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CHAPTER 7
Business Statistics
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7-1
Learning Outcomes
 Interpret and draw:
– A bar graph.
– A line graph.
– A circle graph.
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Interpret and draw a bar graph
7-1-1
Section 7-1
Graphs and Charts
 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.
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Interpret and draw a bar graph
7-1-1
Section 7-1
Graphs and Charts
Corky's Barbecue Business
June
May
April
March
February
January
0
10000
20000
30000
40000
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50000
60000
70000
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Interpret and draw a line graph
7-1-2
Section 7-1
Graphs and Charts
 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.
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Interpret and draw a line graph
7-1-2
Section 7-1
Graphs and Charts
Neighborhood Grocery Daily
2500
2000
Sales
1500
1000
500
0
Monday
Tuesday
Wednesday
Thursday
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Friday
Saturday
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Interpret and draw a circle graph
7-1-3
Section 7-1
Graphs and Charts
 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.
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Interpret and draw a circle graph
7-1-3
Section 7-1
Graphs and Charts
Family Take-Home Pay
Food, 400
Housing, 400
Insurance, 80
Contributions, 160
Education, 80
Personal, 80
Savings, 160
Miscellaneous, 80
Clothing , 160
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7-2
Learning Outcomes
 Find the mean, median & mode.
 Make and interpret a frequency distribution.
 Find the mean of grouped data.
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Key Terms…
Section 7-2
Measures of Central Tendency
 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.
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Find the mean
7-2-1
Section 7-2
Measures of Central Tendency
 A common statistic we may calculate for a data
set is its mean.
– The statistical term for the ordinary arithmetic
average.
 To find the mean, or arithmetic average, divide
the sum of the values by the total number of
values.
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Find the mean
7-2-1
Section 7-2
Measures of Central Tendency
Find the sum of the values.
Divide the sum by the
total number of values.
sum of values
Mean =
number of values
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Data Sets
Section 7-2
Measures of Central Tendency
 A business records its daily sales, and these
values are an example of a data set.
 Data sets can be used to:
– Observe patterns
– Interpret information
– Make predictions about future activity
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Key Terms…
Section 7-2
Measures of Central Tendency
 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.
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An Example…
Section 7-2
Measures of Central Tendency
Sales figures for the last week for the
Western Region have been as follows:
Monday
Tuesday
Wednesday
Thursday
Friday
$4,200
$3,980
$2,400
$3,100
$4,600
What is the average daily sales figure?
(4,200 + 3,980 + 2,400 + 3,100 + 4,600) ÷ 5 = $3,656
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Examples…
Section 7-2
Measures of Central Tendency
 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
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Find the median
7-2-1
Section 7-2
Measures of Central Tendency
 A second kind of average is a statistic called
the median.
 To find the median of a data set, order the values
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.
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An Example…
Section 7-2
Measures of Central Tendency
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, $4,700
Calculate the average of the two middle values:
(9,200 + 8,900) ÷ 2 = $9,050 or the median price
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An Example…
Section 7-2
Measures of Central Tendency
 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
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Find the mode
7-2-3
Section 7-2
Measures of Central Tendency
 Find the mode in a data set by counting the
number of times each value occurs.
– Identify the value or values that occurring 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.
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An Example…
Section 7-2
Measures of Central Tendency
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.
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An Example…
Section 7-2
Measures of Central Tendency
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.
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.
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An Example…
Section 7-2
Measures of Central Tendency
Find the mean, median and mode in this example.
Name
Jack:
Michelle:
Bill:
Jackie:
Jason:
Larissa:
Tony:
Melanie:
Art:
Sheila:
Hours
10
14
5
2
20
12
2
18
1
0
The mean is 8.4.
The median is 7.5.
The mode is 2.
Of the three values, which one
or one(s) would help you make
a realistic assessment of the
number of service hours?
Why?
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7-2-4
Make and interpret a frequency distribution
Section 7-2
Measures of Central Tendency
 Identify appropriate intervals for the data.
 Tally the data for the intervals.
 Count the number in each interval.
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Key Terms…
Section 7-2
Measures of Central Tendency
 Class intervals
– Special categories for grouping the values in
a data set.
 Tally
– A mark 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.
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An Example…
Section 7-2
Measures of Central Tendency
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, 90-94, and 95-99.
Class
Interval
75-79
80-84
85-89
90-94
95-99
Total
Class
Frequency
2
1
2
3
2
10
Calculations
2/10
1/10
2/10
3/10
2/10
10/10
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Relative
Frequency
20%
10%
20%
30%
20%
100%
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Find the mean of grouped data
7-2-4
Section 7-2
Measures of Central Tendency
 Make a frequency distribution.
 Find the products of the midpoint of the interval.
– Find the sum of the products.
 Find the frequency for each interval, for all
intervals.
– Find the sum of the frequencies.
 Divide the sum of the products by the sum
of the frequencies.
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An Example…
Section 7-2
Measures of Central Tendency
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, 90-94, and 95-99.
Class
Interval
75-79
80-84
85-89
90-94
95-99
Total
Class
Frequency
2
1
2
3
2
10
Midpoint
77
82
87
92
97
Product
MP & Freq.
154
82
174
276
194
880
Mean of the grouped data: 880 ÷ 10 = 88
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7-3
Learning Outcomes
 Find the range.
 Find the standard deviation.
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7-3-1
Measures of dispersion
Section 7-3
Measures of Dispersion
 Another group of statistical measures is
measures of variation or dispersion.
 The variation or dispersion of a set of data
may also be referred to as the spread.
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Key Terms…
Section 7-3
Measures of Dispersion
 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.
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Key Terms…
Section 7-3
Measures of Dispersion
 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.
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Key Terms…
Section 7-3
Measures of Dispersion
 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 factor that was multiplied by itself to result in
the number. 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.
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Deviation
Section 7-3
Measures of Dispersion
 The deviation from the mean of a data value is
the difference between the value and the mean.
– A clearer picture is given by examining how much
each data point differs or deviates from the mean.
Find the mean of a set of data.
Find the amount that each data value
deviates or is different from the mean.
Deviation from the mean = data value – mean
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Deviation
Section 7-3
Measures of Dispersion
 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.
 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.
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An Example…
Section 7-3
Measures of Dispersion
Find the highest and lowest values.
Find the difference between the two.
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.
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An Example…
Section 7-3
Measures of Dispersion
What can you learn by analyzing the sum of the deviations?
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
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An Example…
Section 7-3
Measures of Dispersion
What can you learn by analyzing the sum of the deviations?
Data set: 38, 43, 45, 44
Mean = 42.5
1st value: 38 – 42.5 = -4.5 below the mean
One value is below the mean and its deviation is -4.5.
2nd value:
43values
– 42.5are= above
0.5 above
the mean
Three
the mean.
sumdeviations
of deviations
0
The
sum
of
those
is
4.5.
average
deviation
=
=0
3rd value:
45 – 42.5
= 2.5 above the=mean
number
values
n
The sum of all deviations
fromof
the
mean is zero.
4th value: 44 – 42.5 = 1.5 above the mean
This is true of all data sets.
Average deviation also does not
We have not gained any statistical insight or new
provide any statistical insight.
information by analyzing the sum of the deviations.
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HOW TO:
Find the standard deviation of a set of data
Section 7-3
Measures of Dispersion
 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).
– The result is called the variance.
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HOW TO:
Find the standard deviation of a set of data
Section 7-3
Measures of Dispersion
STEP 1
Find the mean.
STEP 2
Find the deviation of each value from the mean.
STEP 3
Square each deviation.
STEP 4
Find the sum of the squared deviations.
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HOW TO:
Find the standard deviation of a set of data
Section 7-3
Measures of Dispersion
STEP 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.
STEP 6
Find the standard deviation by taking the square
root of the variance.
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An Example…
Section 7-3
Measures of Dispersion
Find the standard deviation for the following data set:
18 22 29 27
Value
18
22
29
27
Mean
24
24
24
24
Deviation
Squares of
from Mean
Deviation
18 – 24 = -6
-6 x -6 = 36
22 – 24 = -2
-2 x -2 = 4
29 – 24 = 5
5 x 5 = 25
27 – 24 = 3
3x 3= 9
Sum of Squared Deviations 74
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An Example…
Section 7-3
Measures of Dispersion
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
24 = -2
-2 x -2 = 4
sum 22
of –squared
deviations
=
29Variance24
29 – 24 = 5
5 x 5 = 25
n 1
27
24
27 – 24 = 3
3x 3= 9
Deviations 74
VarianceSum
= 74of
÷ 3Squared
= 24.666667
Standard deviation = square root of the variance
Standard deviation = 4.97 rounded
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Exercises Set A
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EXERCISE SET A
Find the range, mean, median, and mode for the
following. Round to the nearest hundredth if
necessary.
2. Sandwiches
$0.95
$1.65
$1.27
$1.97
$1.65
$1.15
Range = $1.97  $0.95 = $1.02
$0.95  $1.27  $1.65  $1.65  $1.97  $1.15
Mean =
6
 $1.44
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EXERCISE SET A
Find the range, mean, median, and mode for the
following. Round to the nearest hundredth if
necessary.
2. Sandwiches
$0.95
$1.65
$1.27
$1.97
$1.65
$1.15
Arrange in order: $0.95, $1.15, $1.27, $1.65, $1.65,
$1.97
$1.27  $1.65
Median =
 $1.46
2
Mode = $1.65
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EXERCISE SET A
4. During the past year, Piazza’s Clothiers sold a
certain sweater at different prices: $42.95, $36.50,
$40.75, $38.25, and $43.25. Find the range,
mean, median, and mode of the selling prices.
Write a statement about the data set based on
your findings.
Range = $43.25  $36.50 = $6.75
$42.95  $36.50  $40.75  $38.25  $43.25
Mean =
5
$201.70

 $40.34
5
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EXERCISE SET A
4. During the past year, Piazza’s Clothiers sold a
certain sweater at different prices: $42.95, $36.50,
$40.75, $38.25, and $43.25. Find the range,
mean, median, and mode of the selling prices.
Write a statement about the data set based on
your findings.
Arrange in order: $36.50, $38.25, $40.75, $42.95,
$43.25
Median = $40.75
There is no mode.
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EXERCISE SET A
4. During the past year, Piazza’s Clothiers sold a
certain sweater at different prices: $42.95, $36.50,
$40.75, $38.25, and $43.25. Find the range,
mean, median, and mode of the selling prices.
Write a statement about the data set based on
your findings.
Statements about the data set may vary. The mean
and median are very similar and there is no mode.
The data clusters near the mean.
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EXERCISE SET A
6. Which period had the highest average enrollment?
Period 5 (10:40 – 11:30) with an average of 801.
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EXERCISE SET A
8. Draw a bar graph representing
the mean enrollment for each
period.
900
800
Mean Enrollment
700
600
500
400
300
200
100
0
1
2
3
4
5
6
7
8
9
10
Period
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‹#›
EXERCISE SET A
Sales for the Family Store, 2010-2011
2010
2011
Girls’ clothing
$74,675
$81,534
Boys’ clothing
65,153
68,324
125,115
137,340
83,895
96,315
Women’s clothing
Men’s clothing
10. What is the least value for 2010 sales? For 2011
sales?
2010: $65,153; 2011: $68,324
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EXERCISE SET A
Sales for the Family Store, 2010-2011
2010
2011
$74,675 $81,534
12. Using the values in the Girls’ clothing
Boys’ clothing
65,153
68,324
table, which of the following Women’s clothing 125,115 137,340
interval sizes would be more Men’s clothing
83,895
96,315
appropriate in making a bar
graph? Why?
a. $1,000 intervals ($60,000, $61,000, $62,000, . . .)
b. $10,000 intervals ($60,000, $70,000, $80,000, . . .)
(b) Intervals of $10,000 are more appropriate
because the data can be shown with fewer intervals
than with (a).
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EXERCISE SET A
14. What three-month period maintained a fairly
constant sales record?
May–July
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EXERCISE SET A
16. What percent of the gross
pay goes into savings?
(Round to tenths.)
$60
(100%)  8.6%
$700
18. What percent of the gross
pay is the take-home pay?
(Round to tenths.)
$394
(100%)  56.3%
$700
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EXERCISE SET A
20. Find the range for the data set:
90, 89, 82, 87, 93, 92, 98, 79, 81, 80.
98  79 = 19
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EXERCISE SET A
22. Find the variance for the scores in the following
data set: 90, 89, 82, 87, 93, 92, 98, 79, 81, 80.
Show that the sum of the deviations is zero.
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EXERCISE SET A
22. Find the variance for the scores in the following
data set: 90, 89, 82, 87, 93, 92, 98, 79, 81, 80.
Show that the sum of the deviations is zero.
368.90 368.90
Variance =

 40.98888888
10  1
9
Sum of deviations = (-8.1) + (-7.1) + (-6.1) + (-5.1) +
(-0.1) + 1.9 + 2.9 + 4.9 + 5.9 + 10.9 = 0
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EXERCISE SET A
24. Use the test scores of 24 students taking
Marketing 235 to complete the frequency
distribution and find the grouped mean rounded
to the nearest whole number:
57 91 76 89 82 59 72 88
76 84 67 59 77 66 56 76
77 84 85 79 69 88 75 58
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EXERCISE SET A
24.
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EXERCISE SET A
24.
1797
Mean of grouped data =
 74.875
24
The grouped mean of the scores is 75 (rounded
to the nearest whole number).
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Practice Test
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PRACTICE TEST
The costs of producing a piece of luggage at ACME
Luggage Company are labor, $45; materials, $40;
overhead, $35.
2. What is the total cost of producing a piece of
luggage?
$45 + $40 + $35 = $120
4. What percent of the total cost is attributed to
materials?
$40
(100%)  33.3%
$120
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PRACTICE TEST
The costs of producing a piece of luggage at ACME
Luggage Company are labor, $45; materials, $40;
overhead, $35.
6. Compute the number of degrees for labor,
materials, and overhead needed for a circle graph.
Round to whole degrees.
labor: 360(0.375) = 135 degrees
materials: 360(0.333) = 120 degrees
overhead: 360(0.292) = 105 degrees
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PRACTICE TEST
8. What is the greatest value of fresh flowers? Of silk
flowers?
fresh flowers: $23,712; silk flowers: $17,892
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PRACTICE TEST
10. What interval size would be most appropriate
when making a bar graph? Why?
a. $100
b. $1,000 c. $5,000 d. $10,000
c. $5,000; other interval sizes would provide too
many or too few intervals.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
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PRACTICE TEST
11. Construct a bar graph.
25000
Thousand Dollars
20000
15000
Fresh
Silk
10000
5000
0
January
February
March
April
May
June
Month
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
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PRACTICE TEST
The totals of the number of laser printers sold in the
years 2006 through 2011 by Smart Brothers
Computer Store are as follows:
2006 2007 2008 2009 2010 2011
983 1,052
1,117
615 250 400
12. What is the smallest value? The greatest value?
smallest: 250; greatest: 1,117
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
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PRACTICE TEST
14. Find the mean, variation, and standard deviation
for the set of average prices for NFL tickets.
Year
Avg
Ticket
Price
2004
2005
2006
2007
2008
2009
$54.75
$59.05
$62.38
$67.11
$72.20
$74.99
54.75  59.05  62.38  67.11  72.20  74.99
Mean:
6
390.48
 $65.08
6
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
‹#›
PRACTICE TEST
Year
Avg Ticket Price
2004
2005
2006
2007
2008
2009
$54.75
$59.05
$62.38
$67.11
$72.20
$74.99
Variation
Score Mean
$54.75 65.08
$59.05 -6.03
$62.38 -2.7
$67.11 2.03
$72.20 7.12
$74.99 9.91
Deviation
-10.33
36.3609
7.29
4.1209
50.6944
98.2081
(Deviation)2
106.7089
Sum of deviation2 = 106.7089 + 36.3609 + 7.29
+ 4.1209 + 50.6944 + 98.2081
= 303.3832
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
‹#›
PRACTICE TEST
Year
Avg Ticket Price
2004
2005
2006
2007
2008
2009
$54.75
$59.05
$62.38
$67.11
$72.20
$74.99
303.3832 303.3832
Variance =

 60.67664
6 1
5
303.3832
S.D. =
 60.67664
5
 7.789521166 or 7.79
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