Section 3.1

HAWKES LEARNING SYSTEMS

Students Matter. Success Counts.

Copyright © 2013 by Hawkes Learning

Systems/Quant Systems, Inc.

All rights reserved.

Section 3.1

Measures of Center






Topics

o Calculate the mean, median, and mode.o Determine the most appropriate measure of center.






Mean

Sample MeanThe sample mean is the arithmetic mean of a set of sample data, given by

where xi is the ith data value and n is the number of data values in the sample.

1 2 n

i

x x xx

nx

n






Mean

Population MeanThe population mean is the arithmetic mean of all the values in a population, given by

where xi is the ith data value in the population and N is the number of values in the population.

1 2 N

i

x x xN

xN

m






Example 3.1: Calculating the Sample Mean

Students were surveyed to find out the number of hours they sleep per night during the semester. Here is a sample of their self-reported responses. Calculate the mean.

5, 6, 8, 10, 4, 6, 9






Example 3.1: Calculating the Sample Mean (cont.)

SolutionBecause we are given a sample of the student responses, we are calculating the sample mean. Add the hours together and then divide by 7, which is the number of students in the sample. At the end of the calculation, round to one decimal place since the data values are whole numbers.







The sample mean for the number of hours that students reported sleeping per night during the semester is 6.9.

5 6 8 10 4

6.85714

6

3 6.47

9

97

8

ixx

n







Alternate Calculator MethodTo find the sample mean on a TI-83/84 Plus

calculator, follow the steps below.• Press .• Choose option 1:Edit and press . • Enter the data in L1.







• Press again. • Choose CALC. • Choose option 1:1-Var Stats.• Press twice. (Note: If your data are not in

L1, before pressing the second time, enter the list where your data are located, such as L3 or L5.)







The first value in the output, seento the right, shows the value of In addition, the calculator displays many other descriptive statistics, not just the sample mean. We will use the above procedure repeatedly to find various descriptive statistics throughout this chapter.

6.857142857.x






Example 3.2: Using the Mean to Find a Data Value

Rutherford downloaded five new songs from the Internet. He knows that, on average (mean), the songs cost $1.23. If four of the songs cost $1.29 each, what was the price of the fifth song he downloaded?Solution Since we are given the value of the mean, we can use algebra on the formula to find the missing value. Substituting all the values we are given into the formula for the mean, we have the following.






Example 3.2: Using the Mean to Find a Data Value (cont.)

So the cost of Rutherford’s fifth song was $0.99.

1 2 3

5

5

5

4

$1.29 $1.29 $1.29 $1.29$1.23

5$5.16

$1.235

5

x

x x xN

x

x x

m

5

5

5

5$5.16$6.15 $5.16

$6.15 $5. $5.1616$0.99 x

xx






Mean

Weighted Mean The weighted mean is the mean of a data set in which each data value in the set does not hold the same relative importance, given by

where xi is the ith data value andwi is the weight of the ith data value.

i i

i

x wx

w






Example 3.3: Calculating a Weighted Mean

The syllabus in Walter’s US history class states that the final grade is determined by tests (40%), homework (20%), quizzes (10%), and a final exam (30%). Two students in the class, Walter and Virginia, want to calculate their final grades. Below are their average grades in each of the categories for tests, homework, and quizzes. They have also individually guessed at what they might score on their final exam.






Example 3.3: Calculating a Weighted Mean (cont.)

a. Calculate what Walter’s final grade would be if these were his ultimate scores.

Tests: 83 Homework: 98 Quizzes: 90 Final Exam: 87







b. Calculate Virginia’s final grade, given her scores below, using a TI-83/84 Plus calculator. Tests: 95 Homework: 45 Quizzes: 66 Final Exam: 90







Solutiona. First, let’s determine which numbers are values of x and which are weights. The grade earned in each

category is weighted by the percentage for that category in the syllabus. For instance, Walter’s

test average of 83 gets a weight of 40%. Thus, in this case the weights are the percentages for the categories. The values for x are then Walter’s

grades in the categories. The weighted mean is calculated as follows.







Therefore, Walter’s final grade for the class would be 87.9.

83 0.4 98 0.2 90 0.1 87 0.30.4

87.9

0.2 0.1 0.387.9

1

i i

i

x wx

w







b. As noted when calculating Walter’s grade, the weights are the percentages for the categories and the values for x are Virginia’s grade in each category.

To calculate a weighted mean using a TI-83/84 Plus calculator, we will follow steps similar to those used for calculating a sample mean. However, we will need to enter two lists of data instead of just one. • Press . • Choose option 1:Edit and press . • Enter the values of x for your data in L1, and enter the weights for your data in L2.







• Press again. • Choose CALC. • Choose option 1:1-Var Stats and

press . • Press to

enter L1,L2 on the screen. This tells the calculator that this is a weighted mean and that the data are in two lists.

• Press .







The output, seen in the screenshot on the right, shows the value of along with the other descriptive statistics for this data set.

6,80.x







Although Virginia didn’t do as well on her homework and quizzes, if she manages to get a 90 on the final exam like she predicts, she’ll end up with an 80.6 for her final grade. Take a moment and consider the effect on her grade if her test score was low, but the homework and quiz scores were high. Would she come out with the same grade?






Example 3.4: Calculating a Weighted Mean

At the end of the semester, Heather knows all of her grades in her sculpting class except for the final exam. Here’s a breakdown of her points and how much each category counts toward the final course grade.

Tests (35%): 78 Class Assignments (20%): 92 Semester Project (35%): 93 Final Exam (10%): ?







a. Calculate the weighted mean for the portion of the grade that she has completed. b. What grade must Heather make on her final exam in order to have a final grade of 90%, which would be the A that she desires? Assume that the final exam is worth 100 points. c. What grade must Heather make on her final exam in order to have a final grade of 80%, which would give her a B for the course? Again, assume that the final exam is worth 100 points.







Solution a. Just like in the previous example, the weights are

the percentages given for each category, and the values for x are the scores she has so far. We will only include the portions of the final grade in

which she has scores, not all of the categories. The weighted mean is then calculated as follows.

(Remember to change the percentages to decimals for the formula.)







So, Heather has 86.9 for her sculpting grade so far without the final exam.

78 0.35 92 0.2 93 0.350.35 0.2 0.35

78.86.94 86.9

250.9

i i

i

x wx

w

Notice that the total in the denominator no longer equals 1 since we are not including all of the components for the final grade.







b. We know that the weights are the percentages given for each category, and the values for x are the scores for each category. However, in this scenario, it is not the sample mean, x, that we don’t know, but the value, x, of the final exam grade. Use the same formula as the one used in part a., but add in the final exam category with a weight of 0.10 and the

unknown value of x. Set the formula equal to 90, and solve the resulting equation for x as follows.







78 0.35 92 0.2 93 0.35 0.190

0.35 0.2 0.35 0.178.25 0.1

901

90 78.25 0.111.75 0.1

117.5

i i

i

x w

x

x

x

x

xw

x







Since x > 100, it appears as though it is mathematically impossible for Heather to make an A

in the course. The highest final grade that Heather could achieve in the class is 88.25. We will leave this for you to verify on your own.c. To find this solution, we will set up the problem just the same as in part b., except now we will set the

formula equal to 80 rather than 90. We will again solve for x, which represents the unknown final exam score. We then have the following.







78 0.35 92 0.2 93 0.35 0.180

0.35 0.2 0.35 0.178.25 0.1

801

80 78.25 0.11.75 0.

17.51

i i

i

x wx

w

x

xx

x

x







Hence, we see that although an A is not possible, Heather must score only 17.5 or higher on the

final exam in order to secure a B in the course.






Median

Finding the Median of a Data Set 1. List the data in ascending (or descending) order, making an ordered array.2. If the data set contains an ODD number of values, the median is the middle value in the ordered array. 3. If the data set contains an EVEN number of values,

the median is the arithmetic mean of the two middle values in the ordered array. Note that this implies that the median may not be a value in the data set.






Example 3.5: Finding the Median

Given the numbers of absences for samples of students in two different classes, find the median for each sample.

a. 3, 4, 6, 7, 2, 8, 9 b. 5, 7, 8, 1, 4, 9, 8, 9 Solution a. First, put the data in order: 2, 3, 4, 6, 7, 8, 9. Since there are an odd number of values, the median is

the number in the middle: 2, 3, 4, 6, 7, 8, 9. Thus, the median for this sample is 6 absences.






Example 3.5: Finding the Median (cont.)

b. First, put the data in order: 1, 4, 5, 7, 8, 8, 9, 9. Since there are an even number of values, the median is the mean of the two numbers in the middle.

1, 4, 5, 7, 8, 8, 9, 9

Thus, the median for this sample is 7.5 absences.

7 87.5

2







Alternate Calculator MethodThe median is one of the descriptive statistics that the TI-83/84 Plus calculator displays when you choose the 1-Var Stats option from the > CALC menu. Recall from Example 3.1 that the steps to find the descriptive statistics are as follows.







• Press . • Choose option 1:Edit and press . • Enter the data in L1. • Press again. • Choose CALC. • Choose option 1:1-Var Stats. • Press twice.







Do you see an output value for the median? Probably not. That is because the median is actually on the second “page” of the output. Use the down arrow to scroll down to the other descriptive statistics. The one labeled “Med=7.5” is the median.






Example 3.6: Finding the Mode

Given the number of phone calls received each hour during business hours for three different companies, find the mode of each, and state if the data set is unimodal, bimodal, or neither.

a. 6, 4, 6, 1, 7, 8, 7, 2, 5, 7 b. 3, 4, 7, 8, 1, 6, 9 c. 2, 5, 7, 2, 8, 7, 9, 3 d. 2, 2, 3, 3, 4, 4, 5, 5






Example 3.6: Finding the Mode (cont.)

Solution a. To make it easier to see which value(s) occur most often, begin by putting the data in numerical order. The ordered data set is: 1, 2, 4, 5, 6, 6, 7, 7, 7, 8. The number 7 occurs more than any other value, so the mode is 7. This data set is unimodal. b. Begin by sorting the data as follows: 1, 3, 4, 6, 7, 8, 9. The values all occur only once, so there is no

mode. This data set is neither unimodal nor bimodal.







c. As before, sort the data: 2, 2, 3, 5, 7, 7, 8, 9. The values 2 and 7 both occur an equal number of times; thus they are both modes. This data set is bimodal.

d. Note that this data set is already sorted for us. All of the data values occur the same number of times, so there is no mode. This data set is neither unimodal nor bimodal.







Note that, although some statistical software packages will identify the mode of a data set, the mode is not one of the descriptive statistics listed by a TI-83/84 Plus calculator.






Example 3.7: Calculating Measures of Center—Mean, Median, and Mode

Given the recent economy and change of attitude in society, many people chose to take on another job after retiring from one. Below is a sample of ages at which people truly retired; that is, they stopped working for pay. Calculate the mean, median, and mode for the data.

84, 80, 82, 77, 78, 80, 79, 42






Example 3.7: Calculating Measures of Center—Mean, Median, and Mode (cont.)

Solution Mean: Remember, the mean is the sum of all the data points divided by the number of points.

84 80 82 77 78 80 79 428

60275 75..25 3

8

ixx

n







Median: We have an even number of values, so we will need the mean of the middle two values in the ordered array.

42, 77, 78, 79, 80, 80, 82, 84

79 802

79.5







Mode: The number 80 occurs more than any other number, so it is the mode.







As you can see in the figure, of the three measures of center, the mean is closest to the outlier while the median and mode are more similar in value and are not affected by the outlier.






Choosing an Appropriate Measure of Center

Determining the Most Appropriate Measure of Center1. For qualitative data, the mode should be used. 2. For quantitative data, the mean should be used,

unless the data set contains outliers or is skewed. 3. For quantitative data sets that are skewed or

contain outliers, the median should be used.






Example 3.8: Choosing the Most Appropriate Measure of Center

Choose the best measure of center for the following data sets. a. T-shirt sizes (S, M, L, XL) of American women b. Salaries for a professional team of baseball players c. Prices of homes in a subdivision of similar homes d. Professor rankings from student evaluations on a

scale of best, average, and worst






Example 3.8: Choosing the Most Appropriate Measure of Center (cont.)

Solution a. T-shirt sizes are ordinal data; since they are qualitative, the mode is the best measure of center. b. The players’ salaries are quantitative data with

outliers, since the superstars on the team make substantially more than the typical players.

Therefore, the median is the best choice. c. The home prices are quantitative data with no

outliers, since the homes are similar. Therefore, the mean is the best choice.






Example 3.8: Choosing the Most Appropriate Measure of Center (cont.)

d. The rankings are ordinal data; since they are qualitative, it’s best to use the mode as a

measure of center.






Graphs and Measures of Center

Graphs and Measures of Center 1. The mode is the data value at which a distribution

has its highest peak.2. The median is the number that divides the area of

the distribution in half. 3. The mean of a distribution will be pulled toward any

outliers.






Example 3.9: Determining Mean, Median, and Mode from a Graph

Determine which letter represents the mean, the median, and the mode in the graph below.






Example 3.9: Determining Mean, Median, and Mode from a Graph (cont.)

Solution Mode: The mode is always located at the peak of a distribution, so it is line A. Median: The median is the value that divides the area of the distribution in half. Here it is represented by line B. Mean: This distribution is skewed to the right, so the mean will be the measure of center farthest

to the right. Here it is represented by line C.

Section 3.1

Documents

Transcript of Section 3.1