Describing Data: Numerical Measures GOALS. Why a Numeric Approach? Chapter 3 Dr. Richard Jerz

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1 Describing Data: Numerical Measures Chapter 3 Dr. Richard Jerz 1 GOALS Calculate the arithmetic mean, weighted mean, median, and mode Explain the characteristics, uses, advantages, and disadvantages of each measure of location. Compute and interpret the range, mean deviation, variance, and standard deviation. Understand the characteristics, uses, advantages, and disadvantages of each measure of dispersion. 2 Why a Numeric Approach? Graphic approach Quick overview of data General relationships and trends Easy for people to interpret Numerical approach Detailed analysis of data Powerful relationships and trends More difficult for people to interpret Helps in calculating probabilities 3 1

2 Concepts & Goals Central Tendency (one number) Mean (Average) Median Mode Dispersion (spread) Range Mean deviation Variance Standard deviation Be able to do this in Excel! 4 Characteristics of the Mean The arithmetic mean is the most widely used measure of location. It requires the interval or ratio scales. Its major characteristics are: All values are used. It is unique. The sum of the deviations from the mean is 0 The mean is affected by unusually large or small data values. It is calculated by summing the values and dividing by the number of values. 5 Graphic of the Arithmetic Mean 6 2

3 Population Mean For ungrouped data, the population mean, μ, is the sum of all the population values divided by the total number of population values: N å i= 1 µ = N X i 7 Example: Population Mean 8 Sample Mean For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values: X = n å i= 1 n X i 9 3

4 Weighted Mean The weighted mean of a set of numbers X 1, X 2,..., X n, with corresponding weights w 1, w 2,...,w n, is computed from the following formula: Note: This is grouped data. 10 Example: Weighted Mean The Carter Construction Company pays its hourly employees $16.50, $19.00, or $25.00 per hour. There are 26 hourly employees, 14 of which are paid at the $16.50 rate, 10 at the $19.00 rate, and 2 at the $25.00 rate. What is the mean hourly rate paid the 26 employees? 11 The Arithmetic Mean of grouped Data Example In Chapter 2, a frequency distribution for the vehicle selling prices was constructed. The information is repeated below. Determine the arithmetic mean vehicle selling price. 12 4

5 The Arithmetic Mean of Grouped Data 13 The Median The median is the midpoint of the values after they have been ordered from the smallest to the largest. Properties of the median: There is a unique median for a data set. There are as many values above the median as below it in the data array. With an even set of values, the median will be the arithmetic average of the two middle numbers. It is not affected by extremely large or small values. It can be computed for ratio-level, interval-level, and ordinal-level data. 14 The Mode The mode is the value of the observation that appears most frequently. 15 5

6 Mean, Median, & Mode Shape of Distribution 16 Dispersion Why Study Dispersion? The mean or the median only describes the center of the data, but it does not tell us anything about the spread of the data. A second reason for studying the dispersion in a set of data is to compare the spread in two or more distributions. 17 Dispersions with Same Mean 18 6

7 Measures of Dispersion Range Mean Deviation å X -X MD = n å( X - µ ) 2 Variance 2 s = Standard Deviation N s = å( X - µ ) 2 N 19 Example: Range The number of cappuccinos sold at the Starbucks location in the Orange Country Airport between 4 and 7 p.m. for a sample of 5 days last year were 20, 40, 50, 60, and 80. Determine the range for the number of cappuccinos sold. Range = Largest Smallest value = = Example: Mean Deviation The number of cappuccinos sold at the Starbucks location in the Orange Country Airport between 4 and 7 p.m. for a sample of 5 days last year were 20, 40, 50, 60, and 80. Determine the mean deviation for the number of cappuccinos sold. 21 7

8 Example: Variance and Standard Deviation The number of traffic citations issued during the last five months in Beaufort County, South Carolina, is 38, 26, 13, 41, and 22. What is the population variance? 22 Example: Sample Variance The hourly wages for a sample of part-time employees at Home Depot are: $12, $20, $16, $18, and $19. What is the sample variance? s 2 = å( X - X) 2 n Why Use the Standard Deviation And Variance? Interesting observations: Chebyshev s Theorem Empirical Rule 24 8

9 Chebyshev s Theorem The arithmetic mean biweekly amount contributed by the Dupree Paint employees to the company s profit-sharing plan is $51.54, and the standard deviation is $7.51. At least what percent of the contributions lie within plus 3.5 standard deviations and minus 3.5 standard deviations of the mean? 25 The Empirical Rule 26 How do we do these calculations in Excel? 27 9