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1 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 2-5 Measures of Variation

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2 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Objective Compute measures of variability.

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3 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Waiting Times of Bank Customers at Different Banks in minutes Jefferson Valley Bank Bank of Providence

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4 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Jefferson Valley Bank Bank of Providence Jefferson Valley Bank Bank of Providence Mean Median Mode Midrange Waiting Times of Bank Customers at Different Banks in minutes

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5 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Section 2.5 Measures of Variation This last example shows that each bank has the same measures of center, but a closer look at the distribution of waiting times shows that the variability of waiting times is not the same.

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6 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Figure 2-14 Dotplots of Waiting Times

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7 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Measures of Variation 1.Range 2.Standard Deviation 3.Variance 4.Interquartile Range

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8 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Measures of Variation Range value greatest least value

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9 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Range Jefferson Valley Range = = 1.2 minutes Providence Range = 10.0 – 4.2 = 5.8 minutes Jefferson ValleyProvidence

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10 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Measures of Variation Interquartile Range IQR = Q3 – Q1

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11 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Interquartile Range (IQR) = Q3-Q1 Jefferson Valley Median = 7.2 Q1 = 6.7 Q3 = 7.7 IQR = = 1.0 Providence Median = 7.2 Q1 = 5.8 Q3 = 8.5 IQR = 8.5 – 5.8 = 2.7 minutes Jefferson ValleyProvidence

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12 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman a measure of variation of the scores about the mean (average deviation from the mean) Measures of Variation Standard Deviation

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13 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Sample Standard Deviation Formula

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14 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Sample Standard Deviation Formula Formula 2-4 ( x - x ) 2 n - 1 S =S =

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15 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Computing Standard Deviation 1.Compute the mean x

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16 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Computing Standard Deviation 1.Compute the mean 2.Subtract the mean from each data value xx-x

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17 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Computing Standard Deviation 1.Compute the mean 2.Subtract the mean from each data value 3.Square the differences xx-x

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18 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Computing Standard Deviation 1.Compute the mean 2.Subtract the mean from each data value 3.Square the differences 4.Sum the squared differences xx-x

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19 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Computing Standard Deviation 1.Compute the mean 2.Subtract the mean from each data value 3.Square the differences 4.Sum the squared differences 5.Divide the sum by (n-1) xx-x

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20 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Computing Standard Deviation 1.Compute the mean 2.Subtract the mean from each data value 3.Square the differences 4.Sum the squared differences 5.Divide the sum by (n-1) 6.Take the square root of this result xx-x

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21 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 1.Compute the mean Computing Standard Deviation Providence Bank #5 pg81

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22 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 1.Compute the mean 2.Subtract the mean from each data value Computing Standard Deviation Providence Bank #5 pg81

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23 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Computing Standard Deviation Providence Bank #5 pg81 1.Compute the mean 2.Subtract the mean from each data value 3.Square the differences

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24 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 1.Compute the mean 2.Subtract the mean from each data value 3.Square the differences 4.Sum the squared differences Computing Standard Deviation Providence Bank #5 pg81

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25 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 1.Compute the mean 2.Subtract the mean from each data value 3.Square the differences 4.Sum the squared differences 5.Divide the sum by (n-1) Computing Standard Deviation Providence Bank #5 pg81

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26 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 1.Compute the mean 2.Subtract the mean from each data value 3.Square the differences 4.Sum the squared differences 5.Divide the sum by (n-1) 6.Take the square root of this result Computing Standard Deviation Providence Bank #5 pg81

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27 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Population Standard Deviation 2 ( x - µ ) N =

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28 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Symbols for Standard Deviation Sample Population x x n s S x x n-1 Book Some graphics calculators Some non-graphics calculators Textbook Some graphics calculators Some non-graphics calculators Articles in professional journals and reports often use SD for standard deviation and VAR for variance.

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29 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Measures of Variation Variance

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30 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Measures of Variation Variance standard deviation squared

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31 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Measures of Variation Variance standard deviation squared s 2 2 } use square key on calculator Notation

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32 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Sample Variance Population Variance ( x - x ) 2 n - 1 s 2 = (x - µ)2 (x - µ)2 N 2 =

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33 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Round-off Rule for measures of variation Carry one more decimal place than is present in the original set of values. Round only the final answer, never in the middle of a calculation.

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34 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Page 81 3, 5

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35 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Standard Deviation from a Frequency Table

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36 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Standard Deviation from a Frequency Table RatingFrequency midpoints

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37 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman RatingFrequency midpoints

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38 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Standard Deviation from a Frequency Table RatingFrequency midpoints

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39 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Standard Deviation from a Frequency Table RatingFrequency midpoints

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40 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Standard Deviation from a Frequency Table

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41 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Page , 11, 29

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42 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Objective: Understanding Standard Deviation Apply the Empirical Rule Apply Chebyshev’s Rule Apply Range Rule of Thumb Identify Unusual Values

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43 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range 4 s or (minimum usual value) (maximum usual value)

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44 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range 4 s or (minimum usual value) (maximum usual value) Range 4 s

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45 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range 4 s or (minimum usual value) (maximum usual value) Range 4 s = highest value - lowest value 4

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46 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Usual Sample Values

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47 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman minimum ‘usual’ value (mean) - 2 (standard deviation) minimum x - 2(s) Usual Sample Values

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48 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman minimum ‘usual’ value (mean) - 2 (standard deviation) minimum x - 2(s) maximum ‘usual’ value (mean) + 2 (standard deviation) maximum x + 2(s) Usual Sample Values

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49 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman x The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-15

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50 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman x - s x x + sx + s 68% within 1 standard deviation 34% The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-15

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51 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman x - 2s x - s x x + 2s x + sx + s 68% within 1 standard deviation 34% 95% within 2 standard deviations The Empirical Rule (applies to bell-shaped distributions ) 13.5% FIGURE 2-15

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52 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman x - 3s x - 2s x - s x x + 2s x + 3s x + sx + s 68% within 1 standard deviation 34% 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean The Empirical Rule (applies to bell-shaped distributions ) 0.1% 2.4% 13.5% FIGURE 2-15

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53 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example Application of the Empirical Rule A set of 1000 test scores has a symmetric, mound-shaped distribution. The mean is 175 and the standard deviation is 10. Approximately what percent of the scores are between 175 and 195?

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54 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 95% of scores are between The shaded area is half of the area within 2 standard deviations of the mean.... so (.5)(.95)=.475

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55 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example Application of the Empirical Rule A set of 1000 test scores has a symmetric, mound-shaped distribution. The mean is 175 and the standard deviation is 10. Approximately how many scores are between 155 and 165?

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56 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example Application of the Empirical Rule The area from 155 to 195 is 0.95 The area from 165 to 185 is 0.68 Subtracting these values gives the area from 155 to 165 and 185 to 195 combined. We need to divide the result by 2 because the symmetry splits this area equally = divided by 2 =

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57 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Chebyshev’s Theorem applies to distributions of any shape. the proportion (or fraction) of any set of data lying within K standard deviations of the mean is always at least 1 - 1/K 2, where K is any positive number greater than 1. at least 3/4 (75%) of all values lie within 2 standard deviations of the mean. at least 8/9 (89%) of all values lie within 3 standard deviations of the mean.

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58 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Measures of Variation Summary For typical data sets, it is unusual for a score to differ from the mean by more than 2 or 3 standard deviations.

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59 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Page , 20, 21, 22, 23, 24, 25, 30 a and b

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