Learning Objectives In this chapter you will learn about measures of central tendency measures of central tendency levels of measurement levels of measurement.

Learning ObjectivesLearning Objectives

In this chapter you will learn aboutIn this chapter you will learn about measures of central tendencymeasures of central tendency levels of measurementlevels of measurement measures of shapemeasures of shape

Uses of StatisticsUses of Statistics

Statistics provide information by organizing and

summarizing data describe the nature of a sample

Description of a data involves measures that best characterize a

frequency distribution

Measures of Central Measures of Central TendencyTendency

Descriptive statistics measures that best characterize a

frequency distribution the scores that are most “typical” these measures describe scores that

group around a central value

Frequency DistributionFrequency Distribution

The next slide shows the number of The next slide shows the number of prisoners executed prisoners executed

Items are listed in order from the Items are listed in order from the highest to the lowest valuehighest to the lowest value

The symbol The symbol xx stands for the value of stands for the value of the variablethe variable xx = the number of inmates executed = the number of inmates executed

f is the number of cases that assume a certain f is the number of cases that assume a certain value. f is the number of states that have value. f is the number of states that have executed a number of inmates.executed a number of inmates.

Here, we see that one state has executed 104 Here, we see that one state has executed 104 inmates. This is by far the highest number.inmates. This is by far the highest number.

fX is the sum of cases. A total of 303 fX is the sum of cases. A total of 303 offenders were executed in the U.S. offenders were executed in the U.S. between 1977 and 1995.between 1977 and 1995.

Table 3.1: Number of Inmates Executed Between 1977 and 1995

X f fX

104 1 104

36 1 36

29 1 29

22 1 22

20 1 20

17 1 17

11 1 11

8 1 8

7 1 7

6 1 6

5 3 15

4 3 12

3 1 3

2 3 6

1 5 5

0 25 0

Central TendencyCentral Tendency

In a distribution, where do most of the In a distribution, where do most of the cases “cluster?”cases “cluster?”

Three measures of central tendencyThree measures of central tendency modemode medianmedian meanmean

The ModeThe Mode

The mode The mode is the score that occurs most frequently is the score that occurs most frequently

in a distributionin a distribution In our table, zero (0) is the modeIn our table, zero (0) is the mode Twenty-five states (25 under the Twenty-five states (25 under the ff

column) did not execute a single column) did not execute a single convicted offender between 1977 and convicted offender between 1977 and 19951995

The ModeThe Mode

Note - the mode IS NOT 104! The mode is the more frequently

occurring category, in this case zero The mode is ALSO NOT 25! A frequency distribution may have more

than one mode

The ModeThe Mode

If another value (number of states) If another value (number of states) had a frequency of 25 in the table, it had a frequency of 25 in the table, it would also have been the mode would also have been the mode frequency distribution with two modes frequency distribution with two modes

is termed bimodalis termed bimodal more than two modes, it is called more than two modes, it is called

multimodal multimodal

Properties of the ModeProperties of the Mode

The mode The mode does not necessarily occur in or near the does not necessarily occur in or near the

center of a distribution center of a distribution can occur anywhere in a distributioncan occur anywhere in a distribution does not indicate the variability between does not indicate the variability between

scores in a distributionscores in a distribution simply indicates the value(s) that occur simply indicates the value(s) that occur

most frequentlymost frequently

The MedianThe Median

In a frequency distributionIn a frequency distribution scores are placed in order from lowest to scores are placed in order from lowest to

highesthighest the median is the middle of the the median is the middle of the

distribution. distribution. It is the 50It is the 50thth percentile percentile 50% of the scores in the frequency 50% of the scores in the frequency

distribution fall below and above the mediandistribution fall below and above the median

Properties of the Properties of the MedianMedian

Attributes of the median Attributes of the median stabilitystability

the median is unaffected by extreme scoresthe median is unaffected by extreme scores it is calculated by counting the number of it is calculated by counting the number of

casescases it does not consider the value of the it does not consider the value of the

casecase

Calculating the MedianCalculating the Median

The median can be calculated easily The median can be calculated easily and determined by inspection and determined by inspection In the table, N (the number of cases) = In the table, N (the number of cases) =

50 - the number of state 50 - the number of state determine where the middle case liesdetermine where the middle case lies

one half of 50 is 25one half of 50 is 25

The MedianThe Median

Example:Example: The first number in the distribution, The first number in the distribution,

zero, has a frequency of 25zero, has a frequency of 25 Therefore, the median is zeroTherefore, the median is zero Half of the states executed no one Half of the states executed no one

during the time period, 1977 to 1995during the time period, 1977 to 1995

The MeanThe Mean

The mean is The mean is the average score in a distributionthe average score in a distribution calculated by adding all the scores calculated by adding all the scores

in a distribution and dividing the in a distribution and dividing the total by the number of cases total by the number of cases

Calculating the MeanCalculating the Mean

Example:Example: a total of 303 inmates (a total of 303 inmates (fx = 303) were fx = 303) were

executed between 1977 and 1995executed between 1977 and 1995 there were 50 states (N = 50)there were 50 states (N = 50) The mean is 6.06 (303/50) The mean is 6.06 (303/50) An average of six inmates were executed An average of six inmates were executed

by each jurisdiction during the period 1977 by each jurisdiction during the period 1977 – 1995– 1995

Characteristics of the Characteristics of the MeanMean

The mean is The mean is unlike the mode and medianunlike the mode and median the mean is sensitive to extreme scoresthe mean is sensitive to extreme scores

Example:Example: Texas executed 104 inmates between Texas executed 104 inmates between

1977 and 1995 1977 and 1995 The next closest jurisdiction executed 36 The next closest jurisdiction executed 36

inmates inmates

104 executions (Texas) were an extreme score 104 executions (Texas) were an extreme score in this distribution. The median for this in this distribution. The median for this distribution was zero. Half of the jurisdictions distribution was zero. Half of the jurisdictions executed one or no inmates during this time executed one or no inmates during this time period. period.

Yet, our mean was 6.06 – over six points Yet, our mean was 6.06 – over six points above the median. The Texas executions above the median. The Texas executions drove up the average number for the time drove up the average number for the time period.period.

This attribute of the mean occurs because it This attribute of the mean occurs because it is computed by using the value of each is computed by using the value of each score in the distribution. score in the distribution.

The mode and median fail to use the value The mode and median fail to use the value of each score in a distribution. The mode is of each score in a distribution. The mode is derived from the frequency of the scores. derived from the frequency of the scores. The median is based on the position of the The median is based on the position of the scores, regardless of their values. scores, regardless of their values.

The mean is amenable to statistical The mean is amenable to statistical analysis and comparisons between analysis and comparisons between distributions while the mode and distributions while the mode and median are not. median are not.

Also, the sum of the deviations from Also, the sum of the deviations from the mean (how far each score stands the mean (how far each score stands in relation to the mean) is zero.in relation to the mean) is zero.

Symmetric DistributionSymmetric Distribution

zero skewnesszero skewness

mode = median = meanmode = median = mean

Positively Skewed Positively Skewed DistributionDistribution

Positively Positively skewedskewed: : Mean and Mean and Median are Median are to the right of to the right of the Modethe Mode

Negatively Skewed Negatively Skewed DistributionDistribution

Negatively Negatively SkewedSkewed: : Mean and Mean and Median are Median are to the left of to the left of the Modethe Mode

Levels of MeasurementLevels of Measurement

Numbers are used to measure Numbers are used to measure conceptsconcepts

like fear of crime like fear of crime support for the police or capital support for the police or capital

punishment. punishment.

The numbers are used as a codeThe numbers are used as a code

Question?Question?

Statistically, the question isStatistically, the question is can we use mathematics to now can we use mathematics to now

analyze this code that we have analyze this code that we have established?established?

does it make sense to treat the does it make sense to treat the numbers as such and perform numbers as such and perform arithmetic operations on them?arithmetic operations on them?

This code is called the This code is called the level of level of measurementmeasurement. It involves converting the . It involves converting the concepts to numerical data. There are four concepts to numerical data. There are four categories and each have different categories and each have different attributes. However, the levels of attributes. However, the levels of measurement are cumulative, kind of like measurement are cumulative, kind of like the steps on a ladder. You have to step on the steps on a ladder. You have to step on the first step to reach the second, and so the first step to reach the second, and so on. on.

Each succeeding level Each succeeding level automatically possesses the automatically possesses the attributes of the level preceding it, attributes of the level preceding it, plus another distinct one.plus another distinct one.

Levels of Levels of MeasurementMeasurement

Nominal levelNominal level:involves the process :involves the process of classifying data into categories. of classifying data into categories. When we classify respondents by When we classify respondents by race or sex, we are using nominal race or sex, we are using nominal measurement (i.e., 1 – Male, 2 – measurement (i.e., 1 – Male, 2 – Female).Female).

Nominal level measurement Nominal level measurement follows follows

three basic rules:three basic rules:1.1. The list of categories must be The list of categories must be

exhaustive and cover all the types of exhaustive and cover all the types of observations made.observations made.

2.2. The categories must be mutually The categories must be mutually exclusive. Each observation can only exclusive. Each observation can only be classified in one way.be classified in one way.

3.3. No ordering (>) is present in the list No ordering (>) is present in the list of of categories. The order is arbitrary and categories. The order is arbitrary and

no one classification is superior to no one classification is superior to another.another.

It does not make sense to discuss the It does not make sense to discuss the mean (average) or median (midpoint) mean (average) or median (midpoint) with nominal data. with nominal data.

It cannot be summed and divided, nor It cannot be summed and divided, nor can it be ranked in order from highest can it be ranked in order from highest to lowest.to lowest.

Example: Table 3.2 from NCVS.Example: Table 3.2 from NCVS.

Table 3.2: Sex of Respondents in the National Crime Survey

Sex of Respondents Frequency Percentage

Male 524 52.1

Female 481 47.9

TOTALS 1005 100

Here we see that the majority of the Here we see that the majority of the respondents are Male (52.1%). respondents are Male (52.1%).

Although everyone knows that women Although everyone knows that women are smarter, we cannot say that the are smarter, we cannot say that the mutually exclusive categories of sex mutually exclusive categories of sex are in rank order, or can we say that are in rank order, or can we say that one sex is “average.”one sex is “average.”

LEVELS OF LEVELS OF MEASUREMENTMEASUREMENT

Ordinal levelOrdinal level: Exists when we can also : Exists when we can also detect degrees of difference between the detect degrees of difference between the categories on the scale. The values of categories on the scale. The values of the variable indicate order or ranking. the variable indicate order or ranking. EXAMPLEEXAMPLE:“Do you favor or oppose the :“Do you favor or oppose the death penalty for persons convicted of death penalty for persons convicted of murder?” Choices:(1) Favor (2) Oppose murder?” Choices:(1) Favor (2) Oppose (3) Neither (4) Don’t Know.(3) Neither (4) Don’t Know.

TransitivityTransitivity

Ordinal level measurement requires Ordinal level measurement requires transitivity.transitivity.

If A is > B and B is > C, A must be greater If A is > B and B is > C, A must be greater than C or ordinal level measurement is than C or ordinal level measurement is not present. not present.

““Favor” is > “Oppose,” “Oppose” is > Favor” is > “Oppose,” “Oppose” is > “Neither,” “Neither is > “Don’t Know,” and “Neither,” “Neither is > “Don’t Know,” and “Favor” is > “Don’t Know.”“Favor” is > “Don’t Know.”

““Do you favor the death Do you favor the death penalty for persons convicted penalty for persons convicted

of murder?”of murder?”

Favor 727 72.6%

Oppose 177 17.6%

Neither 79 7.9%

Don’tKnow

19 1.9%

LEVELS OF LEVELS OF MEASUREMENTMEASUREMENT

Interval levelInterval level: assumes that the : assumes that the difference between each item on the difference between each item on the scale have equal units (or intervals) scale have equal units (or intervals) of measurement between them. It of measurement between them. It also assumes that this unit has a also assumes that this unit has a common recognized meaning.common recognized meaning.

LEVELS OF LEVELS OF MEASUREMENTMEASUREMENT

Ratio levelRatio level: Data possessing a natural : Data possessing a natural zero point and organized into measures zero point and organized into measures for which differences are meaningful.for which differences are meaningful.

ExamplesExamples:A year is a common, constant :A year is a common, constant unit of measurement. Before birth, a unit of measurement. Before birth, a person is considered to have zero years person is considered to have zero years of age. with ratio level measurement.of age. with ratio level measurement.

For example, analysis of the age of For example, analysis of the age of the respondents to the National the respondents to the National Crime Survey revealed that the Crime Survey revealed that the mean was 45.6 years. mean was 45.6 years.

The median or midpoint was 42. The median or midpoint was 42. The mode was also 42. The mode was also 42.

We can also compare groups of We can also compare groups of respondents according to their age. respondents according to their age.

The fourteen survey respondents were 44 The fourteen survey respondents were 44 (lets call them “Group A”) and twenty (lets call them “Group A”) and twenty one respondents were 22 (Group B). one respondents were 22 (Group B).

We draw the following conclusions about We draw the following conclusions about Respondents from Groups A and B:Respondents from Groups A and B:

They have different ages (Nominal They have different ages (Nominal Measurement).Measurement).

Members of Group A are 22 years older Members of Group A are 22 years older than members of Group B (Ordinal and than members of Group B (Ordinal and Interval Measurement).Interval Measurement).

Members of Group A are twice as old as Members of Group A are twice as old as members of Group B (Ratio members of Group B (Ratio Measurement).Measurement).