VARIABILITY:

Range

Variance

Standard Deviation

Measures of Variability

Describe the extent to which scores in a

distribution differ from each other.

Distance Between the Locations of

Scores in Three Distributions

Three Variations of the Normal

Curve

The Range, Variance, and

Standard Deviation

The Range

• The range indicates the distance between the

two most extreme scores in a distribution

Range = highest score – lowest score

Variance and Standard Deviation

• The variance and standard deviation are two

measures of variability that indicate how

much the scores are spread out around the

mean

• We use the mean as our reference point since

it is at the center of the distribution

The Sample Variance and the Sample Standard Deviation

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22 )(

Sample Variance

• The sample variance is the average of the

squared deviations of scores around the

sample mean

• Definitional formula

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N

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2

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Sample Variance

• Computational formula

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Sample Standard Deviation

• The sample standard deviation is the

square root of the sample variance

• Definitional formula

Sample Standard Deviation

• Computational formula

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N

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22 )(

The Standard Deviation

• The standard deviation indicates the “average

deviation” from the mean, the consistency in

the scores, and how far scores are spread out

around the mean

Normal Distribution and

the Standard Deviation

Normal Distribution and

the Standard Deviation

Approximately 34% of the scores in a perfect

normal distribution are between the mean and

the score that is one standard deviation from

the mean.

Standard Deviation and Range

For any roughly normal distribution, the

standard deviation should equal about one-sixth

of the range.

The Population Variance and the Population Standard

Deviation

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Population Variance

• The population variance is the true or

actual variance of the population of scores.

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Population Standard Deviation

• The population standard deviation is the

true or actual standard deviation of the

population of scores.

The Estimated Population Variance

and The Estimated Population

Standard Deviation

Estimating the Population

Variance and Standard Deviation

• The sample standard deviation is a

biased estimator of the population

standard deviation.

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• The sample variance is a biased

estimator of the population variance.

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XS

1

)( 22

N

XXsX

Estimated Population Variance

• By dividing the numerator of the sample

variance by N - 1, we have an unbiased

estimator of the population variance.

• Definitional formula

Estimated Population Variance

• Computational formula

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)( 22

2

N

N

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sX

1

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N

XXsX

Estimated Population

Standard Deviation

• By dividing the numerator of the sample

standard deviation by N - 1, we have an

unbiased estimator of the population

standard deviation.

• Definitional formula

Estimated Population

Standard Deviation

• Computational formula

1

)( 22

N

N

XX

sX

Unbiased Estimators 2

Xs2• is an unbiased estimator of

Xs

• is an unbiased estimator of

• The quantity N - 1 is called the degrees of

freedom

2

Xs2

XS XS XsUses of , , , and

2

XS

XS• Use the sample variance and the

sample standard deviation to

describe the variability of a sample. 2

Xs

Xs• Use the estimated population variance

and the estimated population standard

deviation for inferential purposes when

you need to estimate the variability in the

population.

Organizational Chart of

Descriptive and Inferential

Measures of Variability

Applying to Research

5 item list 10-item list 15-item list

3

4

2

5

5

8

9

11

7

82.

3

xS

X

41.1

6

xS

X

63.1

9

xS

X

• The standard deviation in each condition tells me about:

1. on “average” the scores differ from each other (i.e.

consistency of scores and behavior)

2. the strength of overall relationship

3. amount of error we have in prediction (Rather, the variance

is the “average error” when using the mean to predict

scores)

Proportion of Variance

Accounted For

• When describing a relationship, we evaluate

its scientific usefulness:

– How important is it?

– What does it “buy” me?

• Using a relationship helps us predict more

accurately

– but “more accurate” compared to what?

Proportion of Variance

Accounted For

• Compare our “average” prediction error

when using the relationship to the “average”

prediction error without using the

relationship

3

5

1

6

2

10

10

4

13 15

6

2

xS

X 5 item 10-item 15-item

3

5

1

6

2

10

10

4

13

Average Error = 10

6X 9X3X

Proportion of Variance

Accounted For

The proportion of variance accounted for by a relationship is:

the proportion of error in our predictions when we

use the overall mean to predict scores that is eliminated

when we use the relationship with another variable to

predict scores

i.e. the improvement that results from using a

relationship to predict scores, compared to not using

that relationship

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Example 1

• Using the following data set, find

– The range,

– The sample variance and standard deviation,

– The estimated population variance and standard deviation

71017

Example1

• The range is the largest value minus the

smallest value.

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44.218

33623406

18

18

)246(3406

2

2

XS

Example 1

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22 )(

56.144.218

18

2463406

2

XS

Example 1

1

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2

N

N

XX

sX

59.217

33623406

17

18

)246(3406

2

2

Xs

Example 1

1

)( 22

N

N

XX

sX

61.159.217

18

2463406

2

Xs

Example 1

Example 2

• For the following sample data, compute the range,

variance and standard deviation

8 8 10 7 9 6 11 9 10 7

11 11 7 9 11 10 11 8 10 7

range= 11-6=5

Variance= 2.60

Standard Deviation= 1.61

Example 3

• For the data set below, calculate the mean,

deviation, sum of squares, variance and

standard deviation by creating a table.

• 15 12 13 15 16 17 13 16 11 18

Example 3 Solution

score mean deviation sum of squares variance standard deviation

11 14,60 -3,60 12,96

12 14,60 -2,60 6,76

13 14,60 -1,60 2,56

13 14,60 -1,60 2,56

15 14,60 0,40 0,16

15 14,60 0,40 0,16

16 14,60 1,40 1,96

16 14,60 1,40 1,96

17 14,60 2,40 5,76

18 14,60 3,40 11,56

46,40 4,64 2,15

Example 4

• For the data set below, calculate the mean,

deviation, sum of squares, variance and

standard deviation by creating a table.

• 1 3 2 2 2 4 3 3 4 1

Example 4 Solution

score mean deviation sum of squares variance standard deviation

1 2,50 -1,50 2,25

1 2,50 -1,50 2,25

2 2,50 -0,50 0,25

2 2,50 -0,50 0,25

2 2,50 -0,50 0,25

3 2,50 0,50 0,25

3 2,50 0,50 0,25

3 2,50 0,50 0,25

4 2,50 1,50 2,25

4 2,50 1,50 2,25

10,50 1,05 1,02

Example 5

• For the data set below, calculate the mean,

deviation, sum of squares, variance and

standard deviation by creating a table.

• 1 3 30 12 15 20 5 13 2 4

Example 5 Solution score mean deviation sum of squares variance standard deviation

1 10,50 -9,50 90,25

2 10,50 -8,50 72,25

3 10,50 -7,50 56,25

4 10,50 -6,50 42,25

5 10,50 -5,50 30,25

12 10,50 1,50 2,25

13 10,50 2,50 6,25

15 10,50 4,50 20,25

20 10,50 9,50 90,25

30 10,50 19,50 380,25

790,50 79,05 8,89