Slide 1Test 2Practice ProblemsAdditional Reading and ExamplesMotivating ExampleSlide 6Measures of Central LocationMeasures of Central LocationMeasures of Central LocationSlide 10B. Measures of SpreadMeasures of SpreadMeasures of SpreadExample 17Example 17B. Measures of SpreadB. Measures of SpreadMeasures of SpreadSample Standard DeviationSample Standard DeviationSample Standard DeviationSample Standard DeviationSample Standard DeviationSample Standard DeviationSample Standard DeviationVarianceVarianceExample 18Example 18Example 18Standard DeviationStandard DeviationExample 19Example 19Example 19Example 19Measures of SpreadMeasures of SpreadMeasures of SpreadMeasures of SpreadIQRIQRIQRIQRIQRIQRIQRIQRIQRIQRExample 20Example 20Example 20Example 20Example 20Example 20Example 21Example 21Example 21Example 21Example 21Example 21Example 21Slide 65TI-83/84 CalculatorMotivating ExampleMotivating Example SolutionSTAT 210Lecture 11Measures of SpreadSeptember 20, 2017Test 2Monday, September 25Sections III - IV (pages 47 - 95)Combination of multiple choice questions and short answer questions and problems.Bring a calculator and writing instrument.Practice ProblemsPages 94 through 97Relevant problems: IV.3, IV.4, IV.5, IV.6 (a) and (c), and 11 (c)Recommended problems: IV.4, IV.6 (c) and IV.11 (c)Additional Reading and ExamplesPages 90 through 93Motivating ExampleA statistics course at a large university provides free to students statistics review sessions that students can use to answer questions, with help solving problems, and with help studying for tests. The course instructor is interested in the number of students who attend each hour of review session, and selects a sample of 15 review session hours spread out over a month’s time. The number of students who attended these 15 review session hours is as follows. This data will be used throughout the rest of this chapter.6 1 8 3 1 5 11 7 4 28 12 9 2 10 13Top Hat 2Measures of Central Location1. Mean (Average)The population mean is denoted by the Greek letter m (read “mu”) and is the sum of all observations divided by how many individuals that there are in the population. This is (usually) an unknown parameter.Measures of Central LocationThe population mean is estimated by the sample mean, denoted by X (read “X-bar”).X = S x = x1 + x2 + x3 + … + xn n nThis is a statistic.Measures of Central Location2. MedianThe population median is usually denoted by the Greek letter h (read “eta”), and is estimated by the sample median, denoted by M.Top HatB. Measures of SpreadIf all the values of a characteristic are the same then the characteristic is a constant, both the mean and median are the constant value, and there is no spread in the data.If, however, all the values are not the same, then the characteristic is called a variable and of interest is to measure the amount of spread (or dispersion or variability) around the central value.Measures of SpreadData set 1: 10, 10, 10Data set 2: 8, 10, 12Data set 3: 5, 10, 15Data set 4: 0, 10, 20Measures of Spread1. Range = maximum value - minimum valueThe range is a measure of overall variation, not variation around the central value. The range can be heavily influenced by outliers.Example 17Without 391:The smallest observation is 113 and the largest observation is 222.So the range is 222 - 113 = 109.Example 17With 391:The smallest observation is 113 and the largest observation is 391.So the range is 391 - 113 = 278.The range went from 109 to 278 due to the existence of one outlier.B. Measures of Spread2. Standard DeviationThe standard deviation is a measure of variability about the mean.A deviation is the amount that an observation is from the mean: x – X.B. Measures of Spread2. Standard DeviationThe standard deviation is a measure of variability about the mean.The population standard deviation is denoted by s (read “sigma”).Measures of Spread2. Standard DeviationThe standard deviation is a measure of variability about the mean.The population standard deviation is denoted by s.Since all subjects of the population are rarely known, the population standard deviation is usually unknown and must be estimated by the sample standard deviation, denoted S.Sample Standard DeviationS = S (xi - X)2 n - 1Sample Standard Deviation1. Calculate the sample mean X.Sample Standard Deviation1. Calculate the sample mean X.2. Compute the n deviations from the mean: x - X.The sum of the deviations (x - X) will always equal 0.S (x - X) = 0Sample Standard Deviation1. Calculate the sample mean X.2. Compute the n deviations from the mean: x - X.3. Square each deviation: (x - X)2.Sample Standard Deviation1. Calculate the sample mean X.2. Compute the n deviations from the mean: x - X.3. Square each deviation: (x - X)2.4. Sum the squared deviations: S (x - X)2.Sample Standard Deviation1. Calculate the sample mean X.2. Compute the n deviations from the mean: x - X.3. Square each deviation: (x - X)2.4. Sum the squared deviations: S (x - X)2.5. Divide this sum by n - 1: S (x - X)2 n - 1 The divisor n - 1 is called the degrees of freedom.Sample Standard Deviation1. Calculate the sample mean X.2. Compute the n deviations from the mean: x - X.3. Square each deviation: (x - X)2.4. Sum the squared deviations: S (x - X)2.5. Divide this sum by n - 1: S (x - X)2 n - 16. Take the square root of the above number.VarianceA measure of spread around the mean that is related to the standard deviation is the variance. The population variance is denoted by s2 (read “sigma squared”) and since the entire population is usually unknown the population variance is estimated using the sample variance s2.s2 = S (x - X)2 n - 1VarianceThe population variance is estimated using the sample variance s2.s2 = S (x - X)2 n – 1The standard deviation is preferred to the variance because while the standard deviation is measured in the units of the original data, the variance is measured in the units squared.Example 18From example 13, the sample mean is X = 162.625.Example 18x x - X (x - X)2128 128-162.625 = -34.625 1198.89150 150-162.625 = -12.625 159.39183 183-162.625 = 20.375 415.14222 222-162.625 = 59.375 3525.39113113-162.625 = -49.625 2462.64154 154-162.625 = - 8.625 74.39201 201-162.625 = 38.375 1472.64150 150-162.625 = -12.625 159.39 0 9467.875Example 18x x -
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