Chapter 5: Variability and Standard (z) Scores

How do we quantify the variability of the scores in a sample?

55 60 65 70 75 80 85 90 95 100 105 110 1150

1

2

3

4

5

Ice Dancing Score

Freq

uenc

y

Method 1: range: difference between the highest and lowest scores

Ice Dancing , compulsory dance scores, Winter Olympics

111.15108.55

106.6103.33100.06

97.3896.6796.1292.7589.6285.3684.5883.8983.1280.47

80.379.3176.7374.2572.0168.8763.7359.64

Example: The range of ice dancing scores is 111.15-59.64 = 51.51 points

The range is easy to calculate, but it really only depends on two scores. So it’s not a very informative or reliable measure of variability.

Method 2: The semi-interquartile range (Q): One half of the distance between P75 and P25.

Ice Dancing , compulsory dance scores, Winter Olympics

Example: ice dancing scores

Q1 = P25Q3 = P75

213 QQQ

Percentile rank111.15 98108.55 93

106.6 89103.33 85100.06 80

97.38 7696.67 7296.12 6792.75 6389.62 5985.36 5484.58 5083.89 4683.12 4180.47 37

80.3 3379.31 2876.73 2474.25 2072.01 1568.87 1163.73 759.64 2

Score

pLpHPLpSLSHSL )(

38.7724282425)73.7631.79(73.76251

PQ

20.9772767275)67.9638.97(67.96753

PQ

91.92

38.7720.972

13

QQQ

Method 3: Variance: the mean of the squares of the deviation scores

deviation score: The difference between a score and the mean of the scores )( XX

Formula for variance of a population of scores

Formula for variance of a sample of scores

Sums of squared deviation scores 2)( XXSSX

NSS

NX XX

X

22 )(

nSS

nXX

s XX

2

2 )(

Example: find the variance of this sample of 7 numbers: 5,3,1,6,2,8,3

4X

nSS

nXX

s XX

2

2 )(

14.5736

711644911

7)1()4()2()2()3()1()1(

7)43()48()42()46()41()43()45()(

2222222

222222222

nXX

SX

Calculating variance this way can be tedious. Fortunately there’s a shortcut for calculating SSx:

Sum of squared deviations from the mean

Sum of squares

Sum squared divided by n

nX

XXXSSX

222)(

Example: from this sample of 7 numbers: 5,3,1,6,2,8,3

4X

nX

XXXSSX

222)(

3611644911)1()4()2()2()3()1()1(

)43()48()42()46()41()43()45()(2222222

22222222

XX

78428)326135( 222 X

36112148

7784148

22 n

XX

1483826135 22222222 X

Example: calculate the variance of this sample of 10 numbers:

8 6 3 7 1 7 7 8 9 10

X X2

2X

X

nSSS X

X2

2X

8 646 363 97 491 17 497 498 649 81

10 100

10

502

66

4356

435.6

66.4

6.64

n =

nX 2

nX

XXXSSX

222)(

n

XXSSX

22

standard deviation: the square root of the variance

Formula for standard deviation for a population of scores

Formula for standard deviation for a sample of scores

The standard deviation has the same units as the original scores (e.g. points, inches, etc.)

NSS

NX XX

X

2)(

nSS

nXX

S XX

2)(

Warning! Point of future confusion!

The definition of variance and standard deviation has an (or N) in the denominator.

Later when we get in to inferential statistics, we’ll start dividing by n-1:

The first definition is the true average of the squared deviance from the mean. But this number a biased estimate of the variance of the population.

Divide by ‘n’ when you just want the standard deviation of our sample (or population). Divide by ‘n-1’ when you want to estimate the standard deviation of the population.

nSS

nXX

S XX

2)(

11)( 2

nSS

nXX

s XX

NSS

NX XX

X

2)(

Example: calculate the standard deviation of this sample of 10 numbers:

8 6 3 7 1 7 7 8 9 10

X X2

2X

X

nSSS X

X2

2X

8 646 363 97 491 17 497 498 649 81

10 100

10

502

66

4356

435.6

66.4

6.64

2.58

n =

nX 2

2XX SS

n

XXSSX

22

Example: calculate the standard deviation of this sample of 20 numbers:

863717789

103234632831

6436

949

149496481

100949

1636

94

6491

X X2

2X

X

nSSS X

X2

2X

n =

nX 2

2XX SS

20

663

101

10201

510.05

152.95

7.65

2.77

n

XXSSX

22

Characteristic RangeSemi-

interquartile range

Standard deviation

Frequency of use Some Very little Almost always

Mathematical tractability

Very little Very little Great

Sampling stability Worst OK Best

Use with skewed distributions

Not so good OK Interpret with caution

Most closely related central

tendency

None Median Mean

Use with open ended

distributions

No OK No

Affected by sample size

Yes No No

Ease of calculation Easy OK OK

Fun facts about the standard deviation:

Adding a constant to each number in a sample does not change the standard deviation (or variance)

SX+b = SX

Multiplying each number in a sample by a constant multiplies the standard deviation by that same constant.

SaX = aSX

How big is a standard deviation?

For a normal (bell-shaped) distribution:

68.2% of the values fall within one standard deviation of the mean95.4% of the values fall within two standard deviations of the mean99.7% of the values fall within three standard deviations of the mean

1 standard deviation above and below the mean is where the bend of the curve switches (the ‘inflection point’)

80 90 100 110 1200

20

40

60

80

100

120

140

160

Score

Guess the mean and standard deviation

80 90 100 110 1200

20

40

60

80

100

120

140

160

Score

Mean= 99, s.d. = 8.0

Guess the mean and standard deviation

0 50 1000

10

20

30

40

50

60

Score

Guess the mean and standard deviation

0 50 1000

10

20

30

40

50

60

Score

Mean= 60, s.d. = 27.3

Guess the mean and standard deviation

-400 -300 -200 -100 0 100 2000

50

100

150

Score

Guess the mean and standard deviation

-400 -300 -200 -100 0 100 2000

50

100

150

Score

Mean= -99, s.d. = 99.7

Guess the mean and standard deviation

497 498 499 500 501 5020

50

100

150

Score

Guess the mean and standard deviation

497 498 499 500 501 5020

50

100

150

Score

Mean= 500, s.d. = 1.0

Guess the mean and standard deviation

Guess the mean and standard deviation

-4 -2 0 2 4 60

20

40

60

80

100

Score

Guess the mean and standard deviation

-4 -2 0 2 4 60

20

40

60

80

100

Score

Mean= 1, s.d. = 1.9

Standard Scores (z scores)

Sometimes it is useful to compare scores across distributions that have different means and standard deviations. A common way to do this is to convert the scores into standard deviation units, or ‘z scores’.The goal is to modify all of the scores so that the new mean is equal to zero, and the new standard deviation equal to one. To make the new mean zero, we subtract the mean from all scores. Remember this shifts the mean but doesn’t change the standard deviation.To make the new standard deviation equal to 1, we divide all scores by the standard deviation. This would normally change the mean, but since it’s zero, it doesn’t change.Here’s the formula for changing a sample of scores, X to z:

XSXXz

Example: Convert the following ten scores to z scores

X X2

2X

X

nSSS X

X2

2X

n =

nX 2

2XX SS

Step 1, calculate the mean and standard deviation:

234

12426293

723

854

52916

144176438448649

49529

642916

10

18504

328

107584

10758.40

7745.60

774.56

27.83

32.80nX

X

n

XXSSX

22

Example: Convert the following ten scores to z scores

X 2XX SS

Step 2, for each score, subtract the mean and divide by the standard deviation

234

12426293

723

854

27.83

32.80nX

X

-0.35-1.03-0.750.331.052.16

-0.93-0.35-0.890.76

-9.80-28.80-20.80

9.2029.2060.20

-25.80-9.80

-24.8021.20

XX XSXXz

Check for yourself that the mean of z is 0, and the standard deviation is 1.

Z-transforming your scores doesn’t affect the shape of the distribution.

0 50 100 1500

20

40

60

80

100

Score

Mean= 80, s.d. = 33.0

-2 -1 0 1 2 30

20

40

60

80

100

z score

Mean= 0, s.d. = 1

The standard normal distribution

The standard normal distribution is a continuous distribution.It has a mean of 0 and a standard deviation of 1The total area under the curve is equal to 1

-4 -3 -2 -1 0 1 2 3 4

Rel

ativ

e fre

quen

cy

z score

-3 -2 -1 0 1 2 3

area =0.1587

z

-3 -2 -1 0 1 2 3

area =0.3413

z

Table A (page 436) gives you the proportion of scores for given ranges in the standard normal

Column 2 Area between 0 and z

Column 3Area above z