this needs to be done on excel, due tonight. lmk
Chapter 3 Describing Data: Numerical Measures Chapter 3 Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 1 Learning Objectives LO3-1 Compute and interpret the mean, the median, and the mode. LO3-2 Compute a weighted mean. LO3-3 Compute and interpret the geometric mean. LO3-4 Compute and interpret the range, variance, and standard deviation. LO3-5 Explain and apply Chebyshev’s theorem and the Empirical Rule. LO3-6 Compute the mean and standard deviation of grouped data. 3-‹#› 2 Measures of Location 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: The arithmetic mean The median The mode LO3-1 Compute and interpret the mean, the median, and the mode. 3-‹#› 3 Characteristics of the Mean 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. LO3-1 3-‹#› 4 . Population Mean For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values: LO3-1 3-‹#› 5 Example – Population Mean LO3-1 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? What is the mean number of miles between exits? 3-‹#› 6 Example – Population Mean LO3-1 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 Parameter versus Statistic PARAMETER A measurable characteristic of a population. STATISTIC A measurable characteristic of a sample. LO3-1 3-‹#› . 8 Properties of the Arithmetic Mean 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. LO3-1 3-‹#› 9 Sample Mean For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values: LO3-1 3-‹#› 10 . Example – Sample Mean LO3-1 3-‹#› 11 The Median Properties of the median: There is a unique median for each data set. It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur. It can be computed for ratio-level, interval-level, and ordinal-level data. It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class. MEDIAN The midpoint of the values after they have been ordered from the minimum to the maximum values. LO3-1 3-‹#› 12 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. LO3-1 3-‹#› 13 The Mode MODE The value of the observation that appears most frequently. LO3-1 3-‹#› 14 Example - Mode LO3-1 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 The Relative Positions of the Mean, Median and the Mode LO3-1 16 Weighted Mean The weighted mean of a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed with the following formula: LO3-2 Compute a weighted mean. 3-‹#› 17 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? LO3-2 3-‹#› 18 The Geometric Mean Useful in finding the average change of percentages, ratios, indexes, or growth rates over time. 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. The geometric mean will always be less than or equal to the arithmetic mean. LO3-3 Compute and interpret the geometric mean. 3-‹#› 19 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? LO3-3 3-‹#› The Geometric Mean: Finding an Average Percent Change Over Time EXAMPLE: During the decade of the 1990s, and into the 2000s, 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? LO3-3 3-‹#› 21 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. LO3-4 Compute and interpret the range, variance, and standard deviation. 3-‹#› 22 Measures of Dispersion Range Variance Standard Deviation LO3-4 3-‹#› 23 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 LO3-4 3-‹#› 24 Variance and Standard Deviation The variance and standard deviations are nonnegative and are zero only if all observations are the same. For populations whose values are near the mean, the variance and standard deviation will be small. For populations whose values are dispersed from the mean, the population variance and standard deviation will be large. The variance overcomes the weakness of the range by using all the values in the population. VARIANCE The arithmetic mean of the squared deviations from the mean. STANDARD DEVIATION The square root of the variance. LO3-4 3-‹#› 25 Computing the Variance Steps in computing the variance: Step 1: Find the mean. Step 2:Find the difference between each observation and the mean, 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. LO3-4 3-‹#› 26 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. LO3-4 3-‹#› 27 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. LO3-4 3-‹#› 28 Sample Variance LO3-4 3-‹#› 29 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? LO3-4 3-‹#› 30 Sample Standard Deviation LO3-4 3-‹#› 31 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? LO3-5 Explain and apply Chebyshev’s theorem and the Empirical Rule. 3-‹#› 32 The Empirical Rule LO3-5 3-‹#› 33 The Arithmetic Mean of Grouped Data LO3-6 Compute the mean and standard deviation of grouped data. 3-‹#› 34 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. LO3-6 3-‹#› 35 Example - The Arithmetic Mean of Grouped Data LO3-6 3-‹#› 36 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. 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