An article on peanut butter reported the following scores (quality ratings on a scale of 0 to 100) for various brands. Construct a comparative stem-and-leaf plot and compare the graphs.

Creamy: 56 44 62 36 39 53 5065 45 40 56 68 41

30 40 50 56 30 22

Crunchy: 62 53 75 42 47 40 3462 52 50 34 42 36

75 80 47 56 62

Creamy: 56 44 62 36 39 53 5065 45 40 56 68 41 3040 50 56 30 22

Crunchy: 62 53 75 42 47 40 3462 52 50 34 42 36 7580 47 56 62

Center: The center of the creamy is roughly 45 whereas the center for crunchy is higher at 51.Shape: Both are unimodal but crunchy is skewed to the right while creamy is more symmetric.Spread: The range for creamy and crunchy are equal at. There doesn’t seem to be any gaps in the distribution.

Variation

Which Brand of Paint is better? Why?

Brand A

10

60

50

30

40

20

Brand B

35

45

30

35

40

25

Standard Deviation

It’s a measure of the typical or average deviation (difference) from the mean.

Variance

This is the average of the squared distance from the mean.

Which Brand of Paint is better? Why?

Brand A

10

60

50

30

40

20

Brand B

35

45

30

35

40

25

Does the Average Help?

Paint A: Avg = 210/6 = 35 months

Paint B: Avg = 210/6 = 35 months

They both last 35 months before fading. No help in deciding which to buy.

Consider the Spread

Paint A: Spread = 60 – 10 = 50 months

Paint B: Spread = 45 – 25 = 20 months

Paint B has a smaller variance which means that it performs more consistently. Choose paint B.

Formula for Population

Variance =

Standard Deviation =

Formula for Sample

Variance =

Standard Deviation =

Formulas for Variance and St. Deviation

Population Sample

22 1xx

Nx

2)(1

xxNx

Variance

Standard Deviation

22

1

1 xx

nsx

2)(1

1xx

nsx

Variance

Standard Deviation

Standard Deviation

A more powerful approach to determining how much individual data values vary.

This is a measure of the average distance of the observations from their mean.

Like the mean, the standard deviation is appropriate only for symmetric data!

The use of squared deviations makes the standard deviation even more sensitive than the mean to outliers!

Standard Deviation

One way to think about spread is to examine how far each data value is from the mean.

This difference is called a deviation.

We could just average the deviations, but the positive and negative differences always cancel each other out! So, the average deviation is always 0 not very helpful!

Finding Variance

To keep them from canceling out, we square each deviation.

Squaring always gives a positive value, so the sum will not be zero!

Squaring also emphasizes larger differences – a feature that turns out to be good and bad.

When we add up these squared deviations and find their average (almost), we call the result the variance.

Finding Standard Deviation

This is the average of the squared distance from the mean.

Variance will play an important role later – but it has a problem as a measure of spread.

Whatever the units of the original data are, the variance is in squared units – we want measures of spread to have the same units as the data, so to get back to the original units, we take the square root of .

The result is, s, is the standard deviation.

Let’s look at the data again on the number of pets owned by a group of 9 children.

987544431 Recall that the mean was 5 pets.

Let’s take a graphical look at the “deviations” from the mean:

Let’s Find the Standard Deviation and Variance of the Data Set of Pets 987544431

Petsx

Deviations Squared Deviations

1

3

4

4

4

5

7

8

9

Sum = Sum = 16

5Mean

1 – 5 = -4

3 – 5 = -2

4 – 5 = -1

4 – 5 = -1

4 – 5 = -1

5 – 5 = 0 7 – 5 = 2

8 – 5 = 3

9 – 5 = 4

164 2

42 2

11 2

11 2 11 2

422

932

1642

0

Find Variance:

5.68

5252

8

11

1

2

22

s

xxn

s

This is the “average” squared deviation.

22

1

1 xx

nsx

Find the Standard Deviation:

This 2.55 is roughly the average distance of the values in the data set from the mean.

2)(1

1 xx

nsx

55.25.62 ss

Find the Standard Deviation and Variance

13191822201314

Values Deviations Squared Deviations

14

13

20

22

18

19

13

65.3

133.132

s

s

Homework

Worksheet