The McGraw-Hill Companies, Inc. 2008McGraw-Hill/IrwinDescribing Data:Numerical MeasuresChapter 3GOALS Calculate the arithmetic mean, weighted mean, median, mode, and geometric mean. Explain the characteristics, uses, advantages, and disadvantages of each measure of location. Identify the position of the mean, median, and mode for both symmetric and skewed distributions. Compute and interpret the range, mean deviation, variance, and standard deviation. Understand the characteristics, uses, advantages, and disadvantages of each measure of dispersion. Understand Chebyshevs theorem and the Empirical Rule as they relate to a set of observations.2Characteristics of the MeanThe arithmetic mean is the most widely used measure of location. It requires the interval scale. Its major characteristics are: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.3Population MeanFor ungrouped data, the population mean is the sum of all the population values divided by the total number of population values:4EXAMPLE Population Mean5Sample MeanlFor ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values: 6EXAMPLE Sample Mean7Properties of the Arithmetic MeanlEvery set of interval-level and ratio-level data has a mean.lAll the values are included in computing the mean.lA set of data has a unique mean.lThe mean is affected by unusually large or small data values.lThe arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero. 8Weighted MeanlThe weighted mean of a set of numbers X1, X2, ., Xn, with corresponding weights w1, w2, .,wn, is computed from the following formula: 9EXAMPLE Weighted MeanThe 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?10The MedianlThe Median is the midpoint of the values after they have been ordered from the smallest to the largest. There are as many values above the median as below it in the data array. For an even set of values, the median will be the arithmetic average of the two middle numbers.11Properties of the MedianlThere is a unique median for each data set.lIt is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur.lIt can be computed for ratio-level, interval-level, and ordinal-level data.lIt can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class. 12EXAMPLES - MedianThe ages for a sample of five college students are:21, 25, 19, 20, 22Arranging 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, 75Arranging the data in ascending order gives: 73, 75, 76, 80. Thus the median is 75.5 13The ModelThe mode is the value of the observation that appears most frequently.14Example - Mode15Mean, Median, Mode Using ExcelTable 24 in Chapter 2 shows the prices of the 80 vehicles sold last month at Whitner Autoplex in Raytown, Missouri. Determine the mean and the median selling price. The mean and the median selling prices are reported in the following Excel output. There are 80 vehicles in the study. So the calculations with a calculator would be tedious and prone to error.16Mean, Median, Mode Using Excel17The Relative Positions of the Mean, Median and the Mode18The Geometric MeanlUseful in finding the average change of percentages, ratios, indexes, or growth rates over time.lIt has a 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, which compound or build on each other. lThe geometric mean will always be less than or equal to the arithmetic mean. lThe geometric mean of a set of n positive numbers is defined as the nth root of the product of n values. lThe formula for the geometric mean is written:19EXAMPLE Geometric MeanSuppose you receive a 5 percent increase in salary this year and a 15 percent increase next year. The average annual percent increase is 9.886, not 10.0. Why is this so? We begin by calculating the geometric mean. 20EXAMPLE Geometric Mean (2)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?21DispersionWhy Study Dispersion?A measure of location, such as the mean or the median, only describes the center of the data. It is valuable from that standpoint, 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.22Samples of Dispersions23Measures of DispersionlRangelMean DeviationlVariance and Standard Deviation24EXAMPLE RangeThe 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.Range = Largest Smallest value = 80 20 = 6025EXAMPLE Mean DeviationThe 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.26EXAMPLE Variance and Standard DeviationThe 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?27EXAMPLE Sample VarianceThe hourly wages for a sample of part-time employees at Home Depot are: $12, $20, $16, $18, and $19. What is the sample variance?28Chebyshevs TheoremThe arithmetic mean biweekly amount contributed by the Dupree Paint employees to the companys 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?29The Empirical Rule30The Arithmetic Mean of Grouped Data31Recall in Chapter 2, we constructed a frequency distribution for the vehicle selling prices. The information is repeated below. Determine the arithmetic mean vehicle selling price.The Arithmetic Mean of Grouped Data - Example32The Arithmetic Mean of Grouped Data - Example33Standard Deviation of Grouped Data34Standard Deviation of Grouped Data - ExampleRefer to the frequency distribution for the Whitner Autoplex data used earlier. Compute the standard deviation of the vehicle selling prices35End of Chapter 336