Download Business Statistics

7.1 Measures of Central Tendency

Find the mean

Find the median
9
8

Find the mode
7
6
5

Make and interpret a
frequency distribution
4
3
2
1

Find the mean of
grouped data
Business Math, Eighth Edition
Cleaves/Hobbs
0
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Key Terms

Data set: a collection of values or
measurements that have a common
characteristic.

Statistic: a standardized, meaningful measure
of a set of data that reveals a certain feature or
characteristic of the data.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Key Terms

Mean: the arithmetic average of a set of
data or sum of the values divided by the
number of values.

Median: the middle value of a data set
when the values are arranged in order of
size.

Mode: the value or values that occur most
frequently in a data set.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
7.1.1 Find the Mean

A business records its daily sales. These
values are an example of a data set.
Data sets can be used to:
 Observe patterns
 Interpret information
 Make predictions about future activity
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Find the mean of a data set
1. Find the sum of the values.
2. Divide the sum by the total number of
values.
Mean =
Business Math, Eighth Edition
Cleaves/Hobbs
sum of values
number of values
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Here’s an example
Sales figures for the last week for the Western
Region have been as follows:

Monday
$4,200
Tuesday
$3,980
Wednesday $2,400
Thursday
$3,100
Friday
$4,600
What is the average daily sales figure?

(4,200 + 3,980 + 2,400 + 3,100 + 4,600) ÷ 5 = 3,656





Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Try these examples

Mileage for the new salesperson has been
243, 567, 766, 422 and 352 this week.
What is the average number of miles
traveled?
 470 miles daily

Prices from different suppliers of 500 sheets
of copier paper are as follows: $3.99, $4.75,
$3.75 and $4.25. What is the average
price?
 $4.19
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
7.1.2 Find the Median

Arrange the values in the data set from
smallest to largest (or largest to smallest)
and select the value in the middle.

If the number of values is odd, it will be
exactly in the middle.

If the number of values is even, identify the
two middle values. Add them together and
divide by two.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Here is an example

A recent survey of the used car market for the particular
model John was looking for yielded several different
prices: $9,400, $11,200, $5,900, $10,000, $4,700,
$8,900, $7,800 and $9,200. Find the median price.

Arrange from highest to lowest:
$11,200, $10,000, $9,400, $9,200, $8,900, $7,800,
$5,900 and $4,700.

Calculate the average of the two middle values.

(9,200 + 8,900) ÷ 2 = $9,050 or the median price
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Try this example

Five local moving companies quoted the following
prices to Bob’s Best Company: $4,900, $3800,
$2,700, $4,400 and $3,300. Find the median
price.
 $3,800
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
7.1.3 Find the Mode

Find the mode in a data set by counting the
number of times each value occurs.

Identify the value or values that occur most
frequently.

There may be more than one mode if the same
value occurs the same number of times as
another value.

If no one value appears more than once, there is
no mode.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Find the mode in this data set

Results of a placement test in mathematics
included the following scores:
65, 80, 90, 85, 95, 85, 80, 70 and 80.

Which score occurred the most frequently?

80 is the mode. It appeared three times.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Use the mean, median and mode

A university recruiter is evaluating the number of
community service hours performed by ten
students who are applying for a job on campus.

Observe the mean, median and mode from this
data set and determine which one or ones might
help the recruiter the most in making a realistic
assessment of the number of service hours
performed last semester.
(see next slide)
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
How many hours?










Jack:
Michelle:
Bill:
Jackie:
Jason:
Larissa:
Tony:
Melanie:
Art:
Sheila:
Business Math, Eighth Edition
Cleaves/Hobbs
10
14
5
2
20
12
2
18
1
0

The mode is 2

The mean is 8.4

The median is 7.5

Of the three values,
which one or one(s)
would help you make a
realistic assessment of
the number of service
hours?

Why?
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
7.1.4 Make and Interpret a
Frequency Distribution



Identify appropriate intervals for the data.
Tally the data for the intervals.
Count the number in each interval.
90
80
70
60
50
40
East
West
North
30
20
10
0
1st Qtr
Business Math, Eighth Edition
Cleaves/Hobbs
2nd Qtr
3rd Qtr
4th Qtr
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Key Terms

Class intervals: special categories for
grouping the values in a data set.

Tally: a mark that is used to count data in class
intervals.

Class frequency: the number of tallies or
values in a class interval.

Grouped frequency distribution: a
compilation of class intervals, tallies, and class
frequencies of a data set.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Look at this example using the
same test scores

Test scores on the last math test were as
follows:
78 84 95 88 99 92 87 94 90 77

Make a relative frequency distribution
using intervals of 75-79, 80-84, 85-89, 9094, and 95-99.
(see next slide)
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Look at this example
Class
Interval
75-79
80-84
85-89
90-94
95-99
Total
78 84 95 88 99 92 87 94 90 77
Class
Relative
Frequency
Calculations
Frequency
2
2/10
20%
1
1/10
10%
2
2/10
20%
3
3/10
30%
2
2/10
20%
10
10/10
100%
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
7.1.5 How to Find the Mean of
Grouped Data





Make a frequency distribution.
Find the products of the midpoint of the
interval and the frequency for each
interval for all intervals.
Find the sum of the frequencies.
Find the sum of the products from step 2.
Divide the sum of the products by the
sum of the frequencies.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Look at this example using the
same test scores

Test scores on the last math test were as
follows:
78 84 95 88 99 92 87 94 90 77

Make a relative frequency distribution
using intervals of 75-79, 80-84, 85-89, 9094, and 95-99.
(see next slide)
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Look at this example
78 84 95 88 99 92 87 94 90 77
Class
Class
Interval Frequency
75-79
2
80-84
1
85-89
2
90-94
3
95-99
2
Total
10
Midpoint
77
82
87
92
97
Product of
Midpoint and
Frequency
154
82
174
276
194
880
Mean of the grouped data: 880 ÷ 10 = 88
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
7.2 Graphs and Charts

Interpret and draw a bar graph.

Interpret and draw a line graph.

Interpret and draw a circle graph.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
7.2.1 Draw and Interpret
a Bar Graph

Write an appropriate title.

Make appropriate labels for bars and scale. The
intervals should be equally spaced and include
the smallest and largest values.

Draw horizontal or vertical bars to represent the
data. Bars should be of uniform width.

Make additional notes as appropriate to aid
interpretation.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Here’s an example
Sales Volume
2001-2004
Product 3
2004
2003
2002
2001
Product 2
Product 1
0
10
20
30
40
50
Thousands of Units
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
7.2.2 Interpret and Draw
a Line Graph

Write an appropriate title.

Make and label appropriate horizontal and
vertical scales, each with equally spaced
intervals. Often, the horizontal scale represents
time.

Use points to locate data on the graph.

Connect data points with line segments or a
smooth curve.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Here’s an example
Thousands of $
First Semester Sales
100
80
60
40
20
0
Jan
Feb
Judy
Business Math, Eighth Edition
Cleaves/Hobbs
Mar
Denise
Apr
May
Linda
Jun
Wally
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
7.2.3 Interpret and Draw
a Circle Graph

Write an appropriate title.

Find the sum of values in the data set.

Represent each value as a fraction or
decimal part of the sum of values.

For each fraction, find the number of degrees
in the sector of the circle to be represented by
the fraction or decimal. (100% = 360°)

Label each sector of the circle as appropriate.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Here’s an example
Local Daycare Market Share
6%
16%
43%
Teddy Bear
La La Land
Little Gems
Other
35%
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
7.3 Measures of Dispersion

Find the range.

Find the standard deviation.
From here to there...
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Key Terms

Measures of central tendency: statistical
measurements such as the mean, median
or mode that indicate how data groups
toward the center.

Measures of variation or dispersion:
statistical measurement such as the range
and standard deviation that indicate how
data is dispersed or spread.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Key Terms

Range: the difference between the highest
and lowest values in a data set. (also called
the spread)

Deviation from the mean: the difference
between a value of a data set and the mean.

Standard variation: a statistical
measurement that shows how data is
spread above and below the mean.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Key Terms

Variance: a statistical measurement that is the
average of the squared deviations of data from
the mean. The square root of the variance is
the standard deviation.

Square root: the quotient of number which is
the product of that number multiplied by itself.
The square root of 81 is 9. (9 x 9 = 81)

Normal distribution: a characteristic of many
data sets that shows that data graphs into a
bell-shaped curve around the mean.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
7.3.1 Find the Range in a Data Set

Find the highest and lowest values.

Find the difference between the two.

Example: The grades on the last exam were
78, 99, 87, 84, 60, 77, 80, 88, 92, and 94.
The highest value is 99.
The lowest value is 60.
The difference or the range is 39.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
7.3.2 Find the Standard Deviation

The deviation from the mean of a data value
is the difference between the value and the
mean.

Get a clearer picture of the data set by
examining how much each data point
differs or deviates from the mean.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Deviations from the mean

When the value is smaller than the mean, the
difference is represented by a negative
number indicating it is below or less than the
mean.

Conversely, if the value is greater than the
mean, the difference is represented by a
positive number indicating it is above or
greater than the mean.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Find the deviation from the mean

Find the mean of a set of data.

Mean = Sum of data values
Number of values

Find the amount that each data value
deviates or is different from the mean.

Deviation from the mean = Data value Mean
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Here’s an example

Data set: 38, 43, 45, 44

Mean = 42.5

1st value: 38 – 42.5 = -4.5 below the mean

2nd value: 43 – 42.5 = 0.5 above the mean

3rd value: 45 – 42.5 = 2.5 above the mean

4th value: 44 – 42.5 = 1.5 above the mean
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Interpret the information

One value is below the mean and its deviation is
-4.5.

Three values are above the mean and the sum
of those deviations is 4.5.

The sum of all deviations from the mean is zero.
This is true of all data sets.

We have not gained any statistical insight or new
information by analyzing the sum of the
deviations from the mean.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Average deviation

Average deviation =
Sum of deviations
Number of values
Business Math, Eighth Edition
Cleaves/Hobbs
=0 =0
n
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Find the standard deviation
of a set of data

A statistical measure called the standard
deviation uses the square of each deviation
from the mean.

The square of a negative value is always
positive.

The squared deviations are averaged
(mean) and the result is called the variance.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Find the standard deviation
of a set of data

The square root is taken of the variance so
that the result can be interpreted within the
context of the problem.

This formula averages the values by
dividing by one less than the number of
values (n-1).

Several calculations are necessary and are
best organized in a table.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Find the standard deviation
of a set of data
1. Find the mean.
2. Find the deviation of each value from the
mean.
3. Square each deviation.
4. Find the sum of the squared deviations.
5. Divide the sum of the squared deviations by
one less than the number of values in the data
set. This amount is called the variance.
6. Find the standard deviation by taking the
square root of the variance.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Find the standard deviation
Find the standard deviation for the following
data set:
18 22 29 27
Deviation
Squares of
Value
Mean
from Mean
Deviation
18
24
18 – 24 = -6
-6 x -6 = 36
22
24
22 – 24 = -2
-2 x -2 = 4
29
24
29 – 24 = 5
5 x 5 = 25
27
24
27 – 24 = 3
3x 3= 9
Sum of Squared Deviations 74
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Find the standard deviation
of a set of data
Variance = sum of squared deviations
n–1
Variance = 74 ÷ 3 = 24.666667
Standard deviation = square root of the variance
Standard deviation = 4.97 rounded
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved