been 100 points, Jackwould have had 80 right.

160 correct200 possible

Percentile rank

× 100 = 75th percentile

100 students

Meaning: If 100 students had takenthe test, Jack would have scoredhigher than 75 of them.

27 students beaten36 total students

80right

20wrong

Below Jack’sscore

AboveJack’sscore

percentage Acomparativesta-tisticthatcomparesascoretoaperfectscoreof100points.

percentilerank Acomparativestatisticthatcomparesascoretootherscoresinanimaginarygroupof100individuals.



Calculating Percentage Scores and Percentile Ranks

Thesetwocommoncom-parativestatisticshavesimilarnamesandarecalculatedwithsimilarformulas,buttheyhavedifferentmeanings.


Figure 3.10

Correlation CoefficientWHAT’S THE POINT?

3-6 What does the correlation coefficient indicate about the relationship between variables?

Another highly useful statistic is the correlation coefficient, a statistical measure of the strength of the relationship between two variables. Variables can be related in two ways— positively or negatively. If both variables increase (or decrease) together, there is a positive correlation. An example of a positive correlation is the one between a weight- lifting conditioning program and strength: Lift more, grow stronger. Negative correlations involve two variables that change in opposite directions— one variable increases as the other decreases. There is a negative cor-relation between tooth flossing and dental decay: Floss more, have fewer cavities.

The calculation of a correlation coefficient is quite complicated, and we won’t go through it here. What’s important to remember is that the number produced by the calculation has a value that always falls between – 1.00 and +1.00. When r = – 1.00 (the letter r stands for correlation coefficient), we have a perfect negative correlation. Every time one variable increases by a certain amount, the other variable would decrease by an equally certain amount. Figure 3.10a shows a per-fect negative correlation (– 1.00).

If r = +1.00, we are looking at a perfect positive correlation between two variables. Every time one vari-able increases by a certain amount, the other variable also increases by an equally certain amount. Similarly, every time one variable decreases by a certain amount, the other variable decreases by an equally certain amount (see Fig-

ure 3.10b).If r = 0.00, there is no correla-

tion whatsoever between two vari-ables. Examples of this seem quite silly, like the relationship between flossing teeth and the temperature in Mexico City, or the relationship between the Los Angeles Lakers’ basketball scores and the num-ber of late arrivals into London’s

Perfect positive correlation (+1.00)(b)

No relationship (0.00)(c)

Perfect negative correlation (–1.00)(a)

correlationcoefficient Asta-tisticalmeasureofthestrengthoftherelationshipbetweentwovariables.

Correlations

Graph(a)displaysaperfectnegativecorrelation(r=–1.00).Eachtimethevariablerep-resentedontheverticalaxisincreasesbyacertainamount,thevariableonthehorizontalaxisdecreasesbyanequallycertainamount.Graph(b)showsaperfectpositivecorrelation(r=+1.00);thevariablesrepresentedonthetwoaxesincreaseinexactproportion.Ingraph(c),thereisnocorrelation(r=0.00).Therandomdotsshowthatthetwovariablesarenotrelated.

No score can have a percentile rank of 100 or of zero. A percen-tile rank of 100 would mean that a person scored better than 100 percent of people who took the test— including the test- taker!

FYi

Percentiles and Standardized Test StatisticsHave your school’s guidance counselor, school psychologist, or psychometrist come to class to discuss percentile ranks and other standardized test statistics. These professionals can clarify what these scores mean and how they are used to evaluate student performance.

active learning


51

3TeaChing TiP TRMAs you teach correlations, you can refer to several activities and enrich-ment lessons from Module 2 relating to this complex topic:

● Technology Application Activity: PsychSim: “Correlations”

● Critical Thinking Activity: Cor-relating Test- Taking Time and Performance

● Critical Thinking Activity: Evalu-ating Media Reports of Research

● Application Activity: Correlation and the Challenger Disaster

● Enrichment Lesson: Understand-ing Correlations

TeaChing TiPHelp students understand that perfect positive and perfect negative correla-tions are extremely rare in psycho-logical research. Research is usually complicated by confounding vari-ables, which prevent most correla-tional research studies from yielding data showing a true +1.00 or – 1.00 correlational coefficient.


Figure 3.9

scores, this is the range from a score of 55 to a score of 145. Statisti-cally, then, it is very unusual for people to have Wechsler test scores below 55 or above 145. No matter what is being tested, in a normal distribution, it is unusual for any individual to exceed three standard deviations from the mean in either direction.



Comparative StatisticsWHAT’S THE POINT?


The two major comparative statistics are percentage and percentile rank. Per-centage, as you probably know, compares a score to a perfect score of 100 points. If a student scores 83 percent on a test, for example, that student would have had 83 right on a test with 100 questions.

The percentile rank compares one score with other scores in an imagi-nary group of 100 individuals. Percentile rank tells you where a particu-lar score stands in that group and how many people had equal or lower scores. If our student scores at the 83rd percentile, it means that score would have equaled or exceeded the score of 83 of every 100 people who took the test. Figure 3.9 shows an example of how percentages and percen-tile ranks are calculated.

Assume Jack gets 160 points on a 200-point test. His score is good enough totop 27 students out of his class of 36 students.

Percentage

× 100 = 80%

100 points

Meaning: If the testhad been 100 points, Jackwould have had 80 right.

160 correct200 possible

Percentile rank

× 100 = 75th percentile

100 students

Meaning: If 100 students had takenthe test, Jack would have scoredhigher than 75 of them.

27 students beaten36 total students

80right

20wrong

Below Jack’sscore

AboveJack’sscore

percentage Acomparativesta-tisticthatcomparesascoretoaperfectscoreof100points.

percentilerank Acomparativestatisticthatcomparesascoretootherscoresinanimaginarygroupof100individuals.



Calculating Percentage Scores and Percentile Ranks

Thesetwocommoncom-parativestatisticshavesimilarnamesandarecalculatedwithsimilarformulas,buttheyhavedifferentmeanings.

TeaAs you teach correlations, you can refer to several activities and enrichment lessons from Module 2 relating to this complex topic:

●

●

●

●

●

TeaHelp students understand that perfect positive and perfect negative correlations are extremely rare in psychological research. Research is usually complicated by ables,tional research studies from yielding data showing a true +1.00 or –correlational coefficient.


Figure 3.10

Correlation CoefficientWHAT’S THE POINT?


Another highly useful statistic is the correlation coefficient, a statistical measure of the strength of the relationship between two variables. Variables can be related in two ways— positively or negatively. If both variables increase (or decrease) together, there is a positive correlation. An example of a positive correlation is the one between a weight- lifting conditioning program and strength: Lift more, grow stronger. Negative correlations involve two variables that change in opposite directions— one variable increases as the other decreases. There is a negative cor-relation between tooth flossing and dental decay: Floss more, have fewer cavities.

The calculation of a correlation coefficient is quite complicated, and we won’t go through it here. What’s important to remember is that the number produced by the calculation has a value that always falls between – 1.00 and +1.00. When r = – 1.00 (the letter r stands for correlation coefficient), we have a perfect negative correlation. Every time one variable increases by a certain amount, the other variable would decrease by an equally certain amount. Figure 3.10a shows a per-fect negative correlation (– 1.00).

If r = +1.00, we are looking at a perfect positive correlation between two variables. Every time one vari-able increases by a certain amount, the other variable also increases by an equally certain amount. Similarly, every time one variable decreases by a certain amount, the other variable decreases by an equally certain amount (see Fig-

ure 3.10b).If r = 0.00, there is no correla-

tion whatsoever between two vari-ables. Examples of this seem quite silly, like the relationship between flossing teeth and the temperature in Mexico City, or the relationship between the Los Angeles Lakers’ basketball scores and the num-ber of late arrivals into London’s

Perfect positive correlation (+1.00)(b)

No relationship (0.00)(c)

Perfect negative correlation (–1.00)(a)

correlationcoefficient Asta-tisticalmeasureofthestrengthoftherelationshipbetweentwovariables.

Correlations

Graph(a)displaysaperfectnegativecorrelation(r=–1.00).Eachtimethevariablerep-resentedontheverticalaxisincreasesbyacertainamount,thevariableonthehorizontalaxisdecreasesbyanequallycertainamount.Graph(b)showsaperfectpositivecorrelation(r=+1.00);thevariablesrepresentedonthetwoaxesincreaseinexactproportion.Ingraph(c),thereisnocorrelation(r=0.00).Therandomdotsshowthatthetwovariablesarenotrelated.


52

3Beyond the Classroom TRMApply Scatterplots are easy to create. In addition to Application Activity: Creating a Scatterplot, you may want to have students create their own scatterplot of data they think might be correlated.

TeaChing TiPConnect the topic of correlation coef-ficients with students’ knowledge of math. In math, students often have to determine the line of best fit in a graph and the slope of that line by calculat-ing the value of the rise of the line in relation to the run of the line. When the slope of the line is negative, the correlation coefficient is negative. When the slope is positive, the coeffi-cient is positive.

want to have students create their own scatterplot of data they think might be

determine the line of best fit in a graph


Figure 3.11

Heathrow Airport. When the first variable changes, we know nothing about what the second variable will do. When graphed, no relationship is apparent, as you can see in Figure 3.10c.

We’ve now illustrated three situations: a perfect negative correlation, a per-fect positive correlation, and no correlation at all. As you might imagine, the real world is rarely so neat and tidy. Positive correlations are usually less than r = +1.00, and negative correlations don’t often reach r = – 1.00. Figure 3.11 shows you what a graph, called a scatterplot, would look like in a more realistic correlation between two variables— height and temperament.

959085807570656055504540353025

8555 60 65 70 75 80

Temperament scoresEmotionally

reactive

Calm

Height in inches

Positive Correlation

Inapositivecorrelation,bothvariablesincrease(ordecrease)together.Themorethispersontrains,thestron-gerhewillbecome.

Scatterplot of a Moderately Positive Correlation

Thesesampledatashowthattallerpeoplearesomewhatmorelikelytobeemotionallyreactivethanshorterpeople.



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Statistical InferenceWHAT’S THE POINT?

3-7 What does it mean when a researcher reports that a research finding is statistically significant?

Most of the statistics we’ve discussed so far are descriptive statistics. They describe data in a way that makes them more meaningful. Another kind of statistics— inferential statistics— lets us make decisions or reach conclu-sions about data. Inferential statistics give psychologists guidelines for decid-ing whether data support hypotheses. Because inferential statistics are more complicated than descriptive statistics to calculate and interpret, we provide only a general discussion here.

Let’s return, one last time, to our music example. (Will you miss it?) Assume we’ve collected grade data from the two groups: the experimental group, whose members did listen to music, and the control group, whose members did not. Recall Figure 3.4, which presents the measures of central tendency we calculated. You can see that the music group did not perform quite as well as the no- music group. By the end of the experiment, there was a 4- point difference between the means of the two groups (81 for the no-music group and 77 for the music group). The key question is whether this difference is statistically significant, a statisti-cal statement of how likely it is that a result occurred by chance alone. In other words, does the 4- point difference represent a real difference, one that would be reflected in real- world conditions? Or is it simply the result of chance— a matter of “luck” that can be accounted for by some difference between our two groups (say, in study skills) despite our efforts to make them the same by using random assignment? We will never know with 100 percent certainty, but we can know how likely it is that this difference is statistically significant.

Most psychologists are willing to accept up to a 5 percent likelihood that an experiment’s results did not occur by chance. This means being at least 95 percent sure that a difference in results is because of the manipulation of the independent variable, which in this case was whether or not students listened to music, and not some other unknown variable. A series of calcula-tions with our results in the imaginary music experiment would tell us that with our 4- point difference we can be about 40 percent sure that the head-phones caused the difference in scores. So, this is not even close to a statisti-cally significant result. For the difference to be statistically significant, the two groups would have to be more clearly separated, with a larger difference or less overlap between them.

There are a number of factors involved in inferential statistics. Here are the three most important:

● The difference between the two groups’ means. If the means are far apart, the result is more likely to be statistically significant.

● The number of participants. If each group has only a few people, the results are not as likely to be statistically significant as they would be if each group has a large number of randomly selected people in it.

inferentialstatistics Statis-ticsthatcanbeusedtomakeadecisionorreachaconclusionaboutdata.

statisticalsignificance Astatisticalstatementofhowlikelyitisthataresultoccurredbychancealone.


Computer ApplicationsInvite a computer- applications teacher to show students how to compute statistics using a software program like Microsoft Excel. Students can take any set of data and compute any of the statistics from this module. Encourage them to also cre-ate graphs and charts of the data.


53

3TeaChing TiPHelp students learn the meaning of inferential statistics by directing them to the root word infer. To infer means “to conclude or judge using evidence.” Inferential statistics help people make judgments based on evi-dence gathered from research.

TeaChing TiPStudents shouldn’t worry too much about calculating statistical signifi-cance, but they do need to understand what the term means conceptually. That way, when they take statistics classes later in life, they will better appreciate the real- world applica-tion of the math associated with significance.

Beyond the Classroom TRMApply Take a set of test grades (from either your current or past classes) and have students conduct some simple statistical analysis on the data. Chal-lenge your students further by first dividing the scores into two groups based on gender, class rank, or some other arbitrary standard and then checking to see if a significant differ-ence is apparent by simply eyeball-ing the data. If you are comfortable, have students conduct an analysis of variance to determine statistical significance.

At this point, you may wish to use Application Activity: Describing Data.


Figure 3.11

Heathrow Airport. When the first variable changes, we know nothing about what the second variable will do. When graphed, no relationship is apparent, as you can see in Figure 3.10c.

We’ve now illustrated three situations: a perfect negative correlation, a per-fect positive correlation, and no correlation at all. As you might imagine, the real world is rarely so neat and tidy. Positive correlations are usually less than r = +1.00, and negative correlations don’t often reach r = – 1.00. Figure 3.11 shows you what a graph, called a scatterplot, would look like in a more realistic correlation between two variables— height and temperament.

959085807570656055504540353025

8555 60 65 70 75 80

Temperament scoresEmotionally

reactive

Calm

Height in inches

Positive Correlation

Inapositivecorrelation,bothvariablesincrease(ordecrease)together.Themorethispersontrains,thestron-gerhewillbecome.

Scatterplot of a Moderately Positive Correlation

Thesesampledatashowthattallerpeoplearesomewhatmorelikelytobeemotionallyreactivethanshorterpeople.



Bo

B D

ae

mm

ric

h/s

toc

k B

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TeaHelp students learn the meaning of inferentialthem to the root word means “to conclude or judge using evidence.” Inferential statistics help people make judgments based on evidence gathered from research.

TeaStudents shouldn’t worry too much about calculating statistical significance, but they do need to understand what the term means conceptually. That way, when they take statistics classes later in life, they will better appreciate the real-tion of the math associated with significance.

Beyond the Classroom Applyeither your current or past classes) and have students conduct some simple statistical analysis on the data. Challenge your students further by first dividing the scores into two groups based on gender, class rank, or some other arbitrary standard and then checking to see if a significant difference is apparent by simply eyeballing the data. If you are comfortable, have students conduct an analysis of variance to determine statistical significance.

Application


Statistical InferenceWHAT’S THE POINT?


Most of the statistics we’ve discussed so far are descriptive statistics. They describe data in a way that makes them more meaningful. Another kind of statistics— inferential statistics— lets us make decisions or reach conclu-sions about data. Inferential statistics give psychologists guidelines for decid-ing whether data support hypotheses. Because inferential statistics are more complicated than descriptive statistics to calculate and interpret, we provide only a general discussion here.

Let’s return, one last time, to our music example. (Will you miss it?) Assume we’ve collected grade data from the two groups: the experimental group, whose members did listen to music, and the control group, whose members did not. Recall Figure 3.4, which presents the measures of central tendency we calculated. You can see that the music group did not perform quite as well as the no- music group. By the end of the experiment, there was a 4- point difference between the means of the two groups (81 for the no-music group and 77 for the music group). The key question is whether this difference is statistically significant, a statisti-cal statement of how likely it is that a result occurred by chance alone. In other words, does the 4- point difference represent a real difference, one that would be reflected in real- world conditions? Or is it simply the result of chance— a matter of “luck” that can be accounted for by some difference between our two groups (say, in study skills) despite our efforts to make them the same by using random assignment? We will never know with 100 percent certainty, but we can know how likely it is that this difference is statistically significant.

Most psychologists are willing to accept up to a 5 percent likelihood that an experiment’s results did not occur by chance. This means being at least 95 percent sure that a difference in results is because of the manipulation of the independent variable, which in this case was whether or not students listened to music, and not some other unknown variable. A series of calcula-tions with our results in the imaginary music experiment would tell us that with our 4- point difference we can be about 40 percent sure that the head-phones caused the difference in scores. So, this is not even close to a statisti-cally significant result. For the difference to be statistically significant, the two groups would have to be more clearly separated, with a larger difference or less overlap between them.

There are a number of factors involved in inferential statistics. Here are the three most important:

● The difference between the two groups’ means. If the means are far apart, the result is more likely to be statistically significant.

● The number of participants. If each group has only a few people, the results are not as likely to be statistically significant as they would be if each group has a large number of randomly selected people in it.

inferentialstatistics Statis-ticsthatcanbeusedtomakeadecisionorreachaconclusionaboutdata.

statisticalsignificance Astatisticalstatementofhowlikelyitisthataresultoccurredbychancealone.

active learning

Inferential StatisticsLead students through the experimental process and show them how to calculate inferen-tial statistics.

1. Divide students into two groups. One group listens to classical music and the other to “white noise” (static or whooshing sounds).

2. Have groups listen while learning a list of nonsense syllables. Then have them recall as many syllables as they can in 30 seconds. Use a software program like Microsoft Excel or SPSS to compute statistics on the data.

You may want to use Application Activity: When Is a Difference Significant?

TRM


54

3assess

Check for UnderstandingAnalyze Have students compare data from two different sports teams (foot-ball, baseball, soccer, and so forth). Then have them calculate the mean, median, mode, and standard deviation for each set of data.

CLose

ReteachSummarize Statistics can be difficult to understand, but knowing how sta-tistics are derived and what they mean can help students make important decisions. You don’t have to know how to calculate the statistics in this mod-ule to know why they are important.

median, mode, and standard deviation

tistics are derived and what they mean

decisions. You don’t have to know how


Figure 3.12

● The standard deviation of the two groups. If the scores of both groups are mostly packed close to the means, the means don’t need to be sepa-rated by as much to produce a statistically significant result. If the scores are widely spread (represented by high standard deviations), the two groups are likely to overlap quite a bit. Many participants will score in the same “overlap range” no matter which group they are assigned to, and the result is not likely to be statistically significant. This is illustrated in Figure 3.12.

There is much to know about statistics, and much of it is beyond the scope of an introductory psychology course. This module has been just a brief introduction. Each formula is a logical application of accepted procedures that can be organized in a series of straightforward steps. The main point to remember is that we use statistics to make research results meaningful. The more you know about statistics, the better equipped you will be to critically evaluate information. Whether it’s sports league player stats or advertisers’ product claims, making sense of statistics in the world around you is a life-long valuable skill.

Not likely to be a significant difference

Likely to be a significant difference

Distribution A Distribution B


Statistical Significance

Aresearchresultissta-tisticallysignificantwhenitisunlikelythattheresultoccurredbychance.Fur-thermore,whentwodistribu-tionsshowlittleoverlap,thedifferencebetweenthemismorelikelytobestatisticallysignificant.




distribution is distorted, or ___________, by some unusually high or low scores.


WHAT’S THE POINT?


● Two ways of looking at the variation of a set of scores are to examine the range (the dif-ference between the highest and the lowest scores) and examine the standard deviation (how the scores are distributed around the mean).

Apply What You Know

5. The standard deviation is a good way to get a sense of the __________ of a distribution of scores.

a. consistency b. center c. accuracy d. reasonableness

6. Two measures of variation are the standard deviation and the __________.

Normal Distribution

WHAT’S THE POINT?


● A graph of a normal distribution is shaped like a symmetrical bell, with most scores falling near the mean and fewer scores at the extremes.

● Of the scores, 68 percent are within one standard deviation of the mean, 96 percent are within two standard deviations, and 99.7 percent are within three standard deviations.


WHAT’S THE POINT?


● A frequency distribution lists a range of scores from highest to lowest and can be easily pre-sented in a graph.

Apply What You Know

1. Which of the following is an example of a frequency distribution?

a. a baseball team roster, organized by jersey number

b. an alphabetized grocery list c. a list of quiz scores from a class, in order

from highest to lowest d. a list of ingredients and amounts in a choco-

late chip cookie recipe

2. What is the advantage of creating a frequency distribution?


WHAT’S THE POINT?


● The mode is the most frequent score or scores in a set of scores.

● The mean is the mathematical average of the set of scores.

● The median is the score in the middle of the fre-quency distribution.

Apply What You Know

3. The __________ is the average of a set of numbers.

4. The median is generally a better representation of the center of a distribution when the

SuMMARY AND FoRMATiVE ASSESSMENTMODULE 3 Thinking About Psychology’s Statistics

Applying Research SkillsLeading students through their own research experiment provides a valuable critical think-ing and learning experience. Help students research and showcase their work. This Portfolio Project can be found in Module 2 of the TRM.

● Choose topics with a rich research background. This will help with literature review and generating hypotheses.

● Have students work cooperatively in groups to encourage teamwork and to decrease the number of studies you must monitor.

● Insist on following ethical guidelines. No research should be conducted without obtain-ing informed consent from the research participants.

● Create a rubric that places emphasis on critical thinking, good research skills, and ethics.

At this point, you may want to use Alternative Assessment/Portfolio Project: Apply-ing Research Skills.

PortFolio Project TRM


55

3Using the Test BankThe Test Bank that accompanies this textbook offers a wide variety of ques-tions in different formats and levels of complexity. Use the software to construct whole tests or to integrate standardized questions into teacher- made tests.

answers Frequency Distributions: Apply What You Know

1. (c)

2. At a glance, you can see the high and low scores for each group, how the scores cluster, and what the most common scores are.

answers Measures of Central Ten-dency: Apply What You Know

3. mean

4. skewed

answers Measures of Variation: Apply What You Know

5. (a)

6. range


Figure 3.12

● The standard deviation of the two groups. If the scores of both groups are mostly packed close to the means, the means don’t need to be sepa-rated by as much to produce a statistically significant result. If the scores are widely spread (represented by high standard deviations), the two groups are likely to overlap quite a bit. Many participants will score in the same “overlap range” no matter which group they are assigned to, and the result is not likely to be statistically significant. This is illustrated in Figure 3.12.

There is much to know about statistics, and much of it is beyond the scope of an introductory psychology course. This module has been just a brief introduction. Each formula is a logical application of accepted procedures that can be organized in a series of straightforward steps. The main point to remember is that we use statistics to make research results meaningful. The more you know about statistics, the better equipped you will be to critically evaluate information. Whether it’s sports league player stats or advertisers’ product claims, making sense of statistics in the world around you is a life-long valuable skill.

Not likely to be a significant difference

Likely to be a significant difference



Statistical Significance

Aresearchresultissta-tisticallysignificantwhenitisunlikelythattheresultoccurredbychance.Fur-thermore,whentwodistribu-tionsshowlittleoverlap,thedifferencebetweenthemismorelikelytobestatisticallysignificant.



Using the Test BankThe textbook offers a wide variety of questions in different formats and levels of complexity. Use the software to construct whole tests or to integrate standardized questions into teacher-made tests.

aApply What You Know

adency: Apply What You Know

aWhat You Know


distribution is distorted, or ___________, by some unusually high or low scores.


WHAT’S THE POINT?


● Two ways of looking at the variation of a set of scores are to examine the range (the dif-ference between the highest and the lowest scores) and examine the standard deviation (how the scores are distributed around the mean).

Apply What You Know

5. The standard deviation is a good way to get a sense of the __________ of a distribution of scores.

a. consistency b. center c. accuracy d. reasonableness

6. Two measures of variation are the standard deviation and the __________.

Normal Distribution

WHAT’S THE POINT?


● A graph of a normal distribution is shaped like a symmetrical bell, with most scores falling near the mean and fewer scores at the extremes.

● Of the scores, 68 percent are within one standard deviation of the mean, 96 percent are within two standard deviations, and 99.7 percent are within three standard deviations.


WHAT’S THE POINT?


● A frequency distribution lists a range of scores from highest to lowest and can be easily pre-sented in a graph.

Apply What You Know

1. Which of the following is an example of a frequency distribution?

a. a baseball team roster, organized by jersey number

b. an alphabetized grocery list c. a list of quiz scores from a class, in order

from highest to lowest d. a list of ingredients and amounts in a choco-

late chip cookie recipe

2. What is the advantage of creating a frequency distribution?


WHAT’S THE POINT?


● The mode is the most frequent score or scores in a set of scores.

● The mean is the mathematical average of the set of scores.

● The median is the score in the middle of the fre-quency distribution.

Apply What You Know

3. The __________ is the average of a set of numbers.

4. The median is generally a better representation of the center of a distribution when the

SuMMARY AND FoRMATiVE ASSESSMENTMODULE 3 Thinking About Psychology’s Statistics


56

3answers Normal Distribution: Apply What You Know

7. (d)

8. 99.7

answers Comparative Statistics: Apply What You Know

9. percentile rank

10. percentage

answers Correlation Coefficient: Apply What You Know

11. There is no correlation.

12. (d)

answers Statistical Inference: Apply What You Know

13. chance

14. (b)

Comparative Statistics: Apply

Correlation Coefficient: Apply


Apply What You Know

7. If you know that scores on an exam in a class are normally distributed, then you know that about __________ of the class scored within one standard deviation of the average score on the exam.

a. 96 percent b. 99.7 percent c. 50 percent d. 68 percent

8. Over __________ percent of all scores fall within three standard deviations of the mean.


WHAT’S THE POINT?


● Percentage refers to a comparison between a score and a perfect score of 100 points (for exam-ple, dividing the number of points earned on an exam by the number of possible points produces the percentage).

● Percentile rank explains where a score falls in an imaginary group of 100 individuals (for example, if your percentile rank on a test is 80, you scored at or better than 80 percent of the people who took the exam).

Apply What You Know

9. Which comparative statistic compares your score to the performance of other people?

10. Which comparative statistic compares your score to the perfect score?


WHAT’S THE POINT?


● A correlation coefficient is a number (rep-resented by r) between – 1.00 and +1.00 that

indicates the strength of a statistical relationship between two variables.

● Positive correlations indicate that as one vari-able increases, it is likely that the other variable will increase.

● Negative correlations indicate that as one vari-able increases, it is likely that the other variable will decrease.

Apply What You Know

11. What does it mean to say that the correlation coefficient is about 0.00?

12. Which of the following pairs of variables is likely to have an r value between 0.00 and – 1.00?

a. height and IQ scores b. studying and GPA c. standard deviation and measures of central

tendency d. smoking and life span


WHAT’S THE POINT?


● Statistical significance indicates that a research finding is most likely the result of the variable the researcher is studying, not ran-dom chance.

Apply What You Know

13. Statistically significant results are not likely to be caused by __________.

14. Which of the following factors has the most influence on whether a result is statistically significant?

a. the experience of the researcher b. the difference between the two groups’

means c. the accuracy of the measurements d. whether the results have been placed in a

frequency distribution


frequency distribution, p. 43

mode, p. 44

mean, p. 45

median, p. 45

skewed, p. 46

range, p. 47

standard deviation, p. 47

normal distribution, p. 48

percentage, p. 50

percentile rank, p. 50

correlation coefficient, p. 51

inferential statistics, p. 53

statistical significance, p. 53

K e y T e r m s


57

3


Apply What You Know

7. If you know that scores on an exam in a class are normally distributed, then you know that about __________ of the class scored within one standard deviation of the average score on the exam.

a. 96 percent b. 99.7 percent c. 50 percent d. 68 percent

8. Over __________ percent of all scores fall within three standard deviations of the mean.


WHAT’S THE POINT?


● Percentage refers to a comparison between a score and a perfect score of 100 points (for exam-ple, dividing the number of points earned on an exam by the number of possible points produces the percentage).

● Percentile rank explains where a score falls in an imaginary group of 100 individuals (for example, if your percentile rank on a test is 80, you scored at or better than 80 percent of the people who took the exam).

Apply What You Know

9. Which comparative statistic compares your score to the performance of other people?

10. Which comparative statistic compares your score to the perfect score?


WHAT’S THE POINT?


● A correlation coefficient is a number (rep-resented by r) between – 1.00 and +1.00 that

indicates the strength of a statistical relationship between two variables.

● Positive correlations indicate that as one vari-able increases, it is likely that the other variable will increase.

● Negative correlations indicate that as one vari-able increases, it is likely that the other variable will decrease.

Apply What You Know

11. What does it mean to say that the correlation coefficient is about 0.00?

12. Which of the following pairs of variables is likely to have an r value between 0.00 and – 1.00?

a. height and IQ scores b. studying and GPA c. standard deviation and measures of central

tendency d. smoking and life span


WHAT’S THE POINT?


● Statistical significance indicates that a research finding is most likely the result of the variable the researcher is studying, not ran-dom chance.

Apply What You Know

13. Statistically significant results are not likely to be caused by __________.

14. Which of the following factors has the most influence on whether a result is statistically significant?

a. the experience of the researcher b. the difference between the two groups’

means c. the accuracy of the measurements d. whether the results have been placed in a

frequency distribution


frequency distribution, p. 43

mode, p. 44

mean, p. 45

median, p. 45

skewed, p. 46

range, p. 47

standard deviation, p. 47

normal distribution, p. 48

percentage, p. 50

percentile rank, p. 50

correlation coefficient, p. 51

inferential statistics, p. 53

statistical significance, p. 53

K e y T e r m s


Lesson PLanning CaLendar - iblog. · PDF fileMode Mean Median Measures of Variation ... Building Vocabulary: Crossword Puzzle offers students an opportunity to improve their understanding

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