MEASURES OF VARIATIONSECTION 3.2

FOCUS POINTS

Find the range, variance, and standard deviation.

Compute the coefficient of variation from raw data. Explain why it is important.

Apply Chebyshev’s theorem to raw data. Explain what a Chebyshev interval tells us.

WHERE WE’VE BEEN…..AVERAGES

Averages summarize data with just one number.

By itself, an average may not be very meaningful.

Need a cross-reference

EXAMPLE: RANGE

A large bakery regularly orders cartons of Maine blueberries. The average weight of the cartons is supposed to be 22 ounces. Random samples of cartons from two suppliers were weighed. The weights in ounces of the cartons were

Supplier I: 17 22 22 22 27

Supplier II: 17 19 20 27 27

A. Compute the range of carton weights from each supplier.

B. Compute the mean weight of cartons from each supplier.

VARIANCE AND STANDARD DEVIATION

Variance

Standard Deviation

HOW TO COMPUTE THE SAMPLE VARIANCE AND SAMPLE DEVIATION

Quantity Description

x The variable x represents a ________________ or outcome.

MEAN This is the _______________________________, or what you “expect” to happen the next time you conduct the statistical experiment.

n is the ____________________________.

This is the ____________________ between what happened and what you expected to happen. This represents a “___________________” away from what you ‘expect’ and is a measure of ___________________.

This expression is called the _______________________________. The _____________ quantity is squared to make it nonnegative.

SUM OF SQUARES

Defining Formula

Computation Formula

Both formulas give the same result.

SAMPLE VARIANCE……….

Defining Formula

Computation Formula

SAMPLE STANDARD DEVIATION,

Defining Formula

Computation Formula

is a measure of deviation or __________

EXAMPLE: STANDARD DEVIATION (DEFINING FORMULA)

Big Blossom Greenhouse was commissioned to develop an extra large rose for the Rose Bowl Parade. A random sample of blossoms from Hybrid A bushes yielded the following diameters (in inches) for mature peak blooms.

2 3 3 8 1010

Use the defining formula to find the sample variance and standard deviation.

EXAMPLE – DEFINING FORMULA (CON’T)

Column I Column II Column IIn = ________

Step 1(fill in col 1)

Step 2:Find the sample mean:

Step 3(Sub. mean from col 1 then fill in col 2) Step 4

(square each value in col 2 and fill in col 3

Step 5: Get the sample variance () – divide the sum of column 3 by ().

Step 6: Get the sample standard deviation (s) – take the square root of the variance ().

EXAMPLE – DEFINING FORMULA (CON’T)

EXAMPLE – COMPUTATION FORMULA

Big Blossom Greenhouse gathered another random sample of mature peak blooms from Hybrid B. The six blossoms had the following widths (in inches):

5 5 5 6 7 8

Column 1 Column 2

5

5

5

6

7

8

n = _______ mean:

= _______

𝑠2=¿

𝑠=¿

SUMMARY OF WHAT THE GREENHOUSE FOUND…

Hybrid A: Mean = ___________inches; standard deviation = __________inches

Hybrid B: Mean = ___________inches; standard deviation = __________inches

STANDARD DEVIATION

The standard deviation, s, of a set of sample values is a measure of variation of values about the mean.

The standard deviation will always be a __________________ value or _______________, never _______________.

Generally, standard deviation is the most important and useful ______________________________.

The value of the standard deviation s can ________________________ dramatically with the inclusion of one or more _______________________.

The units of the standard deviation s are the same as the units of the ________________________.

Population standard deviation is denoted by .

VARIANCE

The variance of a set of values is a measure of variation equal to the __________________ of the standard deviation.

Disadvantage that makes it more difficult to interpret:

The sample variance is denoted by _________.

The population variance is denoted by ________.

PROCEDURE – HOW TO FIND STANDARD DEVIATION USING THE DEFINING FORMULA

STEP 1: Compute the mean

STEP 2: Subtract the mean from each individual value to get a list of deviations of the form

STEP 3: Square each of the differences obtained from Step 2. This produces numbers of

the form

STEP 4: Add all of the squares obtained from Step 3. This is the value of .

STEP 5: Divide the total from STEP 4 by the number , which is 1 less than the total number

of values present.

STEP 6: Find the square root of the result of STEP 5.

USE THE DEFINING FORMULA TO FIND THE STANDARD DEVIATION OF THE WAITING TIMES AT A BANK. THOSE TIMES (IN MINUTES) ARE 1 3 14.

Calculating Standard Deviation

s =

The computing formula requires that we first find values for _______, ___________, and __________.

USE THE COMPUTING FORMULA TO FIND THE STANDARD DEVIATION OF THE WAITING TIMES AT A BANK. THOSE TIMES (IN MINUTES) ARE 1 3 14.

NOTATION

s = sample standard deviation

= sample variance

population standard deviation

= population variance

Note: professional journals and reports often use SD for standard deviation and VAR for variance.

A#3.21 DUE MONDAY, OCTOBER 25

Pages 96-97 Numbers 1, 2, 6, 7 plus worksheet problems 5-8

POPULATION VS. SAMPLE

Most of the time we will use data from a sample.

However, sometimes we will have data from the entire population. We can calculate the following:

Population mean =

Population variance =

Population standard deviation =

N = number of data values in the population.

x = the individual data values of the population.

CALCULATOR NOTE – STANDARD DEVIATION

Most tech programs (TI calculators, Excel, etc.) display on the SAMPLE standard deviation, s.

Use the following formula to calculate the population,

The mean given in the display can be interpreted as the sample mean, , or the population mean, , as appropriate.

ROUND-OFF RULE

Use this rule for rounding final results:

Carry one more decimal place than is present in the original set of values.

Round only the FINAL answer, never in the middle of the calculation. If it becomes absolutely necessary to round in the middle, carry at least twice as many decimal

places as will be used in the final answer.

COEFFICIENT OF VARIATION

Disadvantage of standard deviation as a comparative measure of variation = units of measurement. Means it is difficult to use the SD to compare measurements from different populations.

Coefficient of Variation: ______________________________________________________.

Sample SD:

Population SD:

See Example 7 on page 93

PRACTICE – COEFFICIENT OF VARIATION

Cabela’s in Sidney, Nebraska, is a very large outfitter that carries a broad selection of fishing tackle. It markets its products natiowide through a catalog service. A random sample of 10 spinners from Cabela’s extensive catalog gave the following prices (in dollars):

1.69 1.49 3.09 1.79 1.39 2.89 1.49 1.39 1.49 1.99

a. Use the calculator to find the sample mean and sample SD.

b. Compute the CV for the spinner prices at Cabela’s.

c. Compare the mean, SD, and CV for the spinner prices at the Grand Mesa Trading Post (Ex. 7) and Cabela’s. Comment on the differences.

A#3.22 DUE TUESDAY, OCT 29

A#3.22Pages 96-97 [#3-5, 8, 9, 11]