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Measure of Shape

(Lesson – 02C) Seeing Details of Data in Numbers

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Measures of Shape

1/A B C

2 Month Wrk. Days

3 Jan-96 8.32

4 Feb-96 8.28

5 Mar-96 8.29

6 Apr-96 8.32

7 May-96 8.36

8 Jun-96 8.35

9 Jul-96 8.34

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

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Measures of Shape

• Normal Distribution

• Most frequently observed distribution of phenomenon in nature.

• Examples: – Coloration of leafs– Wing span of flies etc.

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Measures of Shape

Frequency distribution (histogram) of sample data can take on different shapes

– Skewness and

– Kurtosisare two widely used tools to measure the shape.

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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.

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Measures of Shape (Cont.)

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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).

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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.

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Measures of Shape (Cont.)

The following two distributions have the same variance approximately the same skew but differ in kurtosis.

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Example: Measure of Shape (cont.)

1/A B C

2 Month Wrk. Days

3 Jan-96 8.32

4 Feb-96 8.28

5 Mar-96 8.29

6 Apr-96 8.32

7 May-96 8.36

8 Jun-96 8.35

9 Jul-96 8.34

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

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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

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Example: Measure of Shape (cont.)

Descriptive StatisticsN Mean Std. DeviationSkewness KurtosisStatistic Statistic Statistic Statistic Std. Error Statistic Std. Error

Working Days 60 7.4585 0.458342 1.248769 0.308694 -0.16469 0.608492Valid N (listwise) 60

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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.

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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

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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.

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Measures of Shape (Cont.)

Perspective:“Majority of

Observations”

Skewness+ 2*ses - 2*ses

Positively

Negatively

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%

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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.

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Example: Measure of C. T. (cont.)

1/A B C D E F G H

2 MonthWorking

Days

3 Jan-96 8.32 2*SE 1*SE 0.5*SE

4 Feb-96 8.28 Skewness = 1.249 0.617 0.309 0.154 05 Mar-96 8.29 Kurtosis = -0.165 1.217 0.608 0.304 06 Apr-96 8.32

7 May-96 8.36

8 Jun-96 8.35 Range = =MAX(B4:B23)-MIN(B4:B23)9 Jul-96 8.34

Data: St-CE-Ch02-x1-Examples-Slide 33

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Meaning: Measure of Shape (cont.)

1 The 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.

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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.).

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Next Lesson

(Lesson – 02D) Frequency Distribution &

Histogram