Making Sense of Measures of Central Tendency

Introduction

Course Objectives

This lesson will address the following course outcomes:

· 6. Demonstrate an understanding of the connection between the distribution of data and various mathematical summaries of data (measures of central tendency and of variation).

Specific Objectives

Students will understand that

· each statistic—the mean, median, and mode—is a different summary of numerical data.

· conclusions derived from statistical summaries are subject to error.

· they can use the measures of central tendency to make decisions.

Students will be able to

· make good decisions using information about data.

· interpret the mean, median, or mode in terms of the context of the problem.

· match data sets with appropriate statistics.

Job Ads

Problem Situation 1: Making Sense of Measures of Central Tendency

Examine these three job advertisements:

Employment Opportunities

Outdoor Sales

Above Average Sales

Home Sales

Sales Positions Available!

We have immediate need for five enthusiastic self-starters who love the outdoors and who love people. Our salespeople make an average of $1,000 per week. Come join the winning team.

Call 555-0100 now!

Are you above average?

Our company is hiring one person this month—will you be that person? We pay the top percentage commission and supply you leads. Half of our sales force makes over $3,000 per month. Join the Above Average Team!

Call 555-0127 now!

NEED A NEW CHALLENGE?

Join a super sales force and make as much as you want. Five of our nine salespeople closed FOUR homes last month. Their average commission was $1,500 on each sale. Do the math—this is the job for you.

Making dreams real—

call 555-0199

#1 Points possible: 12. Total attempts: 5

Consider the phrase from each advertisement. Which measure of central tendency is it most likely describing?

“Our salespeople make an average of $1,000 per week.”

“Half of our sales force makes over $3,000 per month.”

“Five of our nine salespeople closed FOUR homes last month.”

#2 Points possible: 5. Total attempts: 5

Consider the set of monthly salaries below. Which company (which Ad) could these salaries have come from?

$1500, $2000, $2000, $2500, $2500, $2500, $6000, $8000, $9000

· Outdoor Sales

· Above Average Sales

· Home Sales

#3 Points possible: 5. Total attempts: 5

For the company “Above Average Sales”, create a set of data for 8 employees that fits the measure of center described in the advertisement. You did this in the previous lesson when you made a list of credit card debts for the six college students.

, , , , , , ,

Home Prices

Problem Situation 2: Understanding Trends in Data

The median and average sales price of new homes sold in the United States from 1963–2008 is shown in the following graphic. Examine the graph.

#4 Points possible: 8. Total attempts: 5

Looking at the graph above, what was the average (mean) and median home price in 2005?

Average (mean): $

Median: $

Five possible data sets for the year 2005 are given in Table 2. Use your knowledge of mean and median to answer the following questions without calculating the mean of the data sets. There may be more than one correct answer to any of the questions. Explain your reasoning.

Table 2: Possible Data Sets for 2005

Set A	Set B	Set C	Set D	Set E
$240,000	$84,000	$120,000	$74,000	$74,000
$245,000	$105,000	$135,000	$95,000	$90,000
$250,000	$125,000	$150,000	$105,000	$120,000
$256,000	$240,000	$168,000	$240,000	$240,000
$267,000	$245,000	$201,000	$242,000	$250,000
$275,000	$469,000	$225,000	$250,000	$635,000
$312,000	$810,000	$336,000	$251,000	$669,000

#5 Points possible: 5. Total attempts: 5

Which of the data sets could have the same mean and median shown in the graph for 2005? (Select all that could)

· Set A

· Set B

· Set C

· Set D

· Set E

#6 Points possible: 5. Total attempts: 5

Which of the data sets would likely have a mean that is less than the median? (Select all that could)

· Set A

· Set B

· Set C

· Set D

· Set E

#7 Points possible: 5. Total attempts: 5

Which of the data sets would likely have a mean and median that are close together? (Select all that could)

· Set A

· Set B

· Set C

· Set D

· Set E

Look back at the home sales price graph and compare 2005 and 2007.

#8 Points possible: 8. Total attempts: 5

Both the mean and median sales prices decreased from 2005 to 2007. How much did each decrease?

The mean decreased by $

The median decreased by $

#9 Points possible: 10. Total attempts: 5

What type of changes of home sales prices occurred from 2005 to 2007 so that the mean and median would change in that way?

Since both the median and mean decreased:

· In general all house prices increased

· In general all house prices decreased

· Only high-end home prices decreased

· Only low-end home prices decreased

Since the mean decreased more than the median:

· All home prices decreased about the same amount

· Low-end home prices decreased more than the high-end home prices decreased

· High-end home prices decreased more than the low-end home prices decreased

#10 Points possible: 5. Total attempts: 5

On the sales price graph, for the years 1999 and onward the mean sales price is greater than the median sales price (which it had been for many years). What is new is that the difference between the mean and median prices of the homes is growing larger, especially so in the years 2003 and 2005.

What type of changes in housing prices in those years after 1999 would make the mean be further above the median?

· Low-end home prices increased faster than high-end home prices

· High-end home prices increased faster than low-end home prices

· All home prices increased at the same rate

HW 2.10

#1 Points possible: 5. Total attempts: 5

Which of the following was one of the main mathematical ideas of the lesson?

· The median is the middle number when a data set is listed in order. If there is an even number of points in the data set, the median is found by finding the mean of the middle two numbers.

· Home prices in 2007 were more than 10 times what they were in the 1960s.

· When using measures of central tendency, it is always important to ask questions about what the different types of measurements do and do not tell you about the data set.

· The mean and median of a data set are always very close together, but the mode might be very different.

#2 Points possible: 5. Total attempts: 5

The first advertisement discussed in class states that the salespeople make an average of $1,000 per week. Suppose there are nine salespeople. What would the ninth person need to earn for the mean to be $1,000 if the other eight salespeople earned $550, $600, $600, $800, $950, $950, $1,000, and $1,100?

#3 Points possible: 5. Total attempts: 5

The second advertisement states that half the salespeople make more than $3,000 per month. Suppose there are eight salespeople. What would the eighth person need to earn for the median to be $3,000 if the other seven salespeople earned $2,400, $2,500, $2,800, $2,800, $3,400, $3,400, and $3,800? $

#4 Points possible: 9. Total attempts: 5

Which statistic (mean, median, or mode) is most appropriate in each of the following situations?

a. Tables in the dining hall are numbered 1 through 12 for students who eat there. The principal calls out a number for the table that will go through the buffet line first. The other tables follow in order of the table numbers. One student is sure the principal calls certain tables more often. She keeps track of which numbers are called over a 21-day period.

· Mean

· Median

· Mode

b. The offensive line of a football team is larger than in previous years. The program will list a statistic to show this fact.

· Mean

· Median

· Mode

c. A reporter is doing a story on the falling prices of homes in a large neighborhood. The reporter wants to demonstrate how the prices have fallen for all homes, not just the most expensive houses.

· Mean

· Median

· Mode

#5 Points possible: 25. Total attempts: 5

At a summer camp for kids, the leader asked all the kids their ages. The results were:

11 12 11 11 10 13 13 13 12 12 12 11 11 13 11 10 13 11 13 12 10 12

The data was summarized into a table. The second column shows the number of kids who are that age.

Age	Frequency
10	3
11	7
12	6
13	6

a. How many kids in the camp are 11 years old?

b. To calculate the mean, you could start by adding up all the ages: 11 + 12 + 11 + 11 + 10 + ···, but that would be tedious. Which of the following calculations would allow you to come up with the same result more quickly?

· 10(3)+11(7)+12(6)+13(6)10(3)+11(7)+12(6)+13(6)

· 10+11+12+1310+11+12+13

· 3+7+6+63+7+6+6

· None of the above

c. To figure out the total number of kids in the camp, you could count all the data values. Which of the following calculations would allow you to come up with the same result more quickly?

· 10(3)+11(7)+12(6)+13(6)10(3)+11(7)+12(6)+13(6)

· 10+11+12+1310+11+12+13

· 3+7+6+63+7+6+6

· None of the above

d. Determine the mean of the data. Round to 1 decimal place.

e. Determine the mode of the data.

#6 Points possible: 9. Total attempts: 5

Three descriptions of measures of central tendency are given below. They are labeled A, B, and C. Descriptions of data sets are listed below that. Match each data set with a description of measures of central tendency by writing the letter in the blank. Choices may be used more than once.

A The mean and median are close together.

B The mean is much higher than the median.

C The median is much higher than the mean.

a. The data have a large range, with only a few very high numbers, but most of the numbers are very small.

b. The data set has a large range with the numbers evenly spaced.

c. The data set has a small range with most of the numbers grouped in the middle.

#7 Points possible: 24. Total attempts: 5

If you lived in Canada in 2008, you might have seen the following headline:

“Canada Below G7 Average for Productivity!”

Here is some information to help you understand this headline.

Productivity is a way to measure the economy of a nation. One way to measure productivity is by Gross Domestic Product (GDP) per worker. You may recall from Lesson 2.8 that GDP is the value of all the goods and services produced in a country.

The G7 is a coalition of the major industrial democracies in the world: United States, United Kingdom, France, Germany, Italy, Canada, and Japan.

a. Which of the following is most likely what the author of the headline wanted the reader to think?

· Canada’s economy is weak and is falling behind other countries in the G7.

· Canada’s economy is strong and is leading other countries in the G7.

· Canada’s economy is very similar to other countries in the G7.

· Canada’s economy should not be compared to other countries.

b. Which of the following can you conclude from the headline?

· Canada is less productive than half of the G7 nations.

· There is at least one G7 nation that is more productive than Canada.

· There is at least one G7 nation that is less productive than Canada.

· None of the above.

A graph of the GDP per worker of the G7 nations is shown below.1