Lecture Practical business math procedures (11/e) - Chapter 22: Business statistics

After studying this chapter you will be able to: define and calculate the mean, explain and calculate a weighted mean, define and calculate the median, define and identify the mode, prepare a frequency distribution, prepare bar, line, and circle graphs, calculate price relatives and cost comparisons, explain and calculate the range, define and calculate the standard deviation, estimate percentage of data by using standard deviations.

BUSINESS STATISTICS

Chapter Twenty-Two

1. Define and calculate the mean.

2. Explain and calculate a weighted mean

3. Define and calculate the median

4. Define and identify the mode

LU22-1: Mean, Median, and Mode

Learning unit objectives

LU 22-2: Frequency Distributions and Graphs

1. Prepare a frequency distribution

2. Prepare bar, line, and circle graphs

3. Calculate price relatives and cost comparisons

1. Explain and calculate the range

2. Define and calculate the standard deviation

LU 22-3: Measures of Dispersion (Optional)

Mean – The average used to indicate a single value that represents an

entire group of numbers

Median - A measurement that indicates the center of the data

(average)

Terminology

Mode - A measurement that records values; the value that occurs most

often

Mean Mean =

The accountant of Bill’s Sport Shop told Bill, the owner, that the average

daily sales for the week were $150.14 The accountant stressed that

$150.14 was an average and did not represent specific daily sales Bill

wanted to know how the accountant arrived at $150.14

Sun Mon Tues Wed Thur Fri Sat

$400 $100 $68 $115 $120 $68 $180

Mean = $400 + $100 + $68 + $115 + $120 +$68 + $180 = $150.14

Sum of all values Number of values

Example:

Weighted Mean Weighted mean

How Jill Rivers calculated her GPA to the nearest tenth:

Intro to Comp 4 A 16 (4 x 4)

English Comp 3 B 9 (3 x 3)

Business Law 3 C 6 (2 x 3)

49

16 = 3.1

= Sum of products Sum of frequencies

Example:

Finding the Median of a

Group of Values

Look below at the following yearly salaries of the employees of Rusty’s Clothing

Shop

Alice Knight $95,000 Jane Wang $67,000

Jane Hess $27,000 Bill Joy $40,000

Joel Floyd $32,000

1. Find the median value of all employees

$95,000 + $27,000 + $32,000 + $67,000 + $40,000

5 = $52,200

Finding the Median of a

Group of Values

Step 1 Orderly arrange values from

the smallest to the largest

Step 2 Find the middle value.

95, 27, 32, 67, 40

27, 32, 40 , 67, 95

32, 40, 67, 95

95, 32, 67, 40

a Odd number of values: Median is the

middle value Divide the total number of

numbers by 2 (5/2 = 2 ½) The next-higher

number is the median

b Even number of values: Median is the

average of the two middle values

40

40 + 67 2 = 53.5

3, 4, 5, 6, 3, 8, 9, 3, 5, 3

Mode is the value that occurs most often

If two or more numbers appear most often, you

may have two or more modes

If all the values are different, there is no mode

3 is the mode since it is listed 4 times

Frequency Distribution

Frequency Distribution A way of collecting

and organizing raw data

Price of Tally Frequency

Computer

$1,000 llll 5 2,000 l 1 3,000 llll 5 4,000 l 1 5,000 ll 2 6,000 ll 2 7,000 l 1 8,000 l 1 9,000 l 1 10,000 l 1

1000 7000 4000 1000 5000 1000 3000

5000 2000 3000 3000 3000 8000 9000

3000 6000 6000 1000 10000 1000

Computer costs

Frequency distribution table

Bar Graph

2000 4000 6000 8000 10000 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Bar Graph

$1000 - $ 3,000.99 11

$3001 - 5,000.99 3

$5001 - 7,000.99 3

$8001 - 9,000.99 2

$9001 - 11,000.99 1

$3,001-$5,000.99 $7,001-$9,000.99

0 5 10 15

$1,000­

$3,000.99 $7,000.99$5,001­ $11,000.99$9,001­

Circle Graph

12.9%

12.9%

56.9%

1st Qtr $20,400 2nd Qtr $27,400 3rd Qtr $90,000 4th Qtr $20,400

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr

Measure of Dispersion

Range = Highest value – Lowest value

Find the range of the following values:

83.6, 77.3, 69.2, 93.1, 85.4, 71.6 Range = 93.1 – 69.2 = 23.9

Measure of Dispersion – A number that describes how the numbers of a set

of data are spread out or dispersed

Range – The difference between the two extreme values (highest and

lowest) in a group of values or a set of data

Index Numbers

Price relative = Current price x 100 Base year’s price

A calculator may cost $9 today relative to a cost of $75 some 30

years ago What is the relative price?

$9

$75 x 100 = .12 = 12%

Example:

Consumer Price Index

(in percent)

Step 1 Find the mean of the set of data.

Step 2 Subtract the mean from each piece of data to find each deviation

Step 3 Square each deviation (multiply the deviation by itself).

Step 4 Sum all squared deviations.

Step 5 Divide the sum of the squared deviations by n - 1, where n

equals the number of pieces of data

Standard deviation is intended to measure the

spread of data around the mean.

Standard Deviation

Step 6 Find the square root of the number obtained in Step 5

This is the standard deviation

Step 1 (1 + 2 + 5 + 10 + 12) = 6 (Mean)

5

Data Data Mean (Data Mean)

1 1- 6 = -5 25

2 2 - 6 = -4 16

5 5 - 6 = -1 1

10 10 - 6 = 4 16

12 12 - 6 = 6 36

Total 0 94 (Step 4)

23.5

5 - 1 4

Data Set A

x x x x x

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Standard Deviation

The standard deviation of data set A is 4.8.

Step 1 (4 + 4 + 5 + 8 + 9) = 6 (Mean)

5

Data Data Mean (Data Mean)

1 4- 6 = -2 4

2 4 - 6 = -2 4

5 5 - 6 = -1 1

10 8 - 6 = 2 4

12 9 - 6 = 3 9

Total 0 22 (Step 4)

5 - 1 4

Data Set B

x

x x x x

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Standard Deviation