Describing Data Numerical Measures Chapter 3 Copyright 2015

Describing Data: Numerical Measures Chapter 3 Copyright © 2015 Mc. Graw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of Mc. Graw-Hill Education.

Learning Objectives LO 3 -1 Compute and interpret the mean, the median, and the mode. LO 3 -2 Compute a weighted mean. LO 3 -3 Compute and interpret the geometric mean. LO 3 -4 Compute and interpret the range, variance, and standard deviation. LO 3 -5 Explain and apply Chebyshev’s theorem and the Empirical Rule. LO 3 -6 Compute the mean and standard deviation of grouped data. 3 -2

LO 3 -1 Compute and interpret the mean, the median, and the mode. Measures of Location n n The purpose of a measure of location is to pinpoint the center of a distribution of data. There are many measures of location. We will consider three: 1. The arithmetic mean 2. The median 3. The mode 3 -3

LO 3 -1 Characteristics of the Mean n The arithmetic mean is the most widely used measure of location. It requires the interval scale. Major characteristics: ¨ All values are used. ¨ It is unique. ¨ The sum of the deviations from the mean is 0. ¨ It is calculated by summing the values and dividing by the number of values. 3 -4

LO 3 -1 Population Mean For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values: 3 -5

LO 3 -1 Example – Population Mean There are 42 exits on I-75 through the state of Kentucky. Listed below are the distances between exits (in miles). 1. Why is this information a population? 2. What is the mean number of miles between exits? 3 -6

LO 3 -1 Example – Population Mean There are 42 exits on I-75 through the state of Kentucky. Listed below are the distances between exits (in miles). Why is this information a population? This is a population because we are considering all of the exits in Kentucky. What is the mean number of miles between exits? 3 -7

LO 3 -1 Parameter versus Statistic PARAMETER A measurable characteristic of a population. STATISTIC A measurable characteristic of a sample. 3 -8

LO 3 -1 Properties of the Arithmetic Mean 1. 2. 3. 4. Every set of interval-level and ratio-level data has a mean. All the values are included in computing the mean. The mean is unique. The sum of the deviations of each value from the mean is zero. 3 -9

LO 3 -1 Sample Mean For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values: 3 -10

LO 3 -1 Example – Sample Mean 3 -11

LO 3 -1 The Median MEDIAN The midpoint of the values after they have been ordered from the minimum to the maximum values. Properties of the median: 1. There is a unique median for each data set. 2. It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur. 3. It can be computed for ratio-level, interval-level, and ordinal-level data. 4. It can be computed for an open-ended frequency distribution if the median does not lie in an openended class. 3 -12

LO 3 -1 Examples - Median The ages for a sample of five college students are: 21, 25, 19, 20, 22 Arranging the data in ascending order gives: 19, 20, 21, 22, 25. Thus the median is 21. The heights of four basketball players, in inches, are: 76, 73, 80, 75 Arranging the data in ascending order gives: 73, 75, 76, 80. Thus the median is 75. 5. 3 -13

LO 3 -1 The Mode MODE The value of the observation that appears most frequently. 3 -14

LO 3 -1 Example - Mode Using the data measuring the distance in miles between exits on I-75 through Kentucky, what is the modal distance? Organize the distances into a frequency table and select the distance with the highest frequency. 3 -15

LO 3 -1 The Relative Positions of the Mean, Median and the Mode

LO 3 -2 Compute a weighted mean. Weighted Mean The weighted mean of a set of numbers X 1, X 2, . . . , Xn, with corresponding weights w 1, w 2, . . . , wn, is computed with the following formula: 3 -17

LO 3 -2 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 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 for the 26 employees? 3 -18

Weighted Mean – Example 2 Wendy’s restaurant sold cokes for the last 30 minutes as; Price No. sold small $ 0. 90 3 medium $1. 25 4 large $1. 50 3 10 What is the mean selling price? n 3 -19

Weighted Mean – Example 2 n 3 -20

LO 3 -3 Compute and interpret the geometric mean. The Geometric Mean n Useful in finding the average change of percentages, ratios, indexes, or growth rates over time. n It has wide application in business and economics because we are often interested in finding the percentage changes in sales, salaries, or economic figures, such as the GDP. n The geometric mean will always be less than or equal to the arithmetic mean. 3 -21

LO 3 -3 The Geometric Mean: Finding the average rate of return over time EXAMPLE: The return on investment earned by Atkins Construction Company for four successive years was: 30 percent, 20 percent, -40 percent, and 200 percent. What is the geometric mean rate of return on investment? 3 -22

LO 3 -3 The Geometric Mean: Finding an Average Percent Change Over Time EXAMPLE: During the decade of the 1990 s, and into the 2000 s, Las Vegas, Nevada, was the fastest-growing city in the United States. The population increased from 258, 295 in 1990 to 584, 539 in 2011. This is an increase of 326, 244 people, or a 126. 3 percent increase over the period. What is the average annual increase? 3 -23

GM Example 1: n 3 -24

GM Example 1: Explanation Raise 1: $3, 000(0. 05) = $150 n Raise 2: $3, 150(0. 15) = $472. 5 Total increase $622. 5 This total salary is equivalent to $3, 000(0. 09886) = $ 296. 58 $3296. 58(0. 09886) = $ 325. 90 Total increase $622. 48 n The small difference arises due to rounding off numbers. 3 -25

LO 3 -4 Compute and interpret the range, variance, and standard deviation. 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 the data. § For example, if your nature guide told you that the river ahead averaged 3 feet in depth, would you want to wade across on foot without additional information? Probably not. You would want to know something about the variation in the depth. § A second reason for studying the dispersion in a set of data is to compare the spread in two or more distributions. 3 -26

LO 3 -4 Measures of Dispersion n Range n Variance n Standard Deviation 3 -27

LO 3 -4 Example – Range The number of cappuccinos sold at the Starbucks location in the Orange County 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 = Maximum value – Minimum value = 80 – 20 = 60 3 -28

LO 3 -4 Variance and Standard Deviation VARIANCE The arithmetic mean of the squared deviations from the mean. STANDARD DEVIATION The square root of the variance. n The variance and standard deviations are nonnegative and are zero only if all observations are the same. n For populations whose values are near the mean, the variance and standard deviation will be small. n For populations whose values are dispersed from the mean, the population variance and standard deviation will be large. n The variance overcomes the weakness of the range by using all the values in the population. 3 -29

LO 3 -4 Computing the Variance Steps in computing the variance: Step 1: Find the mean. Stepdifference 2: the between Find observation eachand the mean, and square that difference. Step 3: Sum all the squared differences found in Step 2. Step number of items in the population. 3 -30

LO 3 -4 Example – Variance and Standard Deviation The number of traffic citations issued during the last twelve months in Beaufort County, South Carolina, is reported below: What is the population variance? Step 1: Find the mean. 3 -31

LO 3 -4 Example – Variance and Standard Deviation Continued What is the population variance? Step 2: Find the difference between each observation and the mean of 29, and square that difference. Step 3: Sum all the squared differences found in Step 2. Step 4: Divide the sum of the squared differences by the number of items in the population. 3 -32

LO 3 -4 Sample Variance 3 -33

LO 3 -4 Example – Sample Variance The hourly wages for a sample of part-time employees at Home Depot are: $12, $20, $16, $18, and $19. The sample mean is $17. What is the sample variance? 3 -34

LO 3 -4 Sample Standard Deviation 3 -35

LO 3 -5 Explain and apply Chebyshev’s theorem and the Empirical Rule. 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? 3 -36

Chebyshev’s Theorem n 3 -37

LO 3 -5 The Empirical Rule 3 -38

Applying the Empirical Rule n 1. 2. 3. A sample of rental rates at Park Apartments approximates a symmetrical, bell-shaped distribution. The sample mean is $500, the standard deviation is $20. Using Empirical Rule answer these questions: About 68% of rental rates are between what two amounts? About 95% of rental rates are between what two amounts? Almost all the rental rates are between what two amounts? 3 -39

Applying the Empirical Rule n 3 -40

LO 3 -6 Compute the mean and standard deviation of grouped data. The Arithmetic Mean of Grouped Data 3 -41

LO 3 -6 Example - The Arithmetic Mean of Grouped Data Recall in Chapter 2, we constructed a frequency distribution for Applewood Auto Group profit data for 180 vehicles sold. The information is repeated in the table. Determine the arithmetic mean profit per vehicle. 3 -42

LO 3 -6 Example - The Arithmetic Mean of Grouped Data 3 -43

LO 3 -6 Example - Standard Deviation of Grouped Data Refer to the frequency distribution for the Applewood Auto Group data used earlier. Compute the standard deviation of the vehicle profits. 3 -44