Comparing the Mode, Median, and MeanThree factors in choosing a measure of central tendency

1. Level of measurement– Nominal– Ordinal– Interval/Ratio

2. Shape or form of the distribution of data– Kurtosis– Skewness– Normality

3. Research Objective

Chapter 4Measures of Variability

Chapter 4 - Introduction• Measures of central tendency alone portray an incomplete

picture• Measures of variability allows us to understand how scores are

distributed

• Three types– Range– Variance– Standard Deviation

Measures of Variability and Dispersion

• For example, two tests were given with the following results:– Test 1: 0 80 85 90 95 100– Test 2: 75 75 75 75 75 75

The Range• Simplest and quickest measure of distribution

dispersion• Range = Difference between highest and

lowest scores in a distribution• In equation form:

• Provides a crude measure of variation• Outliers severely affect the range

LHR R = rangeH = highest score in a distributionL = lowest score in a distribution

1, 2, 2, 4, 5, 5, 8, 9, 9, 10, 10, 10

The Inter-Quartile Range• Inter-quartile range manages effects of extreme

outliers• In equation form:

• The larger the size of IQR, the greater the variability

13 QQIQR IQR = inter-quartile rangeQ1 = score at the 1st quartile, 25% below, 75% aboveQ3 = score at the 3rd quartile, 75% below, 25% above

Illustration: Range and IQR

• Probation Officer A– 18, 18, 19, 19, 20, 20, 22, 23

• Probation Officer B– 18, 18, 19, 19, 20, 22, 43

The Variance and the Standard Deviation

• Deviation: distance of any given raw score from its mean

• Deviation• Most appropriate for interval/ratio data• Used to divide and discuss a normal curve• Need a measure of variability that takes into

account every score

)( XX

The Raw-Score Formula for Variance and Standard Deviation

• In equation form:– Variance

– Standard deviation

22

2 XN

Xs

22

XN

Xs

= sum of the squared raw scores 2X2

X = mean squared

N = total number of scores

Illustration: Using Raw Scores

X986421

Illustration: Using Raw Scores

X986421

Step 1: Square each raw score and sum both columns

Step 2: Obtain the mean and square it

X X2

9 81

8 64

6 36

4 16

2 4

1 1

ΣX = 30 ΣX2 = 202

1. 30 / 6 = 52. 52 = 25

Step 3: Insert results from Step 1 and 2 into the formulas

22

XN

Xs N = 6

ΣX2 = 202Xbar = 25

Illustration: Variance and Standard Deviation

• On a 20 item measure of self-esteem (higher scores reflect greater self-esteem), five teenagers scored as follows: 16, 5, 18, 9, 11.

1. Calculate the range2. Calculate the variance3. Calculate the standard deviation

The Meaning of the Standard Deviation

• Standard deviation converts the variance to units we can understand.

• But, how do we interpret?– Standard deviation represents the average variability in

a distribution.– It is the average of deviations from the mean.– The greater the variability – the larger the standard

deviation– Allows for comparison between a given raw score in a

set against a standardized measure

The Normal Curve

Variance and Standard Deviation of a Frequency Distribution

# of Class f

6 1

5 7

4 9

3 3

N = 20

• The table on the right is a simple frequency distribution of the number of courses taken by each full time student in a particular class.

Variance and Standard Deviation of a Grouped Distribution

Class Interval f

30-32 2

27-29 3

24-26 5

21-23 6

18-20 9

N = 25

• The table on the right is a grouped frequency distribution of 25 individuals and their ages when first married.

Step 1:Find each midpoint and multiply it (m) by the frequency (f) in the class interval to obtain the fm products, and then sum the fm column.

Class Interval Midpoint(m)

f fm

30-32 31 2 (31*2) = 62

27-29 28 3 (28* 3) = 84

24-26 25 5 125

21-23 22 6 132

18-20 19 9 171

---- ----- 25 Σfm = 574

Step 2:Square each midpoint and multiply the frequency of the class interval to obtain the f(m2) products, and then sum the f(m2) columns.

Class Interval Midpoint(m)

f fm f(m2)

30-32 31 2 62 1,922

27-29 28 3 84 2,352

24-26 25 5 125 3,125

21-23 22 6 132 2,904

18-20 19 9 171 3,249

---- ----- 25 Σfm = 574 Σf(m2) = 13,552

Step 3:Obtain the mean and square it

• Xm = Σ fm N

• Xm = 574 / 25 = 22.96

• Xm2 = 22.962

• Xm2 = 527.16

• s2 = (13,552/25) – 527.16• s2 = 542.08-527.16• s2 = √14.92• s = 3.86

Step 4:Calculate the variance using the results from the previous steps

s = Σfm2 N

- Xm2

From the table:N = 25Σfm = 574Σf(m2) = 13,552

Selecting the Most Appropriate Measure of Dispersion

• It is harder to determine the most appropriate measure of dispersion than it is to determine the most appropriate measure of central tendency

Not as “tied” to level of measurement• Range can always be used– Regardless of data level or distribution form– Limited in information

• Variance and standard deviation are good for interval and some ordinal data

Comparing Measures of Variability

• Range is simple to calculate but not reliable• The standard deviation can never be greater

than the range.• Standard deviation: – reflects the effect of all scores– requires data at the interval level

Summary

• Measures of variability allow distributions of data to be described more completely

• No widely accepted measures of variability for categorical data

• Variance and standard deviation can be used to measure deviation or dispersion within a variable