2011 Pearson Education, Inc

Transcription

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2 Statistics for Business and Economics Chapter 2 Methods for Describing Sets of Data

3 Summary of Central Tendency Measures Measure Formula Description Mean x i / n Balance Point Median ( n +1) Middle Value Position 2 When Ordered Mode none Most Frequent

4 Shape 1. Describes how data are distributed 2. Measures of Shape Skew = Symmetry Left-Skewed Mean Median Symmetric Mean = Median Right-Skewed Median Mean

5 2.5 Numerical Measures of Variability

6 Range 1. Measure of dispersion 2. Difference between largest & smallest observations Range = x largest x smallest 3. Ignores how data are distributed Range = 10 7 = 3 Range = 10 7 = 3

7 Variance & Standard Deviation 1. Measures of dispersion 2. Most common measures 3. Consider how data are distributed 4. Show variation about mean (x or μ) x =

8 Standard Notation Measure Sample Population Mean Standard Deviation x s Variance s 2 2 Size n N

9 Sample Variance Formula n x i x 2 s 2 i 1 n 1 x 1 x 2 x 2 x 2 L x n x 2 n 1 n 1 in denominator!

10 Sample Standard Deviation Formula s s 2 n i 1 x i x 2 n 1 x 1 x 2 x 2 x 2 L x n x 2 n 1

11 Variance Example Raw Data: s s 2 2 i n ( x i x ) 1 i 1 n 1 2 where x 8. 3 n ( ) ( ) 8. 3 ( 7. 7 ) n x i

12 Thinking Challenge You re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. What are the variance and standard deviation of the stock prices?

13 Sample Variance Variation Solution* Raw Data: s s 2 2 n ( x x ) 2 i x i i 1 i 1 where x n 1 n ( 17 ) ( ) ( ) n

14 Variation Solution* Sample Standard Deviation x i x 2 s s 2 i 1 n 1 n

15 Summary of Variation Measures Measure Formula Description Range X largest X smallest Total Spread Standard Deviation (Sample) n i 1 x i x 2 n 1 Dispersion about Sample Mean Standard Deviation (Population) Variance (Sample) n i 1 n i 1 x i µ x 2 N x i x 2 n 1 Dispersion about Population Mean Squared Dispersion about Sample Mean

16 2.6 Interpreting the Standard Deviation

17 Interpreting Standard Deviation: Chebyshev s Theorem Applies to any shape data set No useful information about the fraction of data in the interval x s to x + s At least 3/4 of the data lies in the interval x 2s to x + 2s At least 8/9 of the data lies in the interval x 3s to x + 3s In general, for k > 1, at least 1 1/k 2 of the data lies in the interval x ks to x + ks

18 Interpreting Standard Deviation: Chebyshev s Theorem x 3s x 2s x s x x s x 2s x 3s No useful information At least 3/4 of the data At least 8/9 of the data

19 Chebyshev s Theorem Example Previously we found the mean closing stock price of new stock issues is 15.5 and the standard deviation is Use this information to form an interval that will contain at least 75% of the closing stock prices of new stock issues.

20 Chebyshev s Theorem Example At least 75% of the closing stock prices of new stock issues will lie within 2 standard deviations of the mean. x = 15.5 s = 3.34 (x 2s, x + 2s) = ( , ) = (8.82, 22.18)

21 Interpreting Standard Deviation: Empirical Rule Applies to data sets that are mound shaped and symmetric Approximately 68% of the measurements lie in the interval x s to x s Approximately 95% of the measurements lie in the interval x 2s to x 2s Approximately 99.7% of the measurements lie in the interval x 3s to x 3s

22 Interpreting Standard Deviation: Empirical Rule x 3s x 2s x s x x + s x +2s x + 3s Approximately 68% of the measurements Approximately 95% of the measurements Approximately 99.7% of the measurements

23 Empirical Rule Example Previously we found the mean closing stock price of new stock issues is 15.5 and the standard deviation is If we can assume the data is symmetric and mound shaped, calculate the percentage of the data that lie within the intervals x + s, x + 2s, x + 3s.

24 Empirical Rule Example According to the Empirical Rule, approximately 68% of the data will lie in the interval (x s, x + s), ( , ) = (12.16, 18.84) Approximately 95% of the data will lie in the interval (x 2s, x + 2s), ( , ) = (8.82, 22.18) Approximately 99.7% of the data will lie in the interval (x 3s, x + 3s), ( , ) = (5.48, 25.52)

25 2.7 Numerical Measures of Relative Standing

26 Numerical Measures of Relative Standing: Percentiles Describes the relative location of a measurement compared to the rest of the data The p th percentile is a number such that p% of the data falls below it and (100 p)% falls above it Median = 50 th percentile

27 Percentile Example You scored 560 on the GMAT exam. This score puts you in the 58 th percentile. What percentage of test takers scored lower than you did? What percentage of test takers scored higher than you did?

28 Percentile Example What percentage of test takers scored lower than you did? 58% of test takers scored lower than 560. What percentage of test takers scored higher than you did? (100 58)% = 42% of test takers scored higher than 560.

29 Numerical Measures of Relative Standing: z Scores Describes the relative location of a measurement compared to the rest of the data Sample z score z x s x Population z score Measures the number of standard deviations away from the mean a data value is located z x µ

30 Z Score Example The mean time to assemble a product is 22.5 minutes with a standard deviation of 2.5 minutes. Find the z score for an item that took 20 minutes to assemble. Find the z score for an item that took 27.5 minutes to assemble.

31 Z Score Example x = 20, μ = 22.5 σ = 2.5 z = x μ σ = 2.5 = 1.0 x = 27.5, μ = 22.5 σ = 2.5 z = x μ σ = 2.5 = 2.0

32 Interpretation of z Scores for Mound-Shaped Distributions of Data 1. Approximately 68% of the measurements will have a z-score between 1 and Approximately 95% of the measurements will have a z-score between 2 and Approximately 99.7% of the measurements will have a z-score between 3 and 3. (see the figure on the next slide)

33 Interpretation of z Scores

34 2.8 Methods for Detecting Outliers: Box Plots and z-scores

35 Outlier An observation (or measurement) that is unusually large or small relative to the other values in a data set is called an outlier. Outliers typically are attributable to one of the following causes: 1. The measurement is observed, recorded, or entered into the computer incorrectly. 2. The measurement comes from a different population. 3. The measurement is correct but represents a rare (chance) event.

36 Quartiles Measure of noncentral tendency Split ordered data into 4 quarters 25% 25% 25% 25% Q 1 Q 2 Q 3 Lower quartile Q L is 25 th percentile. Middle quartile m is the median. Upper quartile Q U is 75 th percentile. Interquartile range: IQR = Q U Q L

37 Quartile (Q 2 ) Example Raw Data: Ordered: Position: Q 2 is the median, the average of the two middle scores ( )/2 = 8.8

38 Quartile (Q 1 ) Example Raw Data: Ordered: Position: Q L or Q 1 is median of bottom half = 6.3

39 Quartile (Q 3 ) Example Raw Data: Ordered: Position: Q U or Q 3 is median of bottom half = 10.3

40 Interquartile Range 1. Measure of dispersion 2. Also called midspread 3. Difference between third & first quartiles Interquartile Range = Q 3 Q 1 4. Spread in middle 50% 5. Not affected by extreme values

41 Thinking Challenge You re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. What are the quartiles, Q 1 and Q 3, and the interquartile range?

42 Quartile Solution* Q 1 : Raw Data: Ordered: Position: Q 1 is the median of the bottom half, the average of the two middle scores ( )/2 = 12.5

43 Quartile Solution* Q 3 : Raw Data: Ordered: Position: Q 3 is the median of the bottom half, the average of the two middle scores ( )/2 = 17.5

44 Interquartile Range Solution* Interquartile Range: Raw Data: Ordered: Position: Interquartile Range = Q 3 Q 1 = = 5

45 Box Plot 1. Graphical display of data using 5-number summary X smallest Q 1 Median Q 3 X largest

46 Box Plot 1. Draw a rectangle (box) with the ends (hinges) drawn at the lower and upper quartiles (Q L and Q U ). The median data is shown by a line or symbol (such as + ). 2. The points at distances 1.5(IQR) from each hinge define the inner fences of the data set. Line (whiskers) are drawn from each hinge to the most extreme measurements inside the inner fence.

47 Box Plot 3. A second pair of fences, the outer fences, are defined at a distance of 3(IQR) from the hinges. One symbol (*) represents measurements falling between the inner and outer fences, and another (0) represents measurements beyond the outer fences. 4. Symbols that represent the median and extreme data points vary depending on software used. You may use your own symbols if you are constructing a box plot by hand.

48 Definition: Boxplot A boxplot is a graph of lines (from lowest point inside the lower inner fence to highest point in the upper inner fence) and boxes (from Lower Quartile to Upper quartile) indicating the position of the median. * Outliers Lowest data Point more than the lower inner fence Lower Quartile Median Upper Quartile Highest data Point less than the upper inner fence 48

49 Suspected outliers and highly Suspected Outliers lie suspected Above 1.5 IQRs but below 3 IQRs from the Upper Quartile Below 1.5 IQRs but above 3 IQRs from the Lower Quartile Highly Suspected Outliers lie Above 3 IQRs from the Upper Quartile Below 3 IQRs from the Lower Quartile. 49

50 Example - Fax 28 data points: 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 6, 6, 8, 10, 13, 25, 37 Min=1, Q L = 2, M = 3, Q U = 6, Max = 37. IQR=6-2=4. Inner Fence extremes: Q IQR=-4, Q IQR =12 Outer Fence extremes: Q 1-3IQR=-10, Q 3 +3 IQR=18 Suspected Outliers: 13 Highly Suspected Outliers: 25, 37 50

51 SPSS Use Analyze/Descriptive Statistics/Explore Under Plots, make certain Box plots is selected 51

52 Shape & Box Plot Left-Skewed Symmetric Right-Skewed Q 1 Median Q 3 Q 1 Median Q 3 Q 1 Median Q 3

53 Detecting Outliers Box Plots: Observations falling between the inner and outer fences are deemed suspect outliers. Observations falling beyond the outer fence are deemed highly suspect outliers. z-scores: Observations with z-scores greater than 3 in absolute value are considered outliers. (For some highly skewed data sets, observations with z-scores greater than 2 in absolute value may be outliers.)