Math Tech IIII, Nov 11

Measures of Variation III – Chebychev’s

Theorem and Understanding Measures of

Variation

Book Sections: 2.4 Essential Questions: How do I compute and use statistical values?

What do I do in variation when the data is skewed? How do I apply

measures of variation?

Standards: DA-4.5, DA-4.6, DA-4.7, DA-4.9, DA-4.10, S.ID.1, .2, .3, .4

Empirical Rule Words and Graph Together

• 68% of the data lies within

one standard deviation of the

mean

• 95% of the data lies between

two standard deviations of the

mean

• 99.7% of the data lies

between three standard

deviations of the mean

Example 1

• A symmetric data set has a mean of 50 and a

standard deviation of 10. What percent of the

data is between 40 and 60?

Example 2

• A symmetric data set has a mean of 75 and a

standard deviation of 15. What is the range of

the middle 95% of the data?

Chebychev’s Theorem Words and Graph Together

• The portion of ANY data

set lying within k standard

deviations (k > 1) of the

mean is at least:

• k = 2: In any data set, at

least 1 – 1/22 = ¾ or 75% of

data are within 2 standard

deviations.

• k = 3: In any data set, at

lease 1 – 1/32 = 8/9 or 88.9%

of the data lie within 3

standard deviations

2

11

k

Example

• The age distributions for Alaska and Florida are shown

above. Decide which is which. Apply Chebychev’s theorem

to both and draw a conclusion.

x = 31.6

s = 19.5

x = 39.2

s = 24.8

Example

x = 31.6

s = 19.5

x = 39.2

s = 24.8

One More Look at the Skew

Skewness is caused by a significant difference between mean

and median.

Mean vs Median

• The median divides a data set in half resulting in two

equal parts. It can be a data value, but is not always.

It is not greatly affected by outliers. The median is

used to find data quartiles and outliers.

• The mean is a unique value computed from all the

data values. It is not used in set division and is not

usually a data value. It is affected by outliers. The

mean is used to compute other statistics, such as

variance and standard deviation.

Variance and Standard Deviation

• The standard deviation is the square root of the

variance.

• The Variance and standard deviation are determined

by the spread of the data. If they are large, the data is

dispersed, if they are small, the data is compact.

• Variance and standard deviation are used to

determine the consistency of a variable. In

manufacturing, the variance of parts must be within a

certain tolerance or parts will not fit together.

Variability – The Concept

• Variability is how spread out a set of data is.

• In comparing two data sets to see which one is more

variable – compute standard deviations. The one with

the largest s is more spread out and is said to be more

variable.

• You can sometimes see variability in a data set. You

can always compute and compare s’s.

Example

• Two brands of paint are tested for durability in

fading with the following results in months. Each has

a mean of 35 months. Which brand is more

consistent?

Brand A Brand B

10 35

60 45

50 30

30 35

40 40

20 25

Examples

The average daily high temps for January for 10 selected cities is:

50, 37, 29, 54, 30, 61, 47, 38, 34, 61

And their normal monthly precipitation for January is:

4.8, 2.6, 1.5, 1.8, 1.8, 3.3, 5.1, 1.1, 1.8, 2.5

Which set is more variable?

Variability by Sight

Which data set is more variable:

0 4

1 5 7

2 3 3 5 9

3 1 3 7 9

4 0 3 6 8 9

5 1 5 7 8

6 0 3 6 8

7 0 7

8 2

Key 1|2 = 12 1 8

2 3

3 1 7

4 1 2 5

5 1 1 3 4 5 5 5 6 7 7 7 8 9

6 4 4 8

7 3 7

8 0

9 5

Variability by Sight

Which data set is more variable:

Class work: Classwork Handout 1-10

Homework: None