Download Measures of variation

MEASURES OF VARIATION
SECTION 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
Both formulas give the
same result.
 Computation Formula
SAMPLE VARIANCE……….𝑠 2
 Defining Formula
 Computation Formula
SAMPLE STANDARD DEVIATION, 𝑠
 Defining Formula
𝑠 is a measure
of deviation or
__________
 Computation Formula
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
10
10
Use the defining formula to find the sample variance and standard deviation.
EXAMPLE – DEFINING FORMULA (CON’T)
n = ________
Column I
Column II
Column II
Step 2:
Find the sample mean:
𝑥=
Step 1
(fill in col 1)
𝑥−𝑥
Step 3
(Sub. mean from col 1 then fill in col 2)
2
=
Step 4
(square each value in col 2 and fill in col 3
EXAMPLE – DEFINING FORMULA (CON’T)
 Step 5: Get the sample variance (𝑠 2 ) – divide the sum of column 3 by (𝑛 − 1).
 Step 6: Get the sample standard deviation (s) – take the square root of the variance (𝑠 2 ).
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):
Column 1
5
5
n = _______
𝑛 − 1 = _______
5
6
mean:
7
8
Column 2
5
5
5
6
7
𝑠2 =
8
𝑥=
𝑠=
𝑥2 =
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 𝜎.
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 𝑥 −
 STEP 3: Square each of the differences obtained from Step 2. This produces numbers of the form 𝑥 −
 STEP 4: Add all of the squares obtained from Step 3. This is the value of
𝑥 −
2.
 STEP 5: Divide the total from STEP 4 by the number 𝑛 − 1 , which is 1 less than the total number of values
present.
 STEP 6: Find the square root of the result of STEP 5.
2
.
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
𝑥
𝑥 −
𝑥 −
s=
2
USE THE COMPUTING FORMULA TO FIND THE STANDARD DEVIATION OF
THE WAITING TIMES AT A BANK. THOSE TIMES (IN MINUTES) ARE 1 3 14.
The computing formula requires that we first find values for _______, ___________, and __________.
NOTATION
s = sample standard deviation
𝑠 2 = sample variance
𝜎 = population standard deviation
𝜎 2 = 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 = 𝜇 =
𝑥
𝑁
2
 Population variance = 𝜎 =
(𝑥−𝜇)2
𝑁
 Population standard deviation = 𝜎 =
(𝑥−𝜇)
𝑁
N = number of data values in the population.
x = the individual data values of the population.
( 2)2
2
=
𝑥2− 𝑁
𝑁
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, 𝜎:
𝑛−1
𝜎=𝑠
𝑛
 The mean given in the display can be interpreted as the sample mean,
appropriate.
, or the population mean, 𝜇, as
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:
𝐶𝑉 =
𝐶𝑉 =
𝑠
𝜎
𝜇
∗ 100
∗ 100
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#3.22
 Pages 96-97 [#3-5, 8, 9, 11]