1. Section 3.2 Measures of Spread

2. Objectives Compute the range of a data set
Compute the variance of a population and a sample
Compute the standard deviation of a population and a sample
Approximate the standard deviation using grouped data
Use the Empirical Rule to summarize data that are unimodal and approximately symmetric
Use Chebyshev’s Inequality to describe a data set
Compute the coefficient of variation

3. Objective 1
Compute the range of a data set

4. The Range
The range of a data set is a measure of spread. That is, it measure how spread out the data are. The range of a data set is the difference between the largest and the smallest value. Range = Largest Value – Smallest Value

5. Source: National Weather Service
Example
The following table presents the average monthly temperature, in degrees Fahrenheit, for the cities of San Francisco and St. Louis. Compute the range for each city.
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
San Francisco 51 54 55 56 58 60 61 63 62 52
St. Louis 30 35 44 57 66 75 79 78 70 59 45
Source: National Weather Service

6. Source: National Weather Service
Solution
The largest value for San Francisco is 63 and the smallest is 51. The range for San Francisco is 63 – 51 = 12. The largest value for St. Louis is 79 and the smallest is 30. The range for St. Louis is 79 – 30 = 49.
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
San Francisco 51 54 55 56 58 60 61 63 62 52
St. Louis 30 35 44 57 66 75 79 78 70 59 45
Source: National Weather Service

7. The Range is Not Used in Practice
Although the range is easy to compute, it is not often used in practice. The reason is that the range involves only two values from the data set; the largest and smallest. The measures of spread that are most often used are the variance and the standard deviation, which use every value in the data set.

8. Objective 2
Compute the variance of a population and a sample

9. Variance
When a data set has a small amount of spread, like the San Francisco temperatures, most of the values will be close to the mean. When a data set has a larger amount of spread, more of the data values will be far from the mean. The variance is a measure of how far the values in a data set are from the mean, on the average. The variance is computed slightly differently for populations and samples. The population variance is presented first.

10. Definition: Population Variance
Let x1, x2, x3, … xN denote the values in a population of size N. Let μ denote the population mean. The population variance, denoted by σ2 , is

11. Procedure for Computing the Population Variance
Following is the procedure for computing the population variance of a data set: Step 1: Compute the population mean μ. Step 2: For each population value xi compute xi – μ. This is called the deviation for the value xi. Step 3: Square the deviations to obtain the quantity (xi – μ)2. Step 4: Sum the squared deviations to obtain the quantity Σ(xi – μ)2. Step 5: Divide the sum obtained in Step 4 by the population size N to obtain the population variance σ2.

12. Example
Compute the population variance for the San Francisco temperatures. Solution: Step 1: Compute the population mean μ. Step 2: For each population value xi compute xi – μ. These values are shown in the second row below.
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
San Francisco 51 54 55 56 58 60 61 63 62 52 Xi 51 54 55 56 58 60 61 63 62 52
xi – μ
–6.5
–3.5
–2.5
–1.5
0.5
2.5
3.5
5.5
4.5
-5.5

13. Solution
Step 3: Square the deviations to obtain the quantity (xi – μ)2. These values are shown in the third row. Step 4: Sum the squared deviations to obtain the quantity Σ(xi – μ)2. Step 5: Divide the sum obtained in Step 4 by the population size N to obtain the population variance σ2. xi 51 54 55 56 58 60 61 63 62 52
xi – μ
–6.5
–3.5
–2.5
–1.5
0.5
2.5
3.5
5.5
4.5
-5.5
(xi – μ)2
42.25
12.25
6.25
2.25
0.25
30.25
20.25

14. Sample Variance
When the data values come from a sample rather than a population, the variance is called the sample variance. The procedure for computing the sample variance is a bit different from the one used to compute a population variance. In the formula, the mean μ is replaced by the sample mean and the denominator is n – 1 instead of N. The sample variance is denoted by s2.

15. Why Divide by n –1?
When computing the sample variance, we use the sample mean to compute the deviations. For the population variance we use the population mean for the deviations. It turns out that the deviations using the sample mean tend to be a bit smaller than the deviations using the population mean. If we were to divide by n when computing a sample variance, the value would tend to be a bit smaller than the population variance. It can be shown mathematically that the appropriate correction is to divide the sum of the squared deviations by n –1 rather than n.

16. Example
A company that manufactures batteries is testing a new type of battery designed for laptop computers. They measure the lifetimes, in hours, of six batteries, and the results are presented in the following table. Find the sample variance of the lifetimes. Solution: Step 1: Compute the sample mean. Step 2: For each sample value xi compute xi – . These values are shown in the second row below.
Battery Lifetime 3 4 6 5 2 xi 3 4 6 5 2
xi – –1 1 –2

17. Solution
Step 3: Square the deviations to obtain the quantity (xi – )2. These values are shown in the third row. Step 4: Sum the squared deviations to obtain the quantity Σ(xi – )2. Step 5: Divide the sum obtained in Step 4 by n –1 to obtain the sample variance s2. xi 3 4 6 5 2
xi – –1 1 –2
(xi – )2

18. Objective 3
Compute the standard deviation of a population and a sample

19. Standard Deviation
Because the variance is computed using squared deviations, the units of the variance are the squared units of the data. For example, in Battery Lifetime example, the units of the data are hours, and the units of variance are squared hours. In most situations, it is better to use a measure of spread that has the same units as the data. We do this simply by taking the square root of the variance. This quantity is called the standard deviation. The standard deviation of a sample is denoted s, and the standard deviation of a population is denoted by σ.

20. Example
Recall that in the Battery Lifetime example, the sample variance was computed as s2 = 2. Find the sample standard deviation. Solution: The sample standard deviation, s, is the square root of the sample variance.
Battery Lifetime 3 4 6 5 2

21. Standard Deviation on the TI-84 PLUS
The following steps will compute the standard deviation for both sample data and population data on the TI-84 PLUS Calculator:
Enter the data into L1 in the data editor.
Run the 1-Var Stats command (the same command used for means and medians), selecting L1 as the location of the data.

22. Example
Using the TI-84 PLUS Calculator, find the sample variance. A company that manufactures batteries is testing a new type of battery designed for laptop computers. They measure the lifetimes, in hours, of six batteries, and the results are presented in the following table. Solution: Running the 1-Var Stats command, we find the sample standard deviation to be s = We square this quantity to find the sample variance: s2 = ( )2 = 2.
Battery Lifetime 3 4 6 5 2

23. Standard Deviation & Resistance
Recall that a statistic is resistant if its value is not affected much by extreme data values. The standard deviation is not resistant. That is, the standard deviation is affected by extreme data values.

24. Objective 4
Approximate the standard deviation using grouped data

25. Approximating the Standard Deviation
Sometimes we don’t have access to the raw data in a data set, but we are given a frequency distribution. In these cases we can approximate the standard deviation.

26. Approximating the Standard Deviation
Following is the procedure for approximating the standard deviation: Step 1: Compute the midpoint of each class and approximate the mean of the frequency distribution. Step 2: For each class, subtract mean from the class midpoint to obtain (Midpoint – Mean). Step 3: For each class, square the differences obtained in Step 2 to obtain (Midpoint – Mean)2, and multiply by the frequency to obtain (Midpoint – Mean)2 x (Frequency). Step 4: Add the products (Midpoint – Mean)2 x (Frequency) over all classes. Step 5: To compute the population variance, divide the sum obtained in Step 4 by n. To compute the sample variance, divide the sum obtained in Step 4 by n –1. Step 6: Take the square root of the variance obtained in Step 5. The result is the standard deviation.

27. Number of Text Messages Sent
Example
The following table presents the number of text messages sent via cell phone by a sample of 50 high school students. Approximate the sample standard deviation number of messages sent.
Number of Text Messages Sent
Frequency
0 – 49 10
50 – 99 5
100 – 149 13
150 – 199 11
200 – 249 7
250 – 299 4

28. Number of Text Messages Sent
Solution
Step 1: Compute the midpoint of each class. Recall from the last section that the sample mean was computed as 137.
Number of Text Messages Sent
Class Midpoint
0 – 49 25
50 – 99 75
100 – 149
125
150 – 199
175
200 – 249
225
250 – 299
275

29. Number of Text Messages Sent
Solution
Step 2: For each class, subtract mean from the class midpoint to obtain (Midpoint – Mean).
Number of Text Messages Sent
Class Midpoint
(Midpoint – Mean)
0 – 49 25
–112
50 – 99 75
–62
100 – 149
125
–12
150 – 199
175 38
200 – 249
225 88
250 – 299
275
138

30. Number of Text Messages Sent (Midpoint – Mean)2 x (Frequency)
Solution
Step 3: For each class, square the differences obtained in Step 2 to obtain (Midpoint – Mean)2, and multiply by the frequency to obtain (Midpoint – Mean)2 x (Frequency).
Number of Text Messages Sent
Frequency
(Midpoint – Mean)
(Midpoint – Mean)2 x (Frequency)
0 – 49 10
–112
125,440
50 – 99 5
–62
19,220
100 – 149 13
–12
1,872
150 – 199 11 38
15,884
200 – 249 7 88
54,208
250 – 299 4
138
76,176

31. (Midpoint – Mean)2 x (Frequency)
Solution
Step 4: Add the products (Midpoint – Mean)2 x (Frequency) over all classes.
(Midpoint – Mean)2 x (Frequency)
125,440
19,220
1,872
15,884
54,208
76,176

32. Solution
Step 5: Since we are computing the sample variance, we divide the sum obtained in Step 4 by n –1. Step 6: Take the square root of the variance to obtain the standard deviation.

33. Grouped Data on the TI-84 PLUS
The same procedure used to compute the mean for grouped data in a frequency distribution may be used to compute the standard deviation.
Enter the midpoint for each class into L1 and the corresponding frequencies in L2. Next, select the 1-Var stats command and enter L1 in the List field and L2 in the FreqList field, if using Stats Wizards. If you are not using Stats Wizards, you may run the1-Var Stats command followed by L1, comma, L2.

34. Example
Class Midpoint
Frequency 25 10 75 5
125 13
175 11
225 7
275 4
The output for the last example on the TI-84 PLUS Calculator is presented below. The value of s represents the approximate sample standard deviation. In this example s = Therefore the approximate standard deviation is

35. Objective 5
Use the Empirical Rule to summarize data that are unimodal and approximately symmetric

36. Bell-Shaped Histogram
Many histograms have a single mode near the center of the data, and are approximately symmetric. Such histograms are often referred to as bell-shaped.

37. The Empirical Rule
When a data set has a bell-shaped histogram, it is often possible to use the standard deviation to provide an approximate description of the data using a rule known as The Empirical Rule.
When a population has a histogram that is approximately bell-shaped, then:
Approximately 68% of the data will be within one standard deviation of the mean.
Approximately 95% of the data will be within two standard deviations of the mean.
All, or almost all, of the data will be within three standard deviations of the mean.

38. Example
The following table presents the U.S. Census Bureau projection for the percentage of the population aged 65 and over for each state and the District of Columbia. Compute the population mean and standard deviation and use The Empirical Rule to describe the data.
Alabama
14.1
Rhode Island
Nevada
12.3
Kentucky
13.1
Arkansas
14.3
Tennessee
13.3
New Mexico
Maryland
12.2
Connecticut
14.4
Vermont
North Dakota
15.3
Minnesota
12.4
Florida
17.8
West Virginia 16
Oregon 13
Montana 15
Idaho 12
Alaska
8.1
South Carolina
13.6
New Hampshire
12.6
Iowa
14.9
California
11.5
Texas
10.5
New York
Louisiana
Delaware
Virginia
Ohio
13.7
Massachusetts
Georgia
10.2
Wisconsin
13.5
Pennsylvania
15.5
Mississippi
12.8
Illinois
Arizona
13.9
South Dakota
14.6
Nebraska
13.8
Kansas
13.4
Colorado
10.7
Utah 9
New Jersey
Maine
15.6
D.C.
Washington
North Carolina
Michigan
Hawaii
Wyoming 14
Oklahoma
Missouri
Indiana
12.7

39. Solution
We first note that the histogram is approximately bell-shaped. We may use the TI-84 PLUS Calculator – or other technology – to compute the population mean and population standard deviation. Mean: µ = Standard Deviation: σ =

40. Solution Approximately 68% of the data values are between these.
Almost all of the data values are between these.

41. Objective 6
Use Chebyshev’s Inequality to describe a data set

42. Any Data Set
When a distribution is bell-shaped, we use The Empirical Rule to approximate the proportion of data within one or two standard deviations. Another rule called Chebyshev’s Inequality holds for any data set.

43. Chebyshev’s Inequality
In any data set, the proportion of the data that is within K standard deviations of the mean is at least 1– 1/K2. Specifically, by setting K = 2 or K = 3, we obtain the following results.
At least 3/4 (75%) of the data are within two standard deviations of the mean.
At least 8/9 (89%) of the data are within three standard deviations of the mean.

44. Example
As part of a public health study, systolic blood pressure was measured for a large group of people. The mean was 120 and the standard deviation was What information does Chebyshev’s Inequality provide about these data?
Solution:
We compute the following:
We conclude:
At least 3/4 (75%) of the people had systolic blood pressures between 100 and 140.
At least 8/9 (89%) of the people had systolic blood pressures between 90 and 150.

45. Objective 7
Compute the coefficient of variation

46. Coefficient of Variation
The coefficient of variation (CV for short) tells how large the standard deviation is relative to the mean. It can be used to compare the spreads of data sets whose values have different units. The coefficient of variation is found by dividing the standard deviation by the mean.

47. Example
National Weather service records show that over a thirty-year period, the annual precipitation in Atlanta, Georgia had a mean of 49.8 inches with a standard deviation of 7.6 inches, and the annual temperature had a mean of 62.2 degrees Fahrenheit with a standard deviation of 1.3 degrees. Compute the coefficient of variation for precipitation and for temperature. Which has greater spread relative to its mean?

48. Solution
We compute the following: The CV for precipitation is larger than the CV for temperature. Therefore precipitation has a greater spread relative to its mean.

49. Do You Know… How to compute the range of a data set?
How to compute the variance of a population and a sample and the appropriate notation?
How to compute the standard deviation of a population and a sample and the appropriate notation?
How to approximate the standard deviation using grouped data?
How to use the Empirical Rule to summarize data?
How to use Chebyshev’s Inequality to describe a data set?
How to compute the coefficient of variation?