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

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

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

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.

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

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

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

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?

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 $ 60,000 $275,000 65,000 80,000 70,000 70,000 80,000 65,000 275,000 60,000 2 2 2 2 3 2

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 $35,000 Illinois $58,000 California 49,100 Louisiana 60,000 Colorado 60,000 Maryland 60,000 New jersey 65,000 Ohio 50,000 Tennessee 60,000 Texas 71,400 2 2 2 2 3 2

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

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

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

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

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 2000. 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

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

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

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 = 112- 90 = 22 pound 2- Arithmetic Mean = 95+103+….90 = 824 = 103 8 8 000 pounds 95 103 105 110 104 105 112 90 2 2 2 3 2 2

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

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 = 103+97+101+106+103 = 102 5 2- 2 2 2 2 3 2

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

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

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 38 +10 100 26 -2 4 13 -15 225 41 +13 169 22 -6 36 140 0 534 µ = 140 = 28 5 σ² = 534 = 106.8 σ = √106.8 = 10.3 Citation 2 2 2 3 2 2

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

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 12 -5 25 20 3 9 16 -1 1 18 +1 1 19 +2 4 85 0 40 x = 85 = $17 5 S² = 40 = 10 5-1 σ = √10 = 3.16 2 2 2 3 2 2

Coefficient of Variation Definition The coefficient of variation is an attribute of a distribution: its standard deviation divided by its mean . Formula σ µ 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

Coefficient of Variation From previous example; find the coefficient variation (x) X- µ (x- µ)² 38 +10 100 26 -2 4 13 -15 225 41 +13 169 22 -6 36 140 0 534 µ = 140 = 28 5 σ² = 534 = 106.8 σ = √106.8 = 10.3 Citation CV = σ / µ = 10.3 = .37 28 2 2 2 3 2 2