SECTION 3.3Measures of Variation

A QUESTION OF TWO SAMPLES

Think of the measures of center that

we have learned about so far. What measure of center do these two

samples have in common?

Both samples have the same mean.

A NEW MEASURE

Clearly these 5 samples have many differences … but

that is not apparent if we start analyzing them with the tools we already know.

We need a new measure.

STANDARD DEVIATION OF A SAMPLE

Definition

The standard deviation of a set of sample values, denoted by s, is a measure of variation

of values about the mean. It is a type of average deviation of values from the mean

that is calculated by using the following formula.

IMPORTANT NOTES

The standard deviation is a measure of variation of all values from the mean.

The value of the standard deviation is usually positive. (It is sometimes zero, but it is never negative).

The units of the standard deviation are the same as the units of the original data values.

CAUTION: Is NOT a resistant measure of center.

APPLICATION

As of 2010, India had 1 satellite used for military and intelligence purposes, Japan has

3, and Russia has 14.

Find the range and the standard deviation for this information.

APPLICATION – SATELLITES & STANDARD DEVIATION

Step 1: Compute the mean

Step 2: Subtract the mean from each individual sample value.

Step 3: Square each of the deviations obtained from step 2.

Step 4: Add all of the squares obtained from step 3.

Step 5: Divide the total from step 4 by the number n-1.

Step 6: Find the square root of the result from step 5.

MATH SWAGG – CALCULATOR SKILLZ

SO … WHY SHOULD WE CARE?

The Range Rule of Thumb

The vast majority (such as 95%) of sample values lie within two standard deviations of the

mean.

Years to Earn Bachelor’s Degree

Listed below are the lengths of time (in years) it took for a random sample of students to earn

bachelor’s degrees (based on data from the U.S National Center for Education Statistics). Based on these results, is it usual for someone to earn

a bachelor’s degree in 12 years?

4, 4, 4, 4, 4, 4, 4.5, 4.5, 4.5, 4.5, 4.5, 4.5, 6, 6, 8, 9, 9, 13, 13, 15

ANOTHER NEW MEASURE - VARIANCE OF A SAMPLE

Definition

The variance of a set of values is a measure of variation equal to the square of the standard

deviation.

Sample variance

IMPORTANT NOTES

The sample variance is an unbiased estimator. Example: Consider an IQ test designed so

that the population variance is 225. If you repeat the process of randomly selecting 100 subjects, giving them IQ tests, and calculating the sample variance in each case, the sample variances you will obtain will tend to center around 225.

The units of the variance are NOT the same as the units of the original data values.

PRACTICE

Pg. 110 #7-9

HOMEWORK QUIZ

Ms. Pobuda find that the times (in seconds) required to complete a homework quiz have a mean of 180 seconds and a standard deviation of 30 seconds. Would it be unfair for Ms. Pobuda to set a time limit of 90 seconds for her homework quizzes? Why or why not?

REAL LIFE APPLICATION – CUSTOMER WAITING TIMES

Do you prefer single waiting lines or multiple wait lines?

REAL LIFE APPLICATION – CUSTOMER WAITING TIMES

Waiting times (in minutes) of customers at the Jefferson Valley Bank (where all

customers enter a single waiting line) and the Bank of Providence (where all customers wait in individual lines at three different teller

windows) are listed below. Determine whether there is a difference between the

two data sets. Jefferson Valley 6.

56.6 6.

76.8

7.1

7.3

7.4

7.7

7.7

7.7

Providence 4.2

5.4 5.8

6.2

6.7

7.7

7.7

8.5

9.3

10

MINI-ACTIVITY

Write your height in inches up on the side of the board. Once everyone’s height is on the board, use your calculator to calculate the standard deviation of our class’ heights.

APPLICATION Would any of the characters of Eclipse have an

“usual” height in our class?

ADDITIONAL PRACTICE

Pg. 111 # 14-16