Chapter 3 Numerically Summarizing Data 3.2 Measures of Dispersion.

Chapter 3Numerically Summarizing

Data

3.2

Measures of Dispersion

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?

1.50 0.79 1.01 1.66 0.94 0.672.53 1.20 1.46 0.89 0.95 0.901.88 2.94 1.40 1.33 1.20 0.843.99 1.90 1.00 1.54 0.99 0.350.90 1.23 0.92 1.09 1.72 2.00

3.50 0.00 0.38 0.43 1.82 3.040.00 0.26 0.14 0.60 2.33 2.541.97 0.71 2.22 4.54 0.80 0.500.00 0.28 0.44 1.38 0.92 1.173.08 2.75 0.36 3.10 2.19 0.23

Wait Time at Wendy’s

Wait Time at McDonald’s

The mean wait time in each line is 1.39 minutes.

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

Find the range of the student data collected from Section 3.1

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 arithmetic mean of the sum of the squared deviations about the population mean.

The population variance is symbolically represented by 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

Compute the population variance of the population data collected in Section 3.1.

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

Compute the sample variance using the sample data from Section 3.1

The population standard deviation is denoted by

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

EXAMPLE Computing a Population Standard Deviation and Sample Standard Deviation

Compute the population and sample standard deviation for the data obtained in Section 3.1

EXAMPLE Comparing Standard Deviations

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

EXAMPLE Comparing 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

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 4462 75 77 58 82 39 85 55 5467 69 69 70 65 72 74 74 7460 60 60 61 62 63 64 64 6454 54 55 56 56 56 57 58 5945 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 33.8 and 81 according to the Empirical Rule. (e) Determine the actual percentage of patients that have serum HDL between 33.8 and 81.

(a) Using a TI83 plus graphing calculator, we find

(b)

57.4 and 11.8

(c) According to the Empirical Rule, approximately 99.7% of the patients will have serum HDL cholesterol levels within 3 standard deviations of the mean. That is, approximately 99.7% of the patients will have serum HDL cholesterol levels greater than or equal to 57.4 - 3(11.8) = 22 and less than or equal to 57.4 + 3(11.8) = 92.8.

57.4 and 11.8

(d) Because 33.8 is 2 standard deviations below the mean (57.4 - 2(11.8) = 33.8) and 81 is 2 standard deviations above the mean (57.4 + 2(11.8) = 81), the Empirical Rule states that approximately 95% of the data will lie between 33.8 and 81.

(e) There are no observations below 33.8. There are 2 observations greater than 81. Therefore, 52/54 = 96.3% of the data lie between 33.8 and 81.

57.4 and 11.8

EXAMPLE Using Chebyshev’s Theorem

Using the data from the previous example, use Chebyshev’s Theorem to

(a) determine the percentage of patients that have serum HDL within 3 standard deviations of the mean.

(b) determine the percentage of patients that have serum HDL between 33.8 and 81.