8.1 means, medians, and modes - mcgraw hill...

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Means, Medians, and Modes8.1

8.1 OBJECTIVES

1. Calculate the mean2. Interpret the mean3. Find the median4. Interpret the median5. Find a mode

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A very useful concept is the average of a group of numbers. An average is a number that istypical of a larger group of numbers. In mathematics we have several different kinds of av-erages that we can use to represent a larger group of numbers. The first of these is the mean.

To find the mean for a group of numbers, follow these two steps:

Step 1 Add all of the numbers in the group.Step 2 Divide that sum by the number of items in the group.

Step by Step: Finding the Mean

Example 1

Finding the Mean

Find the mean of the group of numbers 12, 19, 15, and 14.

Step 1 Add all of the numbers.

12 � 19 � 15 � 14 � 60

Step 2 Divide that sum by the number of items.

60 � 4 � 15 There are four items in this group.

The mean of this group of numbers is 15.

C H E C K Y O U R S E L F 1

Find the mean of the group of numbers 17, 24, 19, and 20.

Let’s apply the mean to a word problem.

Example 2

Finding the Mean

The ticket prices (in dollars) for the nine concerts held at the Civic Arena this school yearwere

33, 31, 30, 59, 30, 35, 32, 36, 56

What was the mean price for these tickets?

Step 1 Add all the numbers.

33 � 31 � 30 � 59 � 30 � 35 � 32 � 36 � 56 � 342

616 CHAPTER 8 DATA ANALYSIS AND STATISTICS

Step 2 Divide by 9.

342 � 9 � 38

The mean ticket price was $38.

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The costs (in dollars) of the six textbooks that Aaron needs for the fall quarter are

75, 69, 57, 87, 76, 80

Find the mean cost of these books.

Finding the Median

Find the median for the following groups of numbers.

(a) 35, 18, 27, 38, 19, 63, 22

Step 1 Rewrite the numbers in order from smallest to largest.

18, 19, 22, 27, 35, 38, 63

Step 2 Count from both ends to find the number in the middle.

Counting from both ends, we find that 27 is the median. There are three numbers above27 and three numbers below it.

(b) 29, 88, 74, 81, 62, 37


29, 37, 62, 74, 81, 88


Counting from both ends, we find that there are two numbers in the middle, 62 and 74.We go on to step 3.

Although the mean is probably the most common way to find an average for a groupof numbers, it is not always the most representative. Another kind of average is called themedian.

The median is the number for which there are as many instances that are abovethat number as there are instances below it. To find the median, follow thesesteps:

Step 1 Rewrite the numbers in order from smallest to largest.Step 2 Count from both ends to find the number in the middle.Step 3 If there are two numbers in the middle, add them together and find

their mean.

Step by Step: Finding the Median

Example 3

NOTE Divide by 9 becausethere are 9 ticket prices.

MEANS, MEDIANS, AND MODES SECTION 8.1 617

Step 3 If there are two numbers in the middle, find their mean.

(62 � 74) � 2 � 136 � 2 � 68

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Find the median for each group of numbers:

(a) 8, 6, 19, 4, 21, 5, 27 (b) 43, 29, 13, 37, 29, 53


The following are Jessica’s phone bills for each month of 2000.

26, 67, 31, 24, 15, 17, 41, 27, 17, 22, 26, 47

(a) Find the mean amount of her phone bills.

(b) Find the median amount of her phone bills.

There are times in which the median is a better representative of a group of numbersthan the mean is. Example 4 illustrates such a case.

Example 4

Comparing the Mean and the Median

The following numbers represent the hourly wage of seven employees of a local chip manu-facturing plant.

12, 11, 14, 16, 32, 13, 14

(a) Find the mean hourly wage.

Step 1 Add all of the numbers in the group.

12 � 11 � 14 � 16 � 32 � 13 � 14 � 112

Step 2 Divide that sum by the number of items in the group.

112 � 7 � 16

The mean wage is $16 an hour.

(b) Find the median wage for the seven workers.


11, 12, 13, 14, 14, 16, 32


The middle number is 14. There are three numbers above it and three numbers belowit. The median salary is $14 per hour. Which salary do you think is more typical of theworkers? Why?


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Example 5

Finding a Mode

Find the mode for the set of numbers given.

22, 24, 24, 24, 24, 27, 28, 32, 32

The mode, 24, is the number that appears most frequently.


Find the mode for the set of numbers given.

7, 7, 7, 9, 11, 13, 13, 15, 15, 15, 15, 21


The following types of computers were available in the lab. Which type was themode?

Apple, IBM, Compaq, Dell, Apple, IBM, Apple, Compaq, Dell, Apple, IBM, Apple, Dell, Apple, Compaq

One advantage of the mode is that it can be used with data that are not a set of numbers.

Example 6

Finding a Mode

Following are the eye colors from a class of 12 students. Which color is the mode?

blue, brown, hazel, blue, brown, brown, brown, brown, blue, brown, hazel, green

Because brown occurs most frequently, it is the mode.

C H E C K Y O U R S E L F A N S W E R S

1. 20 2. $74 3. (a) 8; (b) 33 4. (a) $30; (b) $26 5. 15 6. Apple

The mode of a set of data is the item or number that appears most frequently.

Definitions: Mode

Another measure used as an “average” is called the mode.

NOTE A set with two differentmodes is called bimodal.

NOTE If each eye colorappeared three times, therewould be no mode! Not everydata set has a mode.

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Exercises

Find the mean for each set of numbers.

1. 6, 9, 10, 8, 12 2. 13, 15, 17, 17, 18

3. 13, 15, 17, 19, 24, 26 4. 41, 43, 56, 67, 69, 72

5. 12, 14, 15, 16, 16, 16, 17, 22, 25, 27 6. 21, 25, 27, 32, 36, 37, 43, 44, 44, 51

7. 5, 8, 9, 11, 12 8. 7, 18, 11, 7, 12

9. 9, 8, 11, 14, 8 10. 21, 23, 25, 27, 22, 20

Find the median for each set of numbers.

11. 2, 3, 5, 6, 10 12. 12, 13, 15, 17, 18

13. 23, 24, 27, 31, 36, 38, 41 14. 1, 4, 9, 16, 25, 36, 49

15. 46, 13, 47, 25, 68, 51, 71 16. 26, 71, 33, 69, 71, 25, 75

Find the mode for each set of numbers.

17. 17, 13, 16, 18, 17 18. 41, 43, 56, 67, 69, 72

19. 21, 44, 25, 27, 32, 36, 37, 44 20. 9, 8, 10, 9, 9, 10, 8

21. 12, 13, 7, 14, 4, 11, 9 22. 8, 2, 3, 3, 4, 9, 9, 3

Solve the following applications.

23. Temperature. High temperatures of 86°, 91°, 92°, 103°, and 98° were recorded forthe first 5 days of July. What was the mean high temperature?

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24. Travel. A salesperson drove 238, 159, 87, 163, and 198 miles (mi) on a 5-day trip.What was the mean number of miles driven per day?

25. Mileage rating. Highway mileage ratings for seven new diesel cars were 43, 29, 51,36, 33, 42, and 32 miles per gallon (mi/gal). What was the mean rating?

26. Enrollments. The enrollments in the four elementary schools of a district are 278,153, 215, and 198 students. What is the mean enrollment?

27. Test scores. To get an A in history, you must have a mean of 90 on five tests. Yourscores thus far are 83, 93, 88, and 91. How many points must you have on thefinal test to receive an A? (Hint: First find the total number of points you need toget an A.)

28. Test scores. To pass biology, you must have a mean of 70 on six quizzes. So far yourscores have been 65, 78, 72, 66, and 71. How many points must you have on the finalquiz to pass biology?

29. Test scores. Louis had scores of 87, 82, 93, 89, and 84 on five tests. Tamika hadscores of 92, 83, 89, 94, and 87 on the same five tests. Who had the higher meanscore? By how much?

30. Heating bills. The Wong family had heating bills of $105, $110, $90, and $67 in thefirst 4 months of 1999. The bills for the same months of 2000 were $110, $95, $75,and $76. In which year was the mean monthly bill higher? By how much?

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Monthly energy use, in kilowatt-hours (kWh), by appliance type for four typical U.S.families is shown below.

ANSWERS

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Wong McCarthy Abramowitz GreggFamily Family Family Family

Electric range 97 115 80 96Electric heat 1200 1086 1103 975Water heater 407 386 368 423Refrigerator 127 154 98 121Lights 75 99 108 94Air conditioner 123 117 96 120Color TV 39 45 21 47

31. Heating. What is the mean number of kilowatt-hours used each month by the fourfamilies for heating their homes?

32. Heating. What is the mean number of kilowatt-hours used each month by the fourfamilies for hot water?

33. Heating. What is the mean number of kilowatt-hours used per appliance by theMcCarthy family?

34. Heating. What is the mean number of kilowatt-hours used per appliance by theGregg family?

Use your calculator for exercises 35 and 36.

35. Utility bills. Fred kept the following records of his utility bills for 12 months: $53,$51, $43, $37, $32, $29, $34, $41, $58, $55, $49, and $58. What was the meanmonthly bill?

36. Test scores. The following scores were recorded on a 200-point final examination:193, 185, 163, 186, 192, 135, 158, 174, 188, 172, 168, 183, 195, 165, 183. What wasthe mean of the scores?

37. The following are eye colors from a class of eight students. Which color is the mode?

Hazel, green, brown, brown, blue, green, hazel, green

38. The weather in Philadelphia over the last seven days was as follows:

Rain, sunny, cloudy, rain, sunny, rain, rain. What type of weather was the mode?

39. List the advantages and disadvantages of the mean, median, and mode.

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40. In a certain math class, you take four tests and the final, which counts as two tests.Your grade is the average of the six tests. At the end of the course, you compute boththe mean and the median.

(a) You want to convince the teacher to use the mean to compute your average. Writea note to your teacher explaining why this is a better choice. Choose numbersthat make a convincing argument.

(b) You want to convince the teacher to use the median to compute your average.Write a note to your teacher explaining why this is a better choice. Choosenumbers that make a convincing argument.

41. Create a set of five numbers such that the mean is equal to the median.

42. Create a set of five numbers such that the mean is greater than the median.

43. Create a set of five numbers such that the mean is less than the median.

Answers1. 9 3. 19 5. 18 7. 9 9. 10 11. 5 13. 31 15. 4717. 17 19. 44 21. No mode 23. 94° 25. 38 mi/gal27. 95 points 29. Louis’s mean score was 87, Tamika’s was 89. Tamika’s averagescore was 2 points higher than Louis’ 31. 1091 kWh 33. 286 kWh35. $45 37. Green 39.

41. Answer will vary 43. Answer will vary

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Using Your Calculator to Findan Average

Most electronic calculators have statistical functions that allow you to calculate means, me-dians, and other statistical values. Because these features vary so much from one calcula-tor to another, you will need to consult your owner’s manual to learn how to access thesefeatures.

In this section, we will focus on using a calculator to compute the mean.

Example 1

Calculating a Mean

Find the mean for the set of numbers below.

45, 48, 53, 59, 67, 76

When entering these numbers in your calculator, keep in mind the order of operations.There are two different techniques you may use.

Method 1

Add the numbers

45 � 48 � 53 � 59 � 67 � 76 � 348

then divide the sum by 6 (there are six numbers)

348 � 6 � 58

Method 2

Use parentheses to find the mean.

(45 � 48 � 53 � 59 � 67 � 76) � 6 � 58


Find the mean for the set of numbers below.

132, 144, 156, 158, 279, 337

Unfortunately, not all calculations result in answers that are whole numbers.

Example 2

Finding a Mean

Find the mean of the set of numbers below.

13, 15, 16, 19, 26, 28, 38

Entering this as a single expression, we have

(13 � 15 � 16 � 19 � 26 � 28 � 38) � 7 � 22 Your calculator probably has a display something like 22.14285714. Remember that this is just another (although closer) approximation.


We could approximate the answer as 22, as we’ve indicated. We could also subtract 22 fromthe calculator display, leaving only the decimal approximation, approximately

0.142857142

If we multiply this by the divisor, 7, we will get the remainder, which is 1. The exactanswer is

221

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Find an approximate and exact average for the set of numbers below.

17, 23, 33

C H E C K Y O U R S E L F A N S W E R S

1. 201 2. Approximately 24, exactly 241

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Calculator ExercisesIn exercises 1 to 10, find the mean for each set of numbers.

1. 48, 50, 51, 52, 49, 50 2. 20, 18, 17, 24, 22, 19

3. 108, 113, 109, 113, 110, 101, 112, 114 4. 211, 213, 215, 208, 209, 220, 215, 221

5. 1560, 1540, 1570, 1555, 1565, 1545, 1557

6. 346, 351, 353, 347, 341, 382, 373, 363

7. 2357, 2361, 2372, 2371, 2357, 2375, 2364, 2371

8. 16,430, 16,211, 16,149, 16,232, 16,317, 16,113

9. 24,637, 24,251, 24,454, 24,580, 24,324, 24,478

10. 311,431, 286,356, 356,090, 292,007, 301,857, 299,005

In exercises 11 to 16, find the approximate and exact mean for each set of numbers.

11. 18, 21, 20, 22 12. 36, 41, 43, 39, 40, 37, 39

13. 125, 121, 129, 126, 128, 123 14. 356, 371, 366, 373, 359, 363

15. 1898, 1913, 1875, 1937 16. 15,865, 16,270, 16,090, 15,904

17. The revenue for the leading apparel companies in the United States in 1997 is givenin the following table.

What is the mean revenue taken in by these companies?

Company Revenue (in millions)

Nike $9187Vanity Fair $5222Liz Claiborne $2413Reebok $3637Fruit of the Loom $2140Nine West $1865Kellwood $1521Warmaio $1437Jones Apparel $1387

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ANSWERS

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18.

Find the mean number of employed and unemployed per year from 1989 to 1997.

The work stoppages (strikes and lockouts) in the United States from 1990 to 1997 aregiven in the following table.

19. Find the mean number of work stoppages per year from 1990 to 1997.

20. Find the mean number of work days idle from 1990 to 1997.

Answers

1. 50 3. 110 5. 1556 7. 2366 9. 24,454 11. 20;

13. 125; 15. 1906; 17. $3,201,000,000 19. 28119053

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Year No. of Stoppages Work Days Idle

1990 185 59261991 392 45841992 364 39891993 182 39811994 322 50201995 192 57711996 273 48891997 339 4493

Unemployment in the United States (in thousands)

Year Employed Unemployed

1989 . . . . . . . . . . . . . . . . . . . . 117,342 6,5281990 . . . . . . . . . . . . . . . . . . . . 118,793 7,0471991 . . . . . . . . . . . . . . . . . . . . 117,718 8,6281992 . . . . . . . . . . . . . . . . . . . . 118,482 9,6131993 . . . . . . . . . . . . . . . . . . . . 120,259 8,9401994 . . . . . . . . . . . . . . . . . . . . 123,060 7,9961995 . . . . . . . . . . . . . . . . . . . . 124,900 7,4041996 . . . . . . . . . . . . . . . . . . . . 126,708 7,2361997 . . . . . . . . . . . . . . . . . . . . 129,558 6,739

8.1 means, medians, and modes - mcgraw hill...

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