Chapter 3 Averages and Variation

Chapter 3 Averages and Variation

Chapter 3 Averages and VariationQuick Talk:Think back to middle school and high school (before taking Mr. Lius awesome statistics class). What do you think of when you heard of statistics?AnswerThe 3 Ms. Mean, median and modeWhat is mode?Mode is the value that occurs most frequentlyGroup Work: What is the mode out of this data set?25, 37, 12, 27, 69, 64, 44, 30, 50, 52, 65, 25, 10, 12, 60, 27, 12, 65, 25Answer12 and 25Group work What is the mode out of this data set?

5 3 7 2 4 4 2 4 8 3 4 3 4 8 6 2 4 5 3 4 5 9 8 7 1Answer 7What is a median?How to find median?Group WorkFind the Median1920 25 36 48 35 21 18Answer18 19 20 21 25 35 36 48

Because it is even, I add (21+25)/2 = 23

So median = 23Group WorkFind the median36, 4, 69, 47, 81, 84, 6, 38, 17, 93, 46Note:Median uses the position rather than the specific value of each data entry. If the extreme values of a data set change, the median usually does not change.

It is often used as the average for house prices.Median RuleExample:If n = 77, then the middle value is (77+1)/2= 39th number.

If n=90, then the middle value is (90+1)/2=45.5th value. Which is in between 45th and 46th number.What do you mean Mean?Group WorkFind the mean: 58, 92, 50, 68, 80, 64Group Work: Word problem Mr. Liu is a Paladin Tank thats low in HP. He only has 2 hits left before the boss unleash his ultra super attack and pwns Mr. Liu. The boss has a total of 750 HP and Mr. Liu has damaged the boss with the following damage points: 98, 50, 65, 71, 120, 84, 93, 27. What is the mean damage of the next 2 hits so that Mr. Liu survives?AnswerMany ways to solve this depend on how you think about it.

The mean is 71 damage points for the next 2 hits.Note: Resistant measureResistant measure is one that is not influenced by extremely high or low data values.Quick Talk:Which of the following data has more of a resistant measure, mean or median?AnswerMedian, although the disadvantage of the median is that it is not sensitive to the specific size of a data value.

Mean is too sensitive to the data value.How do we make mean more resistant yet sensitive?It is called 10% trimmed mean

Procedures:1) Order the data from smallest to largest2) Delete the bottom 5% or the data and the top 5% of the data. Note: If the calculation of 5% of the number of data values does not produce a whole number, round down to the nearest integerEx: if you get .75, you round to 0Ex: if you get 1.95, you round to 13) Compute the mean of the remaining 90% of the dataExample situationHere are some sample class sizes for introductory lecture courses14, 30, 45, 11, 20, 80, 65, 34, 18, 60, 100, 42, 40, 40, 78, 31, 52, 35, 120, 19

A) compute the meanB) compute the 10% trimmed meanC) Find the medianD) Find the median of the 10% trimmed data set. Does the median change when you trim the data?E) Is the trimmed mean or the original mean closer to the median?AnswerA) 46.7B) 44.61C) 40D)40E) Trimmed mean is closer to the medianGroup WorkHow old are basketball players? Here are some of the ages: 40, 38, 21, 18, 24, 23, 28, 25, 21, 27, 26, 25, 31, 32, 19, 29, 33, 24, 23, 35, 24, 23, 25, 32, 39, 26, 31, 28, 30, 31, 22, 23, 24, 25, 26, 27, 38, 27, 26, 22

A) compute the meanB) compute the 10% trimmed meanC) Find the medianD) Find the median of the 10% trimmed data set. Does the median change when you trim the data?E) Is the trimmed mean or the original mean closer to the median?

AnswerA) 27.275B) 27.083 C) 26D) 26E) CloserWeighted AverageExampleGroup Work: Mr. Lius grading weightsTest/Quizzes 30 %Participation/bell work 10 %Notebook 15 % Projects 20 %Homework 15 %Finals 10 %

You scored 85% on test/quizzesYou scored 60% on HomeworksYou scored 100% on participation/bell workYou scored 80% on projectsYou scored 98% on notebookYou scored 50% on finals

What is your grade for the class?

Answer.85(.30)+.60(.15)+1(.10)+.80(.20)+.98(.15)+.50(.10)=.255+.09+.10+.16+.147+.05=.802 or 80.2% which is a B-Homework PracticeP83 #1-5, 8, 12, 14, 17Note:An average is an attempt to summarize a set of data using just one number. But it itself may not always be very meaningful.

We need a statistical cross-reference that measures the spread of the data.The first VariationRange: is the difference between the largest and smallest values of a data distributionRemember how to find range?Tell meGroup WorkAverage tennis balls suppose to weigh around 56g. Random balls are selected from two suppliers to see which of them are more accurate. Supplier 1: 46 56 56 60 62Supplier 2: 46 54 58 60 62

A) Compute the range of each supplierB) Compute of mean weight from both supplierC) If you are competing in a competition, which supplier would you choose from?AnswerA) Supplier 1= 16Supplier 2= 16

B) Supplier 1 mean=56Supplier 2 mean= 56

C) You would choose supplier 1 because more balls are around the mean. More clustered at the meanVariance and Standard DeviationNow the fun begins.formulas!!Are you prepared?

Note: I will be covering both sample statistics and population parametersDefining Formulas (Sample Statistic)PracticeRandom Sample5 7 8 12 15 15

Find the variance and standard deviationAnswerGroup WorkRandom Sample5 5 5 6 7 8

Find the variance and standard deviationAnswerPopulation ParametersNote:Samples are usually in english alphabets

Populations are usually in greek alphabets

Samples are in lower case

Population are in upper caseGroup WorkAnswerDisadvantage of Standard DeviationDisadvantage of Standard Deviation is that it depends on the unit of measurement. It is difficult to use the standard deviation to compare measurements from different populations.So what do we do?We use coefficient of variation which expresses the standard deviation as percentage of the sample or population mean. Coefficient of Variation formulaThink!Remember the difference between sample and population!SituationTrading Post on Grand Mesa is a small, family-run store in a remote part of Colorado. They sell spinners (a type of fishing lure). The store only offer 8 types of spinners. The prices are:

2.10 1.95 2.60 2.00 1.85 2.25 2.15 2.25

A)Find mean and SDB) Compute the CV of prices and explain what it means.AnswerSituation 2Cabelas in Sidney, Nebraska, is a very large outfitter that carries a broad selection of fish spinners. A random sample of 10 spinners from Cabelas are of the following prices:

1.69 1.49 3.09 1.79 1.39 2.89 1.49 1.39 1.49 1.99

a)find the mean and standard deviationb)Find the CVC) Compare the mean, SD and CV from this situation from the last oneAnswerChebyshevs TheoremMeaning (this refer to the minimum percentage of data)ExampleNumerical ExampleGroup Work: Compute a 88.9% Chebyshev intervalIf the mean is 96.5 and standard deviation is 17.8Special Formula for standard deviationExampleAge range (years)1-1011-2021-3031-40Number of individuals34181711How to do it (answer)Homework PracticePg 96 #1-4, 10, 11, 17 Percentiles and Box-and-Whisker PlotsQuick Talk:Have you guys taken your SAT yet? Remember how you look at the score and it says your percentile? What does it mean?If you are in the 70th percentile, it means that you scored the same or better than 70% of the people who took the SAT. 30% Scored the same or above you.PencentileGroup WorkA) If your score is at the 27th percentile, what percentage of scores are at or below yours?

B) If the score ranged from 1 to 100 and your score is 90%, does that necessarily mean that your score is at the 90th percentile?AnswerA) 27%

B) No because percentile gives an indication of relative position of the scores.Note:Median is an average computed by using relative position of the data. Median is an example of a percentile; it is the 50th percentile.SAT scores also give you QuartilesWhat is a Quartiles? (Look at the word itself, what does it sound like?)This is a quartileQuartile sounds like quarters so it has 4 intervals

How to Compute Quartiles1) Order the data from smallest to largest2) Find the median (Q2). This is the second quartile (Q2)3) The first quartile Q1 is then the median of the lower half of the data; that is, it is the median of the data falling below the Q2 position (and not including Q2)4) The third quartile Q3 is the median of the upper half of the data; that is, it is the median of the data falling above the Q2 position (and not including Q2)

Practice: These number are random. Find the quartiles and interquartile range1617182025333335373840475050506384979799100103107107107108123Answer1) Where is the median (Q2)? Since there are 27 data, It is at the14th number, which is 50

2) Where is Q1? Since Q2 is at 14th spot, there are 13 numbers below. The median between Q1 and Q2 is at the 7th spot, which is 33

3) Where is Q3? Since Q2 is at the 14th spot, there are 13 numbers above it. The median between Q2 and Q3 is at the 21th spot, which is 100

4) Interquartile is Q3-Q1, which is 100-33=67Group Work: Here are the points Mr. Liu Scored in basketball games. Construct the quartiles and IQR.715236123530146212539516925217121031AnswerQ1=8Q2=15Q3=25IQR=17Group Work: Here are sample SAT MATH scores. Construct the quartiles and IQR.540570520550490430490490570380510640310400380390410340390350330350AnswerQ1=380Q2=420Q3=520IQR=140Box-and-Whiskers PlotsQuartiles together with the low and high data values give us a very useful five-number (nonot $5 foot long) summary of the data and their spread.

Note:The longer the whisker is, the more skewed it is toward that value.

If the median is closer to the lower part of the box, it means that data are more concentrated toward the lower end.

If the median is closer to the upper part of the box, it means that data are more concentrated toward the upper end.

How to make a box-and-whisker plot1) Draw a vertical/horizontal scale to include the lowest and highest data values.2) To the right/top of the scale, draw a box from Q1 to Q33) Include a solid line through the box at the median level4) Draw vertical/horizontal lines, called whiskers, from Q1 to the lowest value and from Q3 to the highest value

(Look at previous slide for examples)Please read guided exercise 8 on page 1075 minutes

Read and explain to your partner to see if it makes sense.Group ActivityDraw box-and-whisker plot for our last 3 quartile examples.

Also, find the following for all the quartile examples:Find the range for all (3) examplesWhich example has the highest median?Which example has the biggest range?Which example has the highest IQR?

It is your group pop-quiz! You have to show them to me before you leave.

TechnologyFor TI 83/84. Press STATPLOT On. Highlight box plot. Use Trace and the arrow keys to display the values of the 5-number summary. The display shows the box-and-whisker plot.

Make sure you input data values in L1Homework PracticePg 109 #1-4, 7-9