1

The Standard Deviation as a Ruler

A student got a 67/75 on the first exam and a 64/75 on the second exam. She was disappointed that she did not score as well on the second exam.

To her surprise, the professor said she actually did better on the second exam, relative to the rest of the class.

2

The Standard Deviation as a Ruler

How can this be?Both exams exhibit variation in the

scores.However, that variation may be

different from one exam to the next.The standard deviation provides a ruler

for comparing the two exam scores.

3

Summarizing Exam ScoresExam 1

– Score: 67– Mean:

– Standard Deviation:

Exam 2– Score: 64– Mean:

– Standard Deviation:

5.59y

61.8s

1.50y

86.11s

4

Standardizing

syyz

Look at the number of standard deviations thescore is from the mean.

5

Standardized Exam ScoresExam 1

– Score: 67

Exam 2– Score: 64

87.061.8

5.5967

z

z

17.186.11

1.5064

z

z

6

Standardized Exam ScoresOn exam 1, the 67 was 0.87

standard deviations better than the mean.

On exam 2, the 64 was 1.17 standard deviations better than the mean.

7

StandardizingShifts the distribution by

subtracting off the mean.Rescales the distribution by

dividing by the standard deviation.

8

Distribution of Low Temps

5

10

15

20

Cou

nt

-10 0 10 20 30 40 50Low Temperature (o F)

9

Shifting the Distribution

5

10

15

20

Cou

nt

-40 -30 -20 -10 0 10 20Low Temperature – 32 (o F)

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ShiftingTemperature (o F)

– Median: 24.0o F– Mean: 24.4o F

– IQR: 16.0o F– Std Dev: 11.22o F

Temp – 32 (o F)– Median: –8o F– Mean: –7.6o F

– IQR: 16.0o F– Std Dev: 11.22o F

11

ShiftingWhen adding (or subtracting) a

constant:– Measures of position and center

increase (or decrease) by that constant.

– Measures of spread do not change.

12

Rescaling

5

10

15

Cou

nt

-20 -15 -10 -5 0 5 10Low Temperature (o C)

13

RescalingTemp – 32 (o F)

– Median: –8o F– Mean: –7.6o F

– IQR: 16.0o F– Std Dev: 11.22o F

Temperature (o C)– Median: –4.4o F– Mean: –4.2o F

– IQR: 8.9o F– Std Dev: 6.24o F

14

RescalingWhen multiplying (or dividing) by

a constant:– All measures of position, center and

spread are multiplied (or divided) by that constant.

15

StandardizingStandardizing does not change the

shape of the distribution.Standardizing changes the center by

making the mean 0.Standardizing changes the spread by

making the standard deviation 1.

16

Normal ModelsOur conceptualization of what the

distribution of an entire population of values would look like.

Characterized by population parameters: μ and σ.

17

40 45 50 55 60 65 70 75 80

0

10

20

30

Height

Per

cent

18

Describe the sampleShape is symmetric and mounded

in the middle.Centered at 60 inches.Spread between 45 and 75 inches.30% of the sample is between 60

and 65 inches.

19

Normal ModelsOur conceptualization of what the

distribution of an entire population of values would look like.

Characterized by a bell shaped curve with population parameters– Population mean = μ– Population standard deviation = σ.

20

40 45 50 55 60 65 70 75 80

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Height

Den

sity

Sample Data

21

40 45 50 55 60 65 70 75 80

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Height (inches)

Den

sity

Normal Model

22

Population – all items of interest.

Example: All children age 5 to 19.

Variable: Height

Sample – afew items from the population.Example: 550

children.

Normal Model

40 45 50 55 60 65 70 75 80

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Height (inches)

Den

sity

40 45 50 55 60 65 70 75 80

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Height

Den

sity

23

Normal ModelHeightCenter:

– Population mean, μ = 60 in.Spread:

– Population standard deviation, σ = 6 in.

24

68-95-99.7 RuleFor Normal Models

– 68% of the values fall within 1 standard deviation of the mean.

– 95% of the values fall within 2 standard deviations of the mean.

– 99.7% of the values fall within 3 standard deviations of the mean.

25

Normal Model - Height68% of the values fall between

60 – 6 = 54 and 60 + 6 = 66.95% of the values fall between

60 – 12 = 48 and 60 + 12 = 72.99.7% of the values fall between

60 – 18 = 42 and 60 + 18 = 78.

26

From Heights to Percentages

What percentage of heights fall above 70 inches?

Draw a picture.How far away from the mean is 70

in terms of number of standard deviations?

27

807570656055504540

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

Height (inches)

Den

sity

Normal Model

Shaded area?

28

Standardizing

67166070 .z

yz

29

Standard Normal Model Table Z: Areas under the standard

Normal curve in the back of your text.

On line:http://davidmlane.com/hyperstat/z_table.html

30

From Percentages to Heights

What height corresponds to the 75th percentile?

Draw a picture.The 75th percentile is how many

standard deviations away from the mean?

31

807570656055504540

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

Height (inches)

Den

sity

Normal Model

25%

50%

25%

32

Standard Normal Model Table Z: Areas under the standard

Normal curve in the back of your text.

On line:http://davidmlane.com/hyperstat/z_table.html

33

Reverse Standardizing

0264606706660670

..*y

y.

yz

34

Do Data Come from a Normal Model?

The histogram should be mounded in the middle and symmetric.

The data plotted on a normal probability (quantile) plot should follow a diagonal line.– The normal quantile plot is an option in

JMP: Analyze – Distribution.

35

Do Data Come from a Normal Model?

Octane ratings – 40 gallons of gasoline taken from randomly selected gas stations.

Amplifier gain – the amount (decibels) an amplifier increases the signal.

Height – 550 children age 5 to 19.

36

.01

.05

.10

.25

.50

.75

.90

.95

.99

-3

-2

-1

0

1

2

3

Nor

mal

Qua

ntile

Plo

t

2

4

6

8

Cou

nt

85 90 95Octane Rating

37

.01

.05

.10

.25

.50

.75

.90

.95

.99

-3

-2

-1

0

1

2

3

Nor

mal

Qua

ntile

Plo

t

5

10

15

20

25

Cou

nt

7.5 8 8.5 9 9.5 10 10.5 11 11.5 12Amplifier Gain (dB)

38

.01

.05

.10

.25

.50

.75

.90

.95

.99

-3

-2

-1

0

1

2

3

Nor

mal

Qua

ntile

Plo

t

50

100

150

Cou

nt

45 50 55 60 65 70 75

39

Nearly normal?Is the histogram basically

symmetric and mounded in the middle?

Do the points on the Normal Quantile plot fall close to the red diagonal (Normal model) line?