2. Dr. C. Ertuna1 Measure of Shape (Lesson – 02C) Seeing Details of Data in Numbers

3. Dr. C. Ertuna2 Measures of Shape The number of days a company takes to pay invoices are given on the left Do they pay more often “late” than “early”? To answer this question we need to measure the shape of distribution first. Data: St-CE-Ch02-x1-Examples-Slide 33

4. Dr. C. Ertuna3 Measures of Shape Normal Distribution Most frequently observed distribution of phenomenon in nature. Examples: –Coloration of leafs –Wing span of flies etc.

5. Dr. C. Ertuna4 Measures of Shape Frequency distribution (histogram) of sample data can take on different shapes –Skewness and –Kurtosis are two widely used tools to measure the shape.

6. Dr. C. Ertuna5 Measures of Shape (Cont.) Skewness measures the degree of asymmetry of a distribution around the mean. (for meaningful results ~30 obs. or above) Skewness of a distributions is considered –high if the value is greater than | 2*ses | –moderate if the value is between | (0.5 to 1)*ses | –symmetric if the value is 0.

7. Dr. C. Ertuna6 Measures of Shape (Cont.)

8. Dr. C. Ertuna7 Measures of Shape (Cont.) Negatively skewed means longer tail on the left In a perfectly symmetric distribution, the mean, median, and mode would all be the same Comparing measure of central tendency reveal information about the shape (mean < mode indicates negative skewness).

9. Dr. C. Ertuna8 Measures of Shape (Cont.) Kurtosis measures the relative peakedness or flatness of a distribution. Zero Kurtosis indicates normal distribution Positive kurtosis indicates a relatively peaked distribution. Negative kurtosis indicates a relatively flat distribution.

10. Dr. C. Ertuna9 Measures of Shape (Cont.) The following two distributions have the same variance approximately the same skew but differ in kurtosis.

11. Dr. C. Ertuna10 Example: Measure of Shape (cont.) The number of days a company takes to pay invoices are given on the left 1. Compute skewness and kurtosis 2. Explain the meaning of the results. Data: St-CE-Ch02-x1-Examples-Slide 33

12. Dr. C. Ertuna11 Example: Measure of Shape (cont.) Analyze/ Descriptive Statistics/Descriptive Select the variable & move to the right pane Select Options, check Mean, Std.Dev, Kurtosis, Skewness. Continue / Ok

13. Dr. C. Ertuna12 Example: Measure of Shape (cont.)

14. Dr. C. Ertuna13 Example: Measure of C. T. (cont.) The the skewness and kurtosis of the data pertaining the number of days Tracway takes to pay invoices are the following: –Skewness = 1.25 –Kurtosis = - 0.17 Explain the meaning of the results.

15. Dr. C. Ertuna14 Measures of Shape (Cont.) ses = Standard error of skewness ses ≈ SQRT(6/N); N = number of Observations sek = Standard error of kurtosis sek ≈ SQRT(24/N); N = number of Observations ses & sek are measures for statistical significance

16. Dr. C. Ertuna15 Measures of Shape (Cont.) Skewness measures the degree of asymmetry of a distribution around the mean. (for meaningful results ~30 obs. or above) Skewness of a distributions is considered –high if the value is greater than | 2*ses | –moderate if the value is between | (0.5 to 1)*ses | –symmetric if the value is 0.

17. Dr. C. Ertuna16 Measures of Shape (Cont.) Perspective: “Majority of Observations” Skewness + 2*ses- 2*ses PositivelyNegatively Strongly Vast Majority Below mean Vast Majority Above mean Moderately a Good Majority Below mean a Good Majority Above mean Slightly Slight Majority Below mean Slight Majority Above mean more than 50%

18. Dr. C. Ertuna17 Measures of Shape (Cont.) Kurtosis measures the relative peakedness or flatness of a distribution. Kurtosis of a distributions is considered –high if the value is greater than | 2*sek | –moderate if the value is between | (0.5 to 1)*sek | –symmetric if the value is 0.

19. Dr. C. Ertuna18 Example: Measure of C. T. (cont.) Data: St-CE-Ch02-x1-Examples-Slide 33

20. Dr. C. Ertuna19 Meaning: Measure of Shape (cont.) 1The skew value suggests that the number of days the company takes to pay invoices are highly asymmetric (greater than | 2*ses |). There are more occasions than one normally expect that the company is stretching its payments substantially. However most of the time the company is paying its invoices earlier than what the mean value suggests.

21. Dr. C. Ertuna20 Meaning: Measure of Shape (cont.) In general this company pays its debt on the invoices, on average, within 7.5 days. This company pays the vast majority of its invoices earlier than 7.5 days (skew=1.25 > 2*ses – majority of obs. pmt.). However, when it pays late than the late payments are far late than one would normally expect (skew=1.25 > 2*ses – asymmetry of obs. pmt.).

22. Dr. C. Ertuna21 Next Lesson (Lesson – 02D) Frequency Distribution & Histogram