Deitz J.E., Southam J.L. Contemporary Business Mathematics for Colleges

Подождите немного. Документ загружается.

Burger King has sold billions of hamburgers. Housing prices are higher in Boston than

in Atlanta. The United States has a trade deficit, which means that the country has been

importing more goods than it has been exporting. Families tend to spend more money

in retail stores during December than during any other single month of the year. These

examples are based on collections of information about businesses. The information is

called business statistics. The word statistics also refers to a field of study that includes

the collection, organization, analysis, and presentation of data. Businesses use statistics for

two primary purposes: (1) to summarize and report the performance of the business

and (2) to analyze their options in making business decisions.

Individuals and groups who want information about the business performance of a

company include the company’s management, board of directors, investors, and govern-

ment agencies like the IRS. Once statistics have been reported, individuals and groups

use the statistics to make business decisions. For example, depending on the amount of

profits, the board of directors decides how much dividend to pay the shareholders. Like-

wise, after hearing about current profits and projected profits, investors decide whether

to purchase or sell shares of the company’s stock. After studying sales figures for its prod-

ucts and those of competitors, management makes decisions about which markets to en-

ter, what products to emphasize, and how to advertise.

If a Burger King analyst wants to report data on sales of hamburgers, she could list

the number of hamburgers sold at every restaurant. But Burger King has so many restau-

rants that there would be too many numbers to be meaningful. To make the data mean-

ingful, the analyst can make some summary calculations and/or organize the data in

tables. To make her presentations of the data more meaningful and easier to interpret,

she may draw charts, diagrams, and/or graphs.

494 Part 6 Corporate and Special Applications

24.1 This chapter is independent of all

other chapters except Chapter 8 where

cost of goods sold, expenses,and profit

were mentioned.

24.2 You might remind students that

the word data is the plural of datum.

It is worth mentioning because of the

effect that it has on grammar.

The objective in reporting statistics is to summarize the data in a simple, yet meaningful,

manner. One way to simplify data is to compute an average. An average is a single num-

ber that is supposed to be “typical” or “representative” of a group of numbers. One

common way to find an average is to add all the data values and divide by the number

of values. In statistics, this particular average is called the mean. When the mean isn’t

typical or representative of an entire group of data, another average might be more rep-

resentative. We also discuss two other averages: the median and the mode.

The mean of a group of values is computed by dividing the sum of the group of

values by the number of values in the group.

EXAMPLE A

Find the mean salary of five employees whose actual salaries are $51,500, $54,400,

$57,600, $62,000, and $64,500.

Sum 5 $51,500 1 $54,400 1 $57,600 1 $62,000 1 $64,500 5 $290,000

Mean 5 $290,000 4 5 5 $58,000

Statistical Averages: Computing the Mean

Learning Objective

Compute the mean.

24.3 The word average is often used

in an imprecise way as a synonym for

the word typical,as in “What kind of

calculator does the average student

use?”The word average has many

meanings,three of which are mean,

median,and mode.

24.4 Bring in—or have students

bring in—examples of news articles

that discuss salaries or housing prices.

Most often the article will mention

the median rather than the mean.One

or two very expensive homes can make

the mean home price too high to be

typical for the average home.

Chapter 24 Buisness Statistics 495

Find the mean for the following set of numbers: 14, 11, 12, 15, 10, 16, 15, 12, 13, 11, 15, 17, 13,

14, 15, 12, 18

There are 17 numbers. The mean equals their sum divided by 17.

Sum 5 233

Mean 5 233 4 17 5 13.706, or 13.7 rounded to one decimal place

✔

CONCEPT CHECK 24.1

Determine the median.

Learning Objective

The median of a group of numbers is determined by arranging the numbers in numeri-

cal order and finding the middle number. The median is useful when one value in the

group is much larger or much smaller than the rest of the numbers.

EXAMPLE B

Find the median salary of five employees whose salaries are $51,500, $54,400, $57,600,

$62,000, and $254,500.

The salaries are already in numerical order; the median is $57,600 because it is the

middle number of the five numbers arranged in order.

In example B, the mean is $480,000 4 5 5 $96,000, but $96,000 is not representative of

the salaries of the five employees. The mean is very large because one employee (perhaps

the owner) has a very large salary compared to the rest of the group. The median salary,

$57,600, is more typical of the group.

If the number of values is even, the median will be halfway between the two middle

values. The median will be the mean of the middle two values.

EXAMPLE C

Find the median salary of six employees whose salaries are $57,600, $64,500, $51,500,

$254,500, $62,000, and $54,400.

Rearranged in numerical order, the salaries are $51,500, $54,400, $57,600, $62,000,

$64,500, and $254,500.

The median is halfway between the middle two numbers, $57,600 and $62,000. The

median is ($57,600 1 $62,000) 4 2, or $119,600 4 2 5 $59,800.

Determining the Median

Find the median for the following set of numbers: 14, 11, 12, 15, 10, 16, 15, 12, 13, 11, 15, 17,

13, 14, 15, 12, 18

The median is the middle number, after all the numbers have been arranged by order of size:

10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 16, 17, 18

The median is the ninth number, or 14.

✔

CONCEPT CHECK 24.2

The mode of a group of numbers is the number that occurs most often. None of exam-

ples A, B, and C has a mode because each number occurs only once. The mode is useful

when the word average implies “most typical” or “happening most often.” Retail busi-

nesses keep track of the items that sell most frequently so that they can avoid shortages

of those items.

EXAMPLE D

Find the mode shoe size of 12 pairs of cross trainer running shoes, sizes 6, 6, , , 8, ,

9, 9, 9, 9, 9, and .

The mode is size 9, because 9 occurs most frequently.

In example D neither the mean nor the median makes any sense. The mean is 98 4 12 5

8.17, or . The median is halfway between sizes and 9, which would be 8.75, or . The

store owner could not buy any shoes in either size or size because those shoe sizes

do not exist. However, the store owner does want to stock several shoes in size 9.

496 Part 6 Corporate and Special Applications

Determining the Mode

Learning Objective

Determine the mode.

Learning Objective

Construct frequency tables.

Find the mode for the following set of numbers: 14, 11, 12, 15, 10, 16, 15, 12, 13, 11, 15, 17, 13,

14, 15, 12, 18

The mode is the number that occurs most often. It is easier to find if you arrange the numbers

by size first:

10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 16, 17, 18

There are four 15s, so the mode is 15.

✔

CONCEPT CHECK 24.3

The data in examples A–D are sometimes called ungrouped data because the numbers

are listed individually. Business applications, such as sales results for all Burger King

restaurants, often involve hundreds or thousands of numbers. Interpreting data that

are literally pages of raw numbers is impossible. To make sense of such data, we organize

the individual values into groups called classes of data or data classes. Adjacent classes

“touch each other,” but cannot overlap, not even by one cent. Also, classes are normally

the same width. In example E, the width of each class is $5,000. The number of values

in each class, called the frequency of the class, is summarized in a table called a

frequency table.

Constructing Frequency Tables

EXAMPLE E

Listed below are the salaries of 25 full-time office employees of a large insurance company.

Make a frequency table with five classes: $40,000 up to but not including $45,000, $45,000

up to but not including $50,000, and so on.

$42,500 $41,300 $53,500 $62,400 $47,500

45,400 54,600 41,000 44,400 59,100

48,000 52,000 57,500 62,500 44,000

53,600 46,200 53,500 51,800 56,400

55,500 46,000 45,200 46,000 60,800

The frequency table for these salaries appears in Figure 24-1.

Chapter 24 Business Statistics 497

to Develop a Frequency Table

1. Determine the classes of data, and list the classes in one column.

2. Tally the data by making one mark for each data item in the column next

to the appropriate class.

3. Count the tally marks for each class and write the number in the column

next to the tally marks.

STEPS

24.5 Continue to emphasize the exact

words of example E:“$40,000 up to

$45,000”means “from $40,000 up to

but not including $45,000.”The class

could also be defined as “$40,000.00

up to and including $44,999.99.”This,

however,is somewhat more awkward

and appears more complicated.We will

try to avoid difficulties by not using

data values,like $45,000 or $50,000,

which are exactly on the boundaries.

Class Tally Frequency (F)

$40,000 up to $45,000 5

$45,000 up to $50,000 7

$50,000 up to $55,000 6

$55,000 up to $60,000 4

$60,000 up to $65,000 13

Total 25

III

IIII

IIII II

IIII

Figure 24-1 Frequency Table

COMPUTING THE MEAN OF LARGE DATA SETS

When a data set contains many numbers, as in example E, a computer spreadsheet is

usually used to compute the mean. If you use a calculator, be sure to check your work.

One way to do so is to add all the numbers twice. One method to add them twice, but in

two different orders, is shown in the following steps.

to Compute the Mean for a Large Data Set

1. Add all the numbers in each column.

2. Add all the numbers in each row.

3. Compute the grand total by adding all the column totals.

4. Check the grand total by adding all the row totals.

5. Divide the grand total by the number of values to get the mean.

STEPS

EXAMPLE F

Compute the mean of the 25 salaries in example E.

$ 42,500 $ 41,300 $ 53,500 $ 62,400 $ 47,500 $ 247,200

45,400 54,600 41,000 44,400 59,100 244,500

48,000 52,000 57,500 62,500 44,000 264,000

53,600 46,200 53,500 51,800 56,400 261,500

1 55,500 1 46,000 1 45,200 1 46,000 1 60,800 1 253,500

$245,000 $240,100 $250,700 $267,100 $267,800 $1,270,700

The sum of the row totals and the sum of the column totals are both $1,270,700.

Mean 5 $1,270,700 4 25 5 $50,828

498 Part 6 Corporate and Special Applications

24.6 In statistics textbooks,authors

often distinguish between class limits

and class boundaries.We do not

mention either.Class limits are the

smallest and largest possible values in

a class.In the first class,the lower class

limit is $40,000.00;the upper class

limit is $44,999.99.The class boundary

is the number that separates two con-

secutive classes.The class boundary

between the first two classes is

$44,999.995,which is halfway

between the upper class limit of

$44,999.99 and the adjacent lower

class limit of $45,000.00.

2,550 3,275 3,410 2,650 3,140

3,480 3,400 2,860 3,810 3,480

1,660 3,280 2,940 2,480 3,325

1,975 4,270 3,520 2,440 2,325

4,110 3,300 2,290 4,140 3,990

2,570 2,150 2,840 4,325 2,720

✔

Make a frequency table for the following set of data. Use the classes 1,500 up to 2,000, 2,000 up to 2,500, and so on.

CONCEPT CHECK 24.4

Class Tally Frequency

1,500 up to 2,000 2

2,000 up to 2,500 5

2,500 up to 3,000 7

3,000 up to 3,500 9

3,500 up to 4,000 3

4,000 up to 4,500 + 4

Total 30

IIII

III

IIII

IIII II

IIII

COMPLETE ASSIGNMENT 24.1.

In business, statistical information is first summarized clearly in tables. For presentation,

the results are then often displayed in charts or graphs. Popular graphs include the his-

togram, the bar graph, the line graph, and the pie chart (circle graph). Histograms, bar

graphs, and line graphs all have perpendicular axes. Labels are placed at the left (the ver-

tical axis) and bottom (the horizontal axis).

A histogram is a diagram that presents the grouped data from a frequency table. The

classes are positioned adjacent to each other along the horizontal axis, and the frequen-

cies are written along the vertical axis. Figure 24-2 shows the histogram for the frequency

table in Figure 24-1. The numbers on the horizontal axis increase from left to right. The

numbers on the vertical axis increase from bottom to top.

Charts and Graphs: Constructing Histograms

Learning Objective

Construct histograms.

24.7 Remind students of the old adage

“A picture is worth a thousand words.”

There are over 100,000 references to it

on Google.Many students will claim

they are not “math-oriented.”Explain

to them that graphs are the artistic

method of conveying the numeric

information from the tables.Some

people who are less “number-oriented”

may be more “graph-oriented.”

A bar graph, or bar chart, resembles the histogram except that there may not be a numeric

scale on the horizontal axis and the bars normally do not touch each other. Sosa’s Markets

has grocery stores in four different towns: Davis, Hubbard, Bay View, and Easton, although

the Davis store just opened last year in July. The table in Figure 24-3 shows the annual sales

revenue, cost of goods sold, operating expenses, and net profits for both the current year

and the previous year. The bar graph in Figure 24-4 illustrates the data from the current

year. Data from the table in Figure 24-3 are used in various examples throughout the

remainder of this chapter.

Note: It does not make sense to have the vertical bars “touch each other” as in a his-

togram. The four stores are distinct objects. If the horizontal axis were “time,” like con-

secutive months of the year, then you could make a bar graph. But it would also make

Chapter 24 Business Statistics 499

24.8 Remind students that in a his-

togram,adjacent classes “touch,”but

do not overlap.Technically the number

that actually separates the first two

rectangles in Figure 24-2 is the ”class

boundary,”$44,999.995.(See Note

24.6.) However,it is more effective and

less confusing on a graph to use simply

a “45.”The label at the bottom explains

that 45 represents $45,000.

lass Frequenc

1,500 up to 2,000 2

2,000 up to 2,500 5

2,500 up to 3,000 7

3,000 up to 3,500 9

3,500 up to 4,000 3

4,000 up to 4,500 1 4

Total 30

✔

Construct a histogram from the following frequency table.

CONCEPT CHECK 24.5

Figure 24-2 Histogram

35 40 45 50 55 60 65 70

Number of

Employees

Employee Salaries (thousands of dollars)

Construct bar graphs.

Learning Objective

Constructing Bar Graphs

24.9 Bar graphs can be drawn

horizontally.Vertical bars are more

common,especially in comparative or

component bar graphs.

1,500 2,000 2,500 3,000 3,500 4,000 4,500

Histogram

sense to use a histogram because last year could touch this year at midnight on December 31.

However, as you will see, we can make some useful variations of bar graphs that we really

cannot do with histograms.

500 Part 6 Corporate and Special Applications

SOSA’S MARKETS SALES DATA FOR THE

CURRENT YEAR (IN MILLIONS OF DOLLARS)

Sales Cost of Operating Net Profit Net Profit

Location Revenue Goods Sold Expenses (This Year) (Last Year)

Davis 1.50 0.75 0.50 0.25 0.15

Hubbard 3.25 1.75 1.00 0.50 0.75

Bay View 2.00 1.00 0.75 0.25 0.50

Easton 4.00 2.00 1.25 0.75 0.50

Figure 24-3 Revenues, Expenses, and Net Profits (in millions of dollars)

Figure 24-4 Bar Graph

4.0

3.0

2.0

1.0

Davis

Dollars

(millions)

Hubbard

Sosa's Markets Sales Revenues—Current Year

Bay View

Easton

GETTY IMAGES

COMPARATIVE BAR GRAPH

Two bar graphs can be combined on one grid to make a comparative bar graph. This

permits the statistician to make a graph that will compare two different sets of data. The

graph for Sosa’s Markets in Figure 24-5 compares each store’s net profit this year with its

net profit last year. Each store has one pair of bars and the two bars in each pair need to

be colored or shaded differently to help the reader distinguish between the two years.

COMPONENT BAR GRAPH

A bar graph constructed to show how certain data are composed of various segments is a

component bar graph. Figure 24-6 shows how the current sales revenue is composed of

cost of goods sold, operating expenses, and net profit. As in the comparative bar graph,

the component parts are colored or shaded differently to permit easier reading.

Chapter 24 Business Statistics 501

Figure 24-5 Comparative Bar Graph

Davis

Dollars

(millions)

Hubbard

Sosa's Markets Net Profits—Current Year and Last Year

Bay View

Easton

Last Year

This Year

1.0

0.75

0.50

0.25

Figure 24-6 Component Bar Graph

Davis

Dollars

(millions)

Hubbard

Sosa's Markets Sales Revenues—Current Year

Bay View

Easton

Net Profit

Cost of Goods Sold

Operating Expenses

4.0

3.0

2.0

1.0

24.10 Discuss with students how

difficult,misleading,or confusing it

would be to try to use a histogram to

illustrate the same information as

either a comparative bar graph or a

component bar graph.

502 Part 6 Corporate and Special Applications

Home Sales L

ast Year

Quart

er 1st 2nd 3rd 4th

Shopping Mall Office 20 60 40 30

Downtown Office 5 20 25 10

Mountain Office 110 13 15 112

Total sales last year 35 83 70 52

Total sales prior year 30 75 65 55

a. Construct a bar graph showing total home sales for each quarter

last year. Make the vertical scale from 0 to 100, and mark the four

quarters on the horizontal scale.

✔

A real estate agency has three offices, all of which sell some homes.The Shopping Mall Office sells homes almost

exclusively; last year it sold 150 homes.The Downtown Office handles mostly commercial property, but it sold

60 homes last year. The Mountain Office primarily manages various resort properties, but it did sell 30 homes.

Following are the numbers of homes sold in each quarter of last year. The first quarter is January through March;

the second quarter is April through June; the third quarter is July through September; and the fourth quarter is

October through December.

CONCEPT CHECK 24.6

100

st 2nd

Home Sales by Quarter—Last Year

rd 4th

b. Construct a comparative bar graph showing total home

sales for each quarter, both last year and the prior year.

c. Construct a component bar graph showing quarterly

home sales for each office last year.

100

st 2nd

Home Sales Last Year vs. Prior Year

rd 4th

Last Year Prior Year

100

st 2nd

Home Sales by Office—Last Year

rd 4th

Shopping Mall

Downtown

Mountain

Businesses often analyze data over time, perhaps monthly or annually. As we mentioned

earlier, both a histogram and a bar graph can be used when time is on the horizontal axis.

Another useful graph for illustrating data over time is the line graph. Plot the heights with

single points above each month (or year). Then connect the consecutive points with straight

line segments. Notice that it would not make sense to put time on the vertical axis.

Constructing Line Graphs

Learning Objective

Construct line graphs.

Following are five months of expenses for materials for the residential and commer-

cial divisions of Solar Metals, Inc., a custom metal-fabricating business.

Jan. Feb. Mar. Apr. May

Residential 24,000 30,000 26,000 36,000 32,000

Commercial 46,000 40,000 50,000 46,000 54,000

Figure 24-7 shows a comparative bar graph for this data. Figure 24-8 shows two line graphs

with one line for the Residential Division and the other for the Commercial Division.

Chapter 24 Business Statistics 503

Figure 24-7 Comparative Bar Graph

Residential

Commercial

Jan. Feb. Mar.

Apr.

May

Dollars

(thousands)

Solar Metals—Materials Expense

Figure 24-8 Line Graph

Jan. Feb. Mar.

April

May

Solar Metals—Materials Expense

Commercial

Residential

Dollars

(thousands)

As we mentioned earlier, there is not a convenient, unconfusing method to make one

histogram show all of the information. If you simply take the comparative bar graph, but

draw the vertical bars all adjacent, the result is NOT a histogram. Histograms are simply

not normally used for this kind of data. Histograms for the Residential and Commercial

Divisions are shown in Figures 24-9 and 24-10. Their only purpose here is for you to see

that the line graph and the comparative bar graph are much better suited to illustrate the

differences in the data.