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Transcript of McGraw-Hill/Irwin © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 10...
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:
Category Code Freq. Pct. Adj. Cum.
Professional 1 37 13.8 14.1 14.1Mgr., Executive 2 62 23.1 23.6 37.6Admin., Clerical 3 69 25.7 26.2 63.9Engr., Technical 4 16 6.0 6.1 70.0Sales, Marketing 5 30 11.2 11.4 81.4Craft, Trade 6 22 8.2 8.4 89.7Semi-Skilled 7 27 10.1 10.3 100.0Missing Data 0 5 1.9 Missing
Total 268 100.0 100.0
10-3
Frequency and Percentage Distributions Report Format
Age Number Percent
Over 50 94 22.4
36 to 50 188 45.4
18 to 35 132 31.9
Age Number Percent
Over 50 94 22.4
36 to 50 188 45.4
18 to 35 132 31.9
10-4
Bar Chart With Frequency Labels
132
188
94
0 50 100 150 200
18 to 35
36 to 50
Over 50
Number
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
10-6
Pie Chart With Percentage Labels
22.7%
45.4%
31.9% Over 50
36 to 50
18 to 35
10-7
Descriptive Statistical Tools
Scale Average Spread ShapeScale Average Spread Shape
Nominal ModeNominal Mode
Ordinal Mode Interquartile RangeMedian Data Range
Minimum, Maximum
Ordinal Mode Interquartile RangeMedian Data Range
Minimum, Maximum
Interval Mode Standard Deviation Skewness& Ratio Mode Interquartile Range Kurtosis
Median Data Range Maximum & Minimum
Interval Mode Standard Deviation Skewness& Ratio Mode Interquartile Range Kurtosis
Median Data 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
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
10-12
Median
Zero Skewness Indicates Symmetry
Mean
Mode
10-13
Mode
Positive Skewness Leans Left
MeanMedian
10-14
Negative Skewness Leans Right
ModeMeanMedian
10-15
Zero Kurtosis Indicates Normality
MedianMean
Mode
10-16
Negative Kurtosis: A Low Peak and High Tails
MedianMean
Mode
10-17
Positive Kurtosis: A High Peak and Low Tails
MedianMean
Mode
10-18
StatisticsMean 3.800 Skewness -0.887Median 4.000 Kurtosis 0.092Mode 4.000 Std. dev. 1.128Number100 Std. err. 0.113
1 5 5.0 5.0 5.02 10 10.0 10.0 15.03 15 15.0 15.0 30.04 40 40.0 40.0 70.05 30 30.0 30.0 100.0
Total 100 100.0 100.0
Code Freq. 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
01020304050
1 2 3 4 5
Line Plot
Mean, Median, and Mode
10-19
Mean 5.66
Median 4
Mode 4
Averages and Outliers
0 5 10 15 20 25 30
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
0
5
10
15
20
25
30
1 6 11 16 21 26 31 36 41 46 51
• 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%13.5% 2.5%2.5% 34% 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%8.0% 1.5%1.5% 40.5% 40.5%
Mean
Distribution with Positive Kurtosis
10-24
More Toward the Center than in the Tails of the Distribution
17%17% 4%4% 29% 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.
10-27
99% C.I.
95% C.I.
Confidence Interval Diagram
• Mean = 50• Standard Error = 5
20 30 40 50 60 70 80
McGraw-Hill/Irwin © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
End of Chapter 10