Skewness

Contents:

• When to use it?
• Formula
• How to calculate?
• Caution - common mistakes?
• Examples
• See Also

When to use it?

If you graph the data using a histogram or bar chart, does it look symmetrical or lop-sided? If your data has more extreme observations to one side of the centre, this long set of data on one side is called a long tail and is measured by the skewness calculation. If there is tail, which means more extreme values on the right, the skew value is positive. If the tail is on your left side when you look at the chart, this skew value is negative.

We can use a calculation called variance or standard deviation to see how much spread or variability is in the data, and the skew value tells us if the data is symmetrical. Normal distributions are symmetrical, and some calculations can be done with normal data distributions that are not suitable with other types of data distributions. Data that is perfectly symmetrical has a skew value of zero.

Positively Skewed:           

Negatively Skewed:         

Symmetrical Distribution:

Formula

Manual Formula

Excel Formula

     

Where:

• s is the sample standard deviation.
• Σ means sum all the values.
• x̄ represents a sample mean.

How to calculate skewness?

The manual formula in most books is: Skew = 3(mean - median) / Standard Deviation. A test question may provide the mean, median, and standard deviation, or you may have to calculate the mean, median, and standard deviation from the raw data.

Excel

Assume you have data values of 1, 2, 3, 4, 5, 6, 7, 8, and 9 in cells A1 to A9.
In Excel, if your teacher accepts the Excel calculation: =SKEW(A1:A9).

If you want Excel to assist in the manual formula, you can get each value for the formula as follows.
In cell A10 for the mean:                            =AVERAGE(A1:A9)
In cell A11 for the median:                          =MEDIAN(A1:A9)
In cell A12 for sample standard deviation: =STDEV(A1:A9)
In cell A13 or the skew calculation:             =3*(A10-A11)/A12

Caution - common mistakes?

You should be aware that computer applications like Minitab or Excel use a different formula for calculating skewness than most books show for manual calculations. Check with your teacher to see which calculation is acceptable.

Examples

1. You are given these numbers: 1, 2, 3, 4, 5, 9, 23, 32, and 69.

Step 1:

Could calculate using Excel SKEW formula: =SKEW(A1:A9)     Answer is 1.95



2. You are given the same numbers above and you use Excel, but do not use the Excel SKEW function.

Step 1:	=AVERAGE(A1:A9) = 16.44 = Mean =MEDIAN(A1:A9)    = 5        = Median =STDEV(A1:A9)      = 22.41 = Standard Deviation
Step 2:	=3*(Mean-Median)/ Standard Deviation = 3*(16.44 - 5) / 22.41 = 1.53



3. The following is a bar chart of a Positive Skewness example, calculated manually, where skewness  =  3(3 - 2) / 2.22  =  1.35 :

       

4. The following is a bar chart of a Negative Skewness example, calculated manually, where skewness  =  3(5 - 6) / 2.19  =  -3 / 2.19  =  -1.37 :

   

5. Following is a bar chart of Symmetrical Distribution, calculated manually, where skewness = 3(4 - 4) / 1.89 = 0 :

   

See Also

The skew formula uses the mean, median, and standard deviation calculations.