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Navigation: STATISTICS WITH PRISM 9 > Descriptive statistics

Interpreting results: Skewness

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Key facts about skewness  

Skewness quantifies how symmetrical the distribution is.

A symmetrical distribution has a skewness of zero.

An asymmetrical distribution with a long tail to the right (higher values) has a positive skew.

An asymmetrical distribution with a long tail to the left (lower values) has a negative skew.

The skewness is unitless.

Any threshold or rule of thumb is arbitrary, but here is one: If the skewness is greater than 1.0 (or less than -1.0), the skewness is substantial and the distribution is far from symmetrical.

How skewness is computed

Skewness has been defined in multiple ways. The steps below explain the method used by Prism, called G1 (one of the most common methods). It is identical to the skew() function in Excel.

1.We want to know about symmetry around the sample mean. So the first step is to subtract the sample mean from each value,  The result will be positive for values greater than the mean, negative for values that are smaller than the mean, and zero for values that exactly equal the mean.

2.To compute a unitless measures of skewness,  divide each of the differences computed in step 1 by the standard deviation of the values (note that when calculating skewness, N should be used instead of N-1 to calculate the standard deviation). These ratios (the difference between each value and the mean divided by the standard deviation) are called z scores.

3.For each value, compute z3. Note that cubing values preserves the sign. The cube of a positive value is still positive, and the cube of a negative value is still negative.

4.Calculate the average for the z3 values (calculate the sum of all of the z3 values, and divide this sum by the number values in the sample). If the distribution is symmetrical, the positive and negative values will balance each other, and the average will be close to zero. If the distribution is not symmetrical, the average will be positive if the distribution is skewed to the right, and negative if skewed to the left. This average value is called the Fisher-Pearson coefficient of skewness, and is sometimes given as "g1".

5.Correct for bias. The average computed in step 4 is biased with small samples -- its absolute value is smaller than it should be. Correct for the bias by multiplying the mean of z3 by the ratio sqrt(N*(N-1))/(N-2). This correction increases the value if the skewness is positive, and makes the value more negative if the skewness is negative. As N becomes larger, this correction becomes closer to a value of 1, resulting in less correction. But with small samples, the correction is more substantial. The final corrected value is sometimes referred to as the "adjusted Fisher-Pearson coefficient of Skewness" and is sometimes given as "G1". This is the value that Prism reports for skewness.

The full formula for the value of skewness that Prism reports is given as:

where

More on skewness and kurtosis

 

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