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VARIABILITY: Range Variance Standard Deviation
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VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

May 22, 2019

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Page 1: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

VARIABILITY:

Range

Variance

Standard Deviation

Page 2: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

Measures of Variability

Describe the extent to which scores in a

distribution differ from each other.

Page 3: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

Distance Between the Locations of

Scores in Three Distributions

Page 4: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

Three Variations of the Normal

Curve

Page 5: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

The Range, Variance, and

Standard Deviation

Page 6: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

The Range

• The range indicates the distance between the

two most extreme scores in a distribution

Range = highest score – lowest score

Page 7: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 8: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

The Sample Variance and the Sample Standard Deviation

Page 9: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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XXSX

22 )(

Sample Variance

• The sample variance is the average of the

squared deviations of scores around the

sample mean

• Definitional formula

Page 10: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

N

N

XX

SX

22

2

)(

Sample Variance

• Computational formula

Page 11: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

N

XXSX

2)(

Sample Standard Deviation

• The sample standard deviation is the

square root of the sample variance

• Definitional formula

Page 12: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

Sample Standard Deviation

• Computational formula

N

N

XX

SX

22 )(

Page 13: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 14: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

Normal Distribution and

the Standard Deviation

Page 15: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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.

Page 16: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

Standard Deviation and Range

For any roughly normal distribution, the

standard deviation should equal about one-sixth

of the range.

Page 17: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

The Population Variance and the Population Standard

Deviation

Page 18: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

N

XX

22 )(

Population Variance

• The population variance is the true or

actual variance of the population of scores.

Page 19: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

N

XX

2)(

Population Standard Deviation

• The population standard deviation is the

true or actual standard deviation of the

population of scores.

Page 20: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

The Estimated Population Variance

and The Estimated Population

Standard Deviation

Page 21: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

Estimating the Population

Variance and Standard Deviation

• The sample standard deviation is a

biased estimator of the population

standard deviation.

)( XS

• The sample variance is a biased

estimator of the population variance.

)( 2

XS

Page 22: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 23: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

Estimated Population Variance

• Computational formula

1

)( 22

2

N

N

XX

sX

Page 24: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

1

)( 2

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

Page 25: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

Estimated Population

Standard Deviation

• Computational formula

1

)( 22

N

N

XX

sX

Page 26: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 27: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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.

Page 28: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

Organizational Chart of

Descriptive and Inferential

Measures of Variability

Page 29: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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)

Page 30: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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?

Page 31: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 32: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 33: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 34: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

71017

Example1

• The range is the largest value minus the

smallest value.

Page 35: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

N

N

XX

SX

22

2

)(

44.218

33623406

18

18

)246(3406

2

2

XS

Example 1

Page 36: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

N

N

XX

SX

22 )(

56.144.218

18

2463406

2

XS

Example 1

Page 37: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

1

)( 22

2

N

N

XX

sX

59.217

33623406

17

18

)246(3406

2

2

Xs

Example 1

Page 38: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

1

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N

N

XX

sX

61.159.217

18

2463406

2

Xs

Example 1

Page 39: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 40: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 41: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 42: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 43: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 44: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 45: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to

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

Page 46: VARIABILITY: Range Variance Standard Deviationpsy116.cankaya.edu.tr/uploads/files/Lectures 8-9 (Variability).pdfS X •Use the sample variance and the sample standard deviation to