Several references I have looked at define skewness as follows:
mean > median: positive, or right-skewness
mean = median: symmetry, or zero-skewness
mean < median: negative, or left-skewness
Now, if I enter the following data into Excel:
-125, -100, -50, -25, -1, 0, 0, 0, 0, 0, 0, 0, 25, 50, 75, 75, 100, 107,
150, 150
You get a mean of 21.55 and a median of 0 so the mean is larger than the
median and the data is right-skewed. Excel returns a skewness of 0.028,
with is positive but barely so.
If I enter the second data set of:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 9, 8, 7, 6, 25, 50, 75, 100, 125
Excel returns a mean of 23.50 and a median of 8.00 so the mean and median
are closer together than data set #1 but the skewness value is 2.035, much
larger than #1. Why should a mean and median that are closer together
generate a skewness measure that is so much larger? Does this mean that the
magnitude of the skewness number has no meaning?
Now, if I delete the two 150's on the end of data set #1 and change the
ranges on the formulae, I get a mean of 7.28 and I still get a median of 0.
Again, the mean is larger than the median so this should be positively
skewed but Excel returns a value of -0.370.
I have verified Excel's calculations manually and they appear to be correct
so it would appear that the commonly used statement that:
mean > median: positive, or right-skewness
mean = median: symmetry, or zero-skewness
mean < median: negative, or left-skewness
is incorrect, or, am I overlooking something?
Excel, and another reference I looked at, state that "The peakeness of a
distribution is measured by its kurtosis. Positive kurtosis indicates a
relatively peaked distribution. Negative kurtosis indicates a relatively
flat distribution."
If that is the case, what does it mean that data set #1 above has a
kurtosis value of zero?
I appreciate any comments you can supply.
Dr. Ronny Richardson
Associate Professor of Management
Southern Polytechnic State University
School of Management
1100 South Marietta Parkway
Marietta, GA 30060-2896
Phone: (770) 528-5542
Fax: (770) 528-4967
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