Measures of Dispersion

Variance and Standard Deviation

Basic Assumptions about Distributions

• We should be able to plot the number of times a specific value occurs on a graph using a line chart or histogram (interval/ratio data)

• Some distributions will be normal or bell-shaped.• Some distributions will be bi-modal or will have data

points distributed irregularly.• Some distributions will be skewed to the right or skewed

to the left. • Theoretically, samples taken from one population, should

over time, approximate a normal distribution. • We should have a normal distribution if we are to use

inferential statistics.

Other reasons to use Measures of Dispersion

• To see if variables taken from two or more samples are similar to one another.

• To see if a variable taken from a sample is similar to the same variable taken from a population – in other words is our sample representative of people in the population at least on that one variable.

Variation in Two SamplesSample 1 Sample 2

1 2

2 3

3 3

4 5

4 7

5 9

6 9

7 10

Mo = 4 Mo, 3, 9

Mdn = 4 Mdn = 6

Mean = 4 Mean = 6

VAR00001

7.06.05.04.03.02.01.0

2.5

2.0

1.5

1.0

.5

0.0

Std. Dev = 2.00

Mean = 4.0

N = 8.00

Sample 2

VAR00001

1097532

Count

2.2

2.0

1.8

1.6

1.4

1.2

1.0

.8

Normal Distributions are Bell-shaped and have the same

number of measures on either side of the mean.

Note: According to Montcalm & Royse only unimodal distributions can be normal distributions.

Normal Distributions

• 50% of all scores are on either side of the mean. • The distribution is symmetrical – same number

of scores fall above and below the mean.• The mean is the midpoint of the distribution.• Mean = median = mode• The entire area under the bell-shaped curve =

100%.

A standard deviation is:

• The degree to which each of the scores in a distribution vary from the mean. (x – mean)

• Calculated by squaring the deviation of each score from the mean.

• Based on first calculating a statistic called the variance.

Formulas are:

• Variance = Sum of each deviation squared divided by (n -1) where n is the number of values in the distribution.

• Standard Deviation = the square root of the sum of squares divided by (n – 1).

Using Sample 1 as an example

1 (1-4) = -3 9  Mean = 4  

2 (2-4) = -2 4    

3 (3-4) = -1 1    

4 (4-4) = 0 0 Variance S.D.

4 (4-4) = 0 0 28/(8-1) Sq Root 4

5 (5-4) = 1 1 4 2

6(6 - 4) =

2 4    

7(7 = 4) =

3 9    

Total 0 28    

Another variance/SD example

1 -5.00 25.00  Mean = 6  

2 -4.00 16.00    

4 -2.00 4.00    

8 2.00 4.00    

10 4.00 16.00Variance =

90/(6-1)SD = sq root of

18

11 5.00 25.00 18.00 4.24

Total 0.00 90.00    

Other Important Terms in This Chapter

• Mean squares – the average of squared deviations from the mean in a set of numbers. (Same as variance)

• Interquartile range – points in a set of numbers that occur between 75% of the scores and 25% of the scores – that is, where the middle 50% of all scores lie (use cumulative percentages)

• Box plot – gives graphic information about minimum, maximum, and quartile scores in a distribution.

Box Plot

258216N =

Gender

MaleFemale

Curr

ent

Sala

ry

160000

140000

120000

100000

80000

60000

40000

20000

0

4314541063410344634318

32

29

2421342774131688072240468348371

Interquartile RangeTest Scores

Frequency Percent Cumulative

Percent

100 3 25% 100%

90 3 25% 75%

80 3 25% 50%

70 1 8.3% 25%

60 2 16.7% 16.7%

Total 12 100.0%

This information is important to our discussion of normal distributions

Central Limit Theorem (we will discuss this in two weeks) specifies

that: • 50% of all scores in a normal distribution are on

either side of the mean. • 68.25% of all scores are one standard deviation

from the mean.• 95.44% of all scores are two standard deviations

from the mean.• 99.74% of all scores in a normal distribution are

within 3 standard deviations of the mean.

Therefore, we will be able to

• Predict what scores are contained within one, two, or three standard deviations from the mean in a normal distribution.

• Compare the distribution of scores in samples.

• Compare the distribution of scores from populations to samples.

To calculate measures of central tendency and dispersion in SPSS

• Select descriptive statistics

• Select descriptives

• Select your variables

• Select options (mean, sd, etc.)

SPSS output

Descriptive Statistics

474 13 8 21 13.49 2.885 8.322474

Educational Level (years)Valid N (listwise)

N Range Minimum Maximum MeanStd.

Deviation Variance