1. CHAPTER 7
Business Statistics

2. Interpret and draw: 7-1 Learning Outcomes A bar graph. A line graph.
A circle graph.

3. Interpret and draw a bar graph
7-1-1
Interpret and draw a bar graph
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.

4. Interpret and draw a bar graph
7-1-1
Interpret and draw a bar graph
Section 7-1
Graphs and Charts

5. Interpret and draw a line graph
7-1-2
Interpret and draw a line graph
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.

6. Interpret and draw a line graph
7-1-2
Interpret and draw a line graph
Section 7-1
Graphs and Charts

7. Interpret and draw a circle graph
7-1-3
Interpret and draw a circle graph
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.

8. Interpret and draw a circle graph
7-1-3
Interpret and draw a circle graph
Section 7-1
Graphs and Charts

9. Find the mean, median & mode.
7-2
Learning Outcomes
Find the mean, median & mode.
Make and interpret a frequency distribution.
Find the mean of grouped data.

10. Mean Median Mode 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.

11. A common statistic we may calculate for a data set is its mean.
7-2-1
Find the mean
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.

12. Find the sum of the values.
7-2-1
Find the mean
Section 7-2
Measures of Central Tendency
Find the sum of the values.
Divide the sum by the total number of values.

13. Data sets can be used to:
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

14. Data set Statistic 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.

15. What is the average daily sales figure?
An Example…
Section 7-2
Measures of Central Tendency
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, , , , ,600) ÷ 5 = $3,656

16. 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 $ What is the average price?
$4.19

17. A second kind of average is a statistic called the median.
7-2-1
Find the median
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.

18. 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, ,900) ÷ 2 = $9,050 or the median price

19. 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

20. 7-2-3
Find the mode
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.

21. Which score occurred the most frequently?
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.

22. Observe the mean, median and mode from this data set.
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.

23. Find the mean, median and mode in this example.
An Example…
Section 7-2
Measures of Central Tendency
Find the mean, median and mode in this example.
Name Hours
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?
Jack: 10
Michelle: 14
Bill: 5
Jackie: 2
Jason: 20
Larissa: 12
Tony: 2
Melanie: 18
Art: 1
Sheila: 0

24. Make and interpret a frequency distribution
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.

25. Grouped frequency distribution
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.

26. An Example… Test scores on the last math test were as follows:
Section 7-2
Measures of Central Tendency
Test scores on the last math test were as follows:
Make a relative frequency distribution using intervals of:
75-79, 80-84, 85-89, 90-94, and
Class Class Relative
Interval Frequency Calculations Frequency
/ %
/ %
/ %
/ %
/ %
Total / %

27. Find the mean of grouped data
7-2-4
Find the mean of grouped data
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.

28. An Example… Test scores on the last math test were as follows:
Section 7-2
Measures of Central Tendency
Test scores on the last math test were as follows:
Make a relative frequency distribution using intervals of:
75-79, 80-84, 85-89, 90-94, and
Class Class Product
Interval Frequency Midpoint MP & Freq.
Total
Mean of the grouped data: 880 ÷ 10 = 88

29. Find the standard deviation.
7-3
Learning Outcomes
Find the range.
Find the standard deviation.

30. Measures of dispersion
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.

31. Measures of central tendency
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.

32. Deviation from the mean
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.

33. Variance Square root Normal distribution 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.

34. Find the mean of a set of data.
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

35. 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.

36. Find the highest and lowest values.
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.

37. 1st value: 38 – 42.5 = -4.5 below the mean
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

38. What can you learn by analyzing the sum of the deviations?
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
One value is below the mean and its deviation is -4.5.
Three values are above the mean. 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.
Average deviation also does not provide any statistical insight.

39. The squared deviations are averaged (mean).
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.

40. STEP 2 Find the deviation of each value from the mean.
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.

41. 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.

42. Find the standard deviation for the following data set:
An Example…
Section 7-3
Measures of Dispersion
Find the standard deviation for the following data set:
Deviation Squares of
Value Mean from Mean Deviation
– 24 = x -6 = 36
– 24 = x -2 = 4
– 24 = x 5 = 25
– 24 = x 3 = 9
Sum of Squared Deviations

43. An Example… Find the standard deviation for the following data set:
Section 7-3
Measures of Dispersion
Find the standard deviation for the following data set:
Value Mean from Mean Deviation
– 24 = x -6 = 36
– 24 = x -2 = 4
– 24 = x 5 = 25
– 24 = x 3 = 9
Deviation Squares of
Sum of Squared Deviations
Variance = 74 ÷ 3 =
Standard deviation = square root of the variance
Standard deviation = 4.97 rounded

44. Exercises Set A

45. 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

46. Arrange in order: $0.95, $1.15, $1.27, $1.65, $1.65, $1.97
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
Mode = $1.65

47. 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 $ 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

48. Arrange in order: $36.50, $38.25, $40.75, $42.95, $43.25
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 $ 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.

49. 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 $ 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.

50. 6. Which period had the highest average enrollment?
EXERCISE SET A
6. Which period had the highest average enrollment?
Period 5 (10:40 – 11:30) with an average of 801.

51. 8. Draw a bar graph representing the mean enrollment for each period.
EXERCISE SET A
8. Draw a bar graph representing the mean enrollment for each period.

52. Sales for the Family Store, 2010-2011
EXERCISE SET A
Sales for the Family Store,
2010
2011
Girls’ clothing
$74,675
$81,534
Boys’ clothing
65,153
68,324
Women’s clothing
125,115
137,340
Men’s clothing
83,895
96,315
10. What is the least value for 2010 sales? For 2011 sales?
2010: $65,153; 2011: $68,324

53. Sales for the Family Store, 2010-2011
EXERCISE SET A
Sales for the Family Store,
2010
2011
Girls’ clothing
$74,675
$81,534
Boys’ clothing
65,153
68,324
Women’s clothing
125,115
137,340
Men’s clothing
83,895
96,315
12. Using the values in the table, which of the following interval sizes would be more 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).

54. 14. What three-month period maintained a fairly constant sales record?
EXERCISE SET A
14. What three-month period maintained a fairly constant sales record?
May–July

55. 16. What percent of the gross pay goes into savings?
EXERCISE SET A
16. What percent of the gross pay goes into savings?
(Round to tenths.)
18. What percent of the gross pay is the take-home pay?

56. 20. Find the range for the data set:
EXERCISE SET A
20. Find the range for the data set:
90, 89, 82, 87, 93, 92, 98, 79, 81, 80.
98  79 = 19

57. 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.

58. Sum of deviations = (-8.1) + (-7.1) + (-6.1) + (-5.1) +
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.
Sum of deviations = (-8.1) + (-7.1) + (-6.1) + (-5.1) +
(-0.1) = 0

59. 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:

60. EXERCISE SET A
24.

61. EXERCISE SET A
24.
The grouped mean of the scores is 75 (rounded to the nearest whole number).

62. Practice Test

63. 2. What is the total cost of producing a piece of luggage?
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?

64. materials: 360(0.333) = 120 degrees overhead: 360(0.292) = 105 degrees
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

65. 8. What is the greatest value of fresh flowers? Of silk flowers?
PRACTICE TEST
8. What is the greatest value of fresh flowers? Of silk flowers?
fresh flowers: $23,712; silk flowers: $17,892

66. 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.

67. PRACTICE TEST
11. Construct a bar graph.

68. 12. What is the smallest value? The greatest value?
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:
983 1,052 1,
12. What is the smallest value? The greatest value?
smallest: 250; greatest: 1,117

69. PRACTICE TEST
14. Find the mean, variation, and standard deviation for the set of average prices for NFL tickets.
Mean:
Year
2004
2005
2006
2007
2008
2009
Avg Ticket Price
$54.75
$59.05
$62.38
$67.11
$72.20
$74.99

70. Score Mean Deviation (Deviation)2 $54.75 65.08 -10.33 106.7089
PRACTICE TEST
Year
2004
2005
2006
2007
2008
2009
Avg Ticket Price
$54.75
$59.05
$62.38
$67.11
$72.20
$74.99
Variation
Score Mean Deviation (Deviation)2 $ $ $ $ $ $
Sum of deviation2 = =

71. PRACTICE TEST
Year
2004
2005
2006
2007
2008
2009
Avg Ticket Price
$54.75
$59.05
$62.38
$67.11
$72.20
$74.99