1. Chapter 3 Describing Data Basic Statistics for Business and Economics
Fifth Edition
Chapter 3
Describing Data
Numerical Measures
Douglas William Samuel
Irwin/McGraw-Hill 1 1 1 2 1 1

2. Topics Covered
Measures of location; arithmetic mean, weighted mean, median, mode, and geometric mean
Dispersion; range, deviation, variance, and Standard deviation 2 2 2 2 3 2

3. Measures of Location Arithmetic mean ; shows the central values
* Population mean; the sum of all values in the population divided by the number of values in the population. The formula;
µ is the population mean
N is the total number of observations.
X is a particular value.
 indicates the operation of adding. 2 2 2 2 3 2

4. Find the mean mileage for the cars.
Example
The Kiers family owns four cars. The following is the current mileage on each of the four cars.
56,000
42,000
23,000
73,000
Find the mean mileage for the cars.

5. Measures of location Arithmetic mean
** Sample Mean; Is the sum of all sampled values divided by the total number of the sampled values.
is the sample mean, read X bar
n is the total number of values in the sample 2 2 2 2 3 2

6. Find the mean bonus for the executives
Example
A sample of five executives received the following bonus last year ($000):
14.0, 15.0, 17.0, 16.0, 15.0
Find the mean bonus for the executives

7. Measures of location Weighted Mean;
occurs when there are several observations of the same value.

8. Example
The carter construction company pays its hourly employees $16.50, $17.50, or $18.50 per hour. There are 26 hourly employees, 14 are paid at the $16.50 rate, 10 at $17.50 rate, and 2 at the $18.50 rate.
What is the mean hourly rate paid the 26 employees?

9. Measures of location The Median
The midpoint of the values after they have been ranked from the smallest to the largest or the vise versa. Example of Odd numbers
Prices ordered Prices ordered
From low to high From high to low
$ 60, $275,000
65, ,000
70, ,000
80, ,000
275, ,000 2 2 2 2 3 2

10. Measures of location The Mode
The value of observation that appears most frequently. Example
The annual salaries of quality-control managers in selected states in
The USA are shown below. What is the Modal annual salary?
State Salary State Salary
Arizona $35, Illinois $58,000
California , Louisiana ,000
Colorado , Maryland ,000
New jersey , Ohio ,000
Tennessee , Texas ,400 2 2 2 2 3 2

11. Measures of location The Mode
- We can determine the mode for all levels of data- nominal, ordinal, interval, and ratio
- sometimes there is no mode ; $18,19,21,23,24.
- We can determine the mode from charts. Example
Which bath oil consumers prefer?? Lamoure 2 2 2 2 3 2

12. Measures of location The Geometric Mean
used in finding the average of percentages, ratios, and growth rate. The formula; 2 2 2 2 3 2

13. The Geometric Mean Example;
The difference between Arithmetic Mean & Geometric Mean
Example;
The interest rate on three bonds were 5, 21,4 percent.
**The arithmetic mean is (5+21+4)/3 =10.0.
***The geometric mean is
The GM gives a more conservative profit figure
because it is not heavily weighted by the rate
of 21percent. 2 2 2 2 3 2

14. The Geometric Mean
Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another. The formula; 2 2 2 2 3 2

15. The Geometric Mean
Example; During the decade of 1990s, Las Vegas was the fastest- growing metropolitan in the United States. The population increased from 852,737 in 1990 to 1,563,282 in This is an increase of 710,545 or (710,545 ÷ 852,737 = 83%) an 83% increase over the period of 10 years. What is the average annual increase?
The population of Las Vegas increased at a rate of 6.25%
per year from 1990 to 2000. 2 2 2 2 3 2

16. Measures of Dispersion
Dispersion; Is the Mean variation in the set of data
Why we study Dispersion?
A measure of location , such as the mean, or the median, only describes the center of the data, but it does not tell us anything about the spread of data. Example; if you would like to cross a river and you are not a good swimmer, if you know that the average depth of the river is 3 feet in depth, you might decide to cross it, but if you have been told that the variation is between one feet to 6 feet, you might decide not to cross.
** Before making a decision about crossing the river, you want information on both the typical depth and the dispersion in the depth of the river.
*** Same thing if you want to buy stocks. 2 2 2 2 3 2

17. Measures of Dispersion
- Range
- Deviation
- Variance
- Standard Deviation 2 2 2 2 3 2

18. Measures of Dispersion
Range = largest value – Smallest value
Example;
The weight of containers being shipped to Ireland
What is the range of the weight?
Compute the arithmetic mean weight.
1- The range = = 22 pound
2- Arithmetic Mean = ….90 = 824 = 103
000 pounds 2 2 2 3 2 2

19. Measures of Dispersion
Deviation
The absolute values of the deviations from the arithmetic mean. Formula
X is the value of each observation
X is the arithmetic mean
n is the number of observations
II indicates the absolute values
The drawbacks of deviation is the use of absolute
values because it’s hard to work with. From this point
came the impertinence of Standard deviation. 2 2 2 3 2 2

20. Measures of Dispersion
Deviation
Example;
The weights of a sample of crates containing books for the bookstore (in pounds ) are:
103, 97, 101, 106, 103
Find the mean deviation.
1- Arithmetic mean = = 102 5 2- 2 2 2 2 3 2

21. Measures of Dispersion
Variance & Standard Deviation
They are also based on the deviations from the mean, however, instead of using the absolute value of the deviation, the variance and the standard deviation square the deviation. Formula
Population Variance
σ² variance is called sigma squared
X is the value of observation in the population
µ is the arithmetic mean of population
N is the number of observations 2 2 2 3 2 2

22. Measures of Dispersion
Variance & Standard Deviation
Population Standard Deviation
σ = 2 2 2 3 2 2

23. Variance & Standard Deviation
Example
The number of traffic citation during the last five months in Beaufort County , South Carolina , is 38,26,13,41,22.
What is the population variance?
What is the standard deviation?
Number
(x) X- µ (x- µ)²
µ = 140 = 28 5
σ² = 534 = 106.8
σ = √106.8 = 10.3 Citation 2 2 2 3 2 2

24. Variance & Standard Deviation
Sample Variance
Sample Standard deviation 2 2 2 2 3 2

25. Variance & Standard Deviation
Example
The hourly wages for the sample of a part-time employees at Fruit Packers, Inc are; $12, 20,16,18, and 19
What is the sample variance?
What is the standard deviation?
Number
(x) X- µ (x- µ)²
x = = $17 5
S² = = 10
5-1
σ = √10 = 3.16 2 2 2 3 2 2

26. Coefficient of Variation
Definition
The coefficient of variation is an attribute of a distribution: its standard deviation divided by its mean . Formula σ
Coefficient of variation is a measure of how much variation exists in relation to the mean, so it represents a highly useful—and easy—concept in data analysis.
CV = 2 2 2 3 2 2

27. Coefficient of Variation
From previous example; find the coefficient variation
(x) X- µ (x- µ)²
µ = 140 = 28 5
σ² = 534 = 106.8
σ = √106.8 = 10.3 Citation
CV = σ / µ
= = .37 28 2 2 2 3 2 2