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McGraw-Hill/Irwin © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 10 Describing Data Distributions

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10-2 Modal and Median Category Categorical Data Print Output Frequency Table Occupational Status: CategoryCodeFreq.Pct.Adj.Cum. Professional13713.814.114.1 Mgr., Executive26223.123.637.6 Admin., Clerical36925.726.263.9 Engr., Technical4166.06.170.0 Sales, Marketing53011.211.481.4 Craft, Trade6228.28.489.7 Semi-Skilled72710.110.3100.0 Missing Data051.9 Missing Total268100.0100.0

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10-3 Frequency and Percentage Distributions Report Format AgeNumberPercent Over 50 9422.4 36 to 5018845.4 18 to 3513231.9 AgeNumberPercent Over 50 9422.4 36 to 5018845.4 18 to 3513231.9

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10-4 Bar Chart With Frequency Labels 132 188 94 050100150200 18 to 35 36 to 50 Over 50 Number

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10-5 Vertical Bar Chart With Percentage Labels 0% 10% 20% 30% 40% 50% 60% 22.7% Over 50 45.4% 36 to 50 31.9% 18 to 35

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10-6 Pie Chart With Percentage Labels 22.7% 45.4% 31.9% Over 50 36 to 50 18 to 35

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10-7 Descriptive Statistical Tools ScaleAverageSpreadShape NominalMode OrdinalModeInterquartile Range MedianData Range Minimum, Maximum OrdinalModeInterquartile Range MedianData Range Minimum, Maximum IntervalModeStandard Deviation Skewness & RatioModeInterquartile Range Kurtosis MedianData Range Maximum & Minimum IntervalModeStandard Deviation Skewness & RatioModeInterquartile Range Kurtosis MedianData Range Maximum & Minimum

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10-8 Choosing an Average Mean The sum divided by the number Inappropriate for highly skewed distributions Overly sensitive to extreme values Median Middle value when arrayed from low to high Unaffected by asymmetry or extreme values Mode Peak of a continuous distribution Category with the highest frequency Only legitimate average for nominal data

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10-9 Median Mode Mean Measures of Central Tendency

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10-10 Spread and Standard Deviation Standard Deviation Root mean squared deviation from the mean Special properties that make it very useful Normal Distributions 68% of data are within ± 1 S.D. of the mean 95% of data are within ± 2 S.D. of the mean 99% of data are within ± 3 S.D. of the mean

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10-11 99% w/i ± 3 S.D. 95% w/i ± 2 S.D. 68% w/i ± 1 S.D. Mean Spread and Standard Deviation

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10-12 Median Zero Skewness Indicates Symmetry Mean Mode

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10-13 Mode Positive Skewness Leans Left Mean Median

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10-14 Negative Skewness Leans Right Mode Mean Median

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10-15 Zero Kurtosis Indicates Normality Median Mean Mode

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10-16 Negative Kurtosis: A Low Peak and High Tails Median Mean Mode

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10-17 Positive Kurtosis: A High Peak and Low Tails Median Mean Mode

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10-18 Statistics Mean3.800Skewness-0.887 Median4.000Kurtosis0.092 Mode4.000Std. dev.1.128 Number100Std. err.0.113 155.05.05.0 21010.010.015.0 31515.015.030.0 44040.040.070.0 53030.030.0100.0 Total100100.0100.0 CodeFreq.Pct.Adj.Cum. Frequency and Percentage Table 26% 42% 16% 11% 5% 0%10%20%30%40%50% 1 2 3 4 5 Bar Chart 0 10 20 30 40 50 12345 Line Plot Mean, Median, and Mode

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10-19 Mean5.66 Median 4 Mode4 Averages and Outliers 051015202530 One Two Three Four Five Six Fifty This bar chart appears at a glance to show a symmetrical distribution. In fact, there is radical asymmetry resulting from 5 outliers with values of 50.

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10-20 Outliers Correctly Shown 0 5 10 15 20 25 30 16111621263136414651 This more clear representation of the distribution makes the radical asymmetry very obvious.

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10-21 Normal Amount of Data to the Left and Right of the Mean 13.5% 2.5% 34% Mean Standard Normal Distribution

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10-22 More Data to the Left than to the Right of the Mean 7.5%9.5%0.3%0.0%47%33% Mean Positively Skewed Distribution

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10-23 More Toward the Center than in the Tails of the Distribution 8.0% 1.5% 40.5% Mean Distribution with Positive Kurtosis

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10-24 More Toward the Center than in the Tails of the Distribution 17% 4% 29% Mean Distribution with Negative Kurtosis

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10-25 Statistical Inference and Confidence Intervals Objective To make inferences about the population based on the sample Sample Statistics Used as estimates of the population parameters Estimates Are Imperfect The probability of error can be determined Confidence Interval The range within which the parameter is likely to be from the sample mean at a given probability

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10-26 Statistical Inference and Confidence Intervals Sampling Distribution of Means The distribution that would result if samples of a given size were taken again and again and the mean of each sample were plotted. Standard Error of the Estimate The standard deviation of the theoretical sampling distribution of means. Confidence Interval Probabilities 68% chance the parameter is within ± 1 S.E 95% chance the parameter is within ± 2 S.E. 99% chance the parameter is within ± 3 S.E.

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10-27 99% C.I. 95% C.I. Confidence Interval Diagram Mean = 50 Standard Error = 5 2030 40 506070 80

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McGraw-Hill/Irwin © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. End of Chapter 10

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