W EIGHTED M EAN AND M EASURES OF G ROUPED D ATA

z-Score

2.3 W EIGHTED M EAN AND M EASURES OF G ROUPED D ATA

Some situations require a special type of mean called a weighted mean. Applications of weighted means include, but are not limited to, calculating GPA, determining average stock recommendation, and approximating the mean of grouped data.

Weighted Mean

The weighted mean of a set of data is x = a wix_i

n (2.20)

where w_i = weight of the i th observation and n = gwi.

One important situation that requires the use of a weighted mean is the calculation of grade point average (GPA).

Example 2.16 Grade Point Average (Weighted Mean)

Suppose that a student who completed 15 credit hours during his first semester of col-lege received one A, one B, one C, and one D. Suppose that a value of 4 is used for an A, 3 for a B, 2 for a C, 1 for a D, and 0 for an F. Calculate the student’s semester GPA.

Solution If each course were given the same number of credit hours, the student’s semester GPA would equal the following:

x = a

n i=1x_i

n = x₁ + x2 + . . . + xn

n = 4 + 3 + 2 + 1

4 = 2.5

However, each course is not worth the same number of credit hours. The A was earned in a 3-credit-hour English course, and the B was earned in a 3-credit-hour math course, but the C was earned in a 4-credit-hour biology lab course, and the D grade, unfortunately, was earned in a 5-credit-hour Spanish class. Computation of the mean is x = 14 + 4 + 42 + 13 + 3 + 32 + 12 + 2 + 2 + 22 + 11 + 1 + 1 + 1 + 12

15 = 34

15 = 2.267 where the numerator is the sum of 14 + 4 + 42 representing the three English credits plus 13 + 3 + 32 for the three math credits plus 12 + 2 + 2 + 22 for the four biology lab credits plus 11 + 1 + 1 + 1 + 12 for the five Spanish credits. Using Equation 2.20 the computation of the GPA is given in Table 2.7.

x = a

n i=1w_ix_i

n = w₁x₁ + w2x₂ + . . . + wnx_n

n = 12 + 9 + 8 + 5

15 = 34

15 = 2.267 Table 2.7 Semester Academic Record

C^OURSE G^RADE C^REDIT H^OURS, wi V^ALUE, xi C^REDIT H^OURS * V^ALUE, wix_i

English A 3 4 12

Math B 3 3 9

Biology lab C 4 2 8

Spanish D 5 1 5

Total 15 34

2.3 Weighted Mean and Measures of Grouped Data 81 A survey may ask respondents to select an age category such as 20–29 rather than giv-ing their specific age. Or respondents may be asked to select a cost category such as $4.00 to under $6.00 for a purchase at a local coffee shop. In these situations exact values of the mean and variance are not possible. However, we are able to approximate the mean and the variance.

Example 2.17 Stock Recommendation (Weighted Mean)

Zack’s Investment Research is a leading investment research firm. Zack’s will make one of the following recommendations with corresponding weights for a given stock:

Strong Buy (1), Moderate Buy (2), Hold (3), Moderate Sell (4), or Strong Sell (5). Sup-pose that on a particular day, 10 analysts recommend Strong Buy, 3 analysts recom-mend Moderate Buy, and 6 analysts recomrecom-mend Hold for a particular stock. Based on Zack’s weights, find the mean recommendation.

Solution Table 2.8 shows the weights for each recommendation and the computation leading to a recommendation based on the following weighted mean recommendation conversion values: if the weighted mean is 1, Strong Buy; 1.1 through 2.0, Moderate Buy; 2.1 through 3.0, Hold; 3.1 through 4.0, Moderate Sell; 4.1 through 5, Strong Sell.

Table 2.8 Computation of Zack’s Investment Research’s Average Brokerage Recommendation

A^CTION N^UMBEROF A^NALYSTS, wi V^ALUE, xi w_ix_i

Strong Buy 10 1 10

Moderate Buy 3 2 6

Hold 6 3 18

Moderate Sell 0 4 0

Strong Sell 0 5 0

x = a

n i=1w_ix_i

n = 10 + 6 + 18 + 0 + 0

19 = 1.79

The weighted mean of 1.79 yielded a Moderate Buy recommendation.

Approximate Mean and Variance for Grouped Data Suppose that data are grouped into K classes, with frequencies f₁, f₂, . . . , f_K. If the midpoints of these classes are m₁, m₂, . . . , m_K, then the sample mean and sample variance of grouped data are approximated in the following manner:

The mean is

x = a

K i=1fimi

n (2.21)

where n = a^K

i=1f_i, and the variance is

s² = a

K

i=1f_i1m_i- x2²

n - 1 (2.22)

82 Chapter 2 Using Numerical Measures to Describe Data

Example 2.18 Cost of Coffee Shop Purchase (Mean and Variance for Grouped Values)

Coffee shop customers were randomly surveyed and asked to select a category that described the cost of their recent purchase. The results were as follows:

Cost 1in USD2 0 6 2 2 6 4 4 6 6 6 6 8 8 6 10 Number of Customers 2 3 6 5 4

Find the sample mean and standard deviation of these costs.

Solution The frequencies are the number of customers for each cost category. The computations for the mean and the standard deviation are set out in Table 2.9.

Table 2.9 Cost of Purchase (Grouped Data Computation)

C^OSTS ($) F^REQUENCY, fi M^IDPOINT, mi 1f_im_i2 1m_i - x2 1m_i- x2² f_i1m_i- x2²

0 6 2 2 1 2 -4.6 21.16 42.32

2 6 4 3 3 9 -2.6 6.76 20.28

4 6 6 6 5 30 -0.6 0.36 2.16

6 6 8 5 7 35 1.4 1.96 9.80

8 6 10 4 9 36 3.4 11.56 46.24

20 112 120.80

n = a^K

i=1f_i = 20 and a^K

i=1f_im_i = 112 The sample mean is estimated by

x = a

K i=1f_im_i

n = 112 20 = 5.6 Since these are sample data, the variance is estimated by

s² = a

K

i=1f_i1m_i - x2²

n - 1 = 120.8

19 = 6.3579 Hence, the sample standard deviation is estimated as

s = 2s² = 26.3579 = 2.52

Therefore, the mean coffee shop purchase price is estimated as $5.60, and the sam-ple standard deviation is estimated to be $2.52.

Exercises 83

E

^XERCISES

Visit www.mymathlab.com/global or www.pearsonglobal editions.com/newbold to access the data files.

Basic Exercises

2.26 Consider the following sample of five values and cor-responding weights:

x_i w_i

4.6 8

3.2 3

5.4 6

2.6 2

5.2 5

a. Calculate the arithmetic mean of the x_i values with-out weights.

b. Calculate the weighted mean of the x_i values.

2.27 Consider the following frequency distribution for a sample of 40 observations:

Class Frequency

0–4 5

5–9 8

10–14 11

15–19 9

20–24 7

a. Calculate the sample mean.

b. Calculate the sample variance and sample standard deviation.

Application Exercises

2.28 An online pharmaceutical company obtained the following frequency distribution of shipping times (number of hours between the time an order is placed and the time the order is shipped) for a random sam-ple of 40 orders. (Be sure to comsam-plete all appropriate columns and show your work).

Number of Hours fi

4 6 ^{10 8}

10 6 16 15

16 6 22 10

22 6 ^{28 7}

a. What is the approximate mean shipping time?

b. What is the approximate variance and standard deviation?

2.29 A manufacturer of portable radios obtained a sample of 50 radios from a week’s output. The radios were checked and the numbers of defects were recorded as follows.

Number of defects 0 1 2 3

Number of radios 12 15 17 6

Calculate the standard deviation.

2.30 A random sample of 50 personal property insurance policies showed the following number of claims over the past 2 years.

Number of claims 0 1 2 3 4 5 6

Number of policies 21 13 5 4 2 3 2 a. Find the mean number of claims per policy.

b. Find the sample variance and standard deviation.

2.31 For a random sample of 25 students from a very large university, the accompanying table shows the amount of time (in hours) spent studying for final exams.

Study time 0 6 4 4 6 8 8 6 12 12 6 16 16 6 20 Number of

students

3 7 8 5 2

a. Estimate the sample mean study time.

b. Estimate the sample standard deviation.

2.32 A sample of 20 financial analysts was asked to provide forecasts of earnings per share of a corporation for next year. The results are summarized in the following table:

Forecast ($ per share) Number of Analysts

$9.95 to under $10.45 2

$10.45 to under $10.95 8

$10.95 to under $11.45 6

$11.45 to under $11.95 3

$11.95 to under $12.45 1

a. Estimate the sample mean forecast.

b. Estimate the sample standard deviation.

2.33 A publisher receives a copy of a 500-page textbook from a printer. The page proofs are carefully read and the number of errors on each page is recorded, pro-ducing the data in the following table:

Number of errors 0 1 2 3 4 5 Number of pages 102 138 140 79 33 8 Find the mean and standard deviation in number of errors per page.

2.34 In Chapter 1, we described graphically using a frequency distribution table and a histogram the time (in seconds) for a random sample of n = 110 employees to complete a particular task. Describe the data numerically based on the frequency distribution given in Table 1.7. The data is stored in the data file Completion Times.

a. Compute the mean using Equation 2.21.

b. Compute the variance using Equation 2.22.

c. Compare your answers to the mean and variance calculated in Exercise 2.23.

84 Chapter 2 Using Numerical Measures to Describe Data