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Section 3.2 Measures of Dispersion

Objectives

- Compute the range of a variable from raw data
- Compute the variance of a variable from raw data
- Compute the standard deviation of a variable from raw data
- Use the Empirical Rule to describe data that are bell shaped
- Use Chebyshev’s Inequality to describe any data set

To order food at a McDonald’s Restaurant, one must choose from multiple lines, while at Wendy’s Restaurant, one enters a single line. The following data represent the wait time (in minutes) in line for a simple random sample of 30 customers at each restaurant during the lunch hour. For each sample, answer the following:

(a) What was the mean wait time?

(b) Draw a histogram of each restaurant’s wait time.

(c ) Which restaurant’s wait time appears more dispersed? Which line would you prefer to wait in? Why?

Wait Time at Wendy’s

1.50 0.79 1.01 1.66 0.94 0.67

2.53 1.20 1.46 0.89 0.95 0.90

1.88 2.94 1.40 1.33 1.20 0.84

3.99 1.90 1.00 1.54 0.99 0.35

0.90 1.23 0.92 1.09 1.72 2.00

Wait Time at McDonald’s

3.50 0.00 0.38 0.43 1.82 3.04

0.00 0.26 0.14 0.60 2.33 2.54

1.97 0.71 2.22 4.54 0.80 0.50

0.00 0.28 0.44 1.38 0.92 1.17

3.08 2.75 0.36 3.10 2.19 0.23

Objective 1

- Compute the range of a variable from raw data

The range, R, of a variable is the difference between the largest data value and the smallest data values. That is

Range = R = Largest Data Value – Smallest Data Value

EXAMPLE Finding the Range of a Set of Data

The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.

23, 36, 23, 18, 5, 26, 43

Find the range.

Range = 43 – 5

= 38 minutes

Objective 2

- Compute the variance of a variable from raw data

The population variance of a variable is the sum of squared deviations about the population mean divided by the number of observations in the population, N.

That is it is the mean of the sum of the squared deviations about the population mean.

The population variance is symbolically represented by σ2 (lower case Greek sigma squared).

Note: When using the above formula, do not round until the last computation. Use as many decimals as allowed by your calculator in order to avoid round off errors.

EXAMPLE Computing a Population Variance

The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.

23, 36, 23, 18, 5, 26, 43

Compute the population variance of this data. Recall that

EXAMPLE Computing a Population VarianceUsing the Computational Formula

The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.

23, 36, 23, 18, 5, 26, 43

Compute the population variance of this data using the computational formula.

The sample variance is computed by determining the sum of squared deviations about the sample mean and then dividing this result by n – 1.

Note: Whenever a statistic consistently overestimates or underestimates a parameter, it is called biased. To obtain an unbiased estimate of the population variance, we divide the sum of the squared deviations about the mean by n - 1.

EXAMPLE Computing a Sample Variance

In Section 3.1, we obtained the following simple random sample for the travel time data: 5, 36, 26.

Compute the sample variance travel time.

square minutes

Objective 3

- Compute the standard deviation of a variable from raw data

The population standard deviation is denoted by

It is obtained by taking the square root of the population variance, so that

The sample standard deviation is denoted by

s

It is obtained by taking the square root of the sample variance, so that

EXAMPLEComputing a Population Standard DeviationThe following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.

23, 36, 23, 18, 5, 26, 43

Compute the population standard deviation of this data.

Recall, from the last objective that σ2 = 129.0 minutes2.

Therefore,

EXAMPLEComputing a Sample Standard Deviation

Recall the sample data 5, 26, 36 results in a sample variance of

square minutes

Use this result to determine the sample standard deviation.

EXAMPLEComparing Standard Deviations

Determine the standard deviation waiting time for Wendy’s and McDonald’s. Which is larger? Why?

Wait Time at Wendy’s

1.50 0.79 1.01 1.66 0.94 0.67

2.53 1.20 1.46 0.89 0.95 0.90

1.88 2.94 1.40 1.33 1.20 0.84

3.99 1.90 1.00 1.54 0.99 0.35

0.90 1.23 0.92 1.09 1.72 2.00

Wait Time at McDonald’s

3.50 0.00 0.38 0.43 1.82 3.04

0.00 0.26 0.14 0.60 2.33 2.54

1.97 0.71 2.22 4.54 0.80 0.50

0.00 0.28 0.44 1.38 0.92 1.17

3.08 2.75 0.36 3.10 2.19 0.23

EXAMPLEComparing Standard Deviations

Determine the standard deviation waiting time for Wendy’s and McDonald’s. Which is larger? Why?

Sample standard deviation for Wendy’s:

0.738 minutes

Sample standard deviation for McDonald’s:

1.265 minutes

Objective 4

- Use the Empirical Rule to Describe Data That Are Bell Shaped

EXAMPLE Using the Empirical Rule

The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor.

41 48 43 38 35 37 44 44 44

62 75 77 58 82 39 85 55 54

67 69 69 70 65 72 74 74 74

60 60 60 61 62 63 64 64 64

54 54 55 56 56 56 57 58 59

45 47 47 48 48 50 52 52 53

(a) Compute the population mean and standard deviation.

(b) Draw a histogram to verify the data is bell-shaped.

(c) Determine the percentage of patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule.

(d) Determine the percentage of patients that have serum HDL between 34 and 69.1 according to the Empirical Rule.

(e) Determine the actual percentage of patients that have serum HDL between 34 and 69.1.

22.3 34.0 45.7 57.4 69.1 80.8 92.5

(c) According to the Empirical Rule, 99.7% of the patients that have serum HDL within 3 standard deviations of the mean.

(d) 13.5% + 34% + 34% = 81.5% of patients will have a serum HDL between 34.0 and 69.1 according to the Empirical Rule.

(e) 45 out of the 54 or 83.3% of the patients have a serum HDL between 34.0 and 69.1.

Objective 5

- Use Chebyshev’s Inequality to Describe Any Set of Data

EXAMPLE Using Chebyshev’s Theorem

- Using the data from the previous example, use Chebyshev’s Theorem to
- determine the percentage of patients that have serum HDL within 3 standard deviations of the mean.
- (b) determine the actual percentage of patients that have serum HDL between 34 and 80.8.

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