Chapter 4Measures of Variability

Measures of Variability and DispersionTwo 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

The Raw-Score Formula for Variance and Standard Deviation

• In equation form:– Variance

– Standard deviation

22

2 XNX

s

22

XNX

s

= sum of the squared raw scores 2X2

X = mean squaredN = total number of scores

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

9 818 646 364 162 41 1ΣX = 30 ΣX2 = 202

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

22

XNX

s N = 6ΣX2 = 202 = 25

On your own: Measures of Variability

• On a 20 item measure of self-esteem (higher scores reflect greater self-esteem), five teenagers scored as follows: 16, 5, 18, 9, 11, 13, 17, 10, 11, 14.

Calculate the..1. Range2. IQR3. Variance4. Standard deviation

The Meaning of the Standard Deviation• Standard deviation converts the variance to units

we can understand. But, how do we interpret it?

Measuring the Base Line in Units of Standard Deviation when the SD is 5 and is 80

End Day 1

Variance and Standard Deviation of a Frequency Distribution

# of Classes f

6 1

5 7

4 9

3 3

• 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

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

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