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

10-2 Modal and Median Category Categorical Data Print Output Frequency Table Occupational Status: CategoryCodeFreq.Pct.Adj.Cum. Professional Mgr., Executive Admin., Clerical Engr., Technical Sales, Marketing Craft, Trade Semi-Skilled Missing Data051.9 Missing Total

10-3 Frequency and Percentage Distributions Report Format AgeNumberPercent Over to to AgeNumberPercent Over to to

10-4 Bar Chart With Frequency Labels to to 50 Over 50 Number

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

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

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

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

10-9 Median Mode Mean Measures of Central Tendency

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

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

10-12 Median Zero Skewness Indicates Symmetry Mean Mode

10-13 Mode Positive Skewness Leans Left Mean Median

10-14 Negative Skewness Leans Right Mode Mean Median

10-15 Zero Kurtosis Indicates Normality Median Mean Mode

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

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

10-18 Statistics Mean3.800Skewness Median4.000Kurtosis0.092 Mode4.000Std. dev Number100Std. err Total CodeFreq.Pct.Adj.Cum. Frequency and Percentage Table 26% 42% 16% 11% 5% 0%10%20%30%40%50% Bar Chart Line Plot Mean, Median, and Mode

10-19 Mean5.66 Median 4 Mode4 Averages and Outliers 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.

10-20 Outliers Correctly Shown This more clear representation of the distribution makes the radical asymmetry very obvious.

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

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

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

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

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

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.

% C.I. 95% C.I. Confidence Interval Diagram Mean = 50 Standard Error =

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