Calculate Karl Pearson's Coefficient of Skewness From The Following Data

What is Skewness?

Skewness refers to the asymmetry of a probability distribution. A distribution is symmetric when it looks the same to the left and right of the center point. When a distribution is skewed, one of its tails is longer than the other.

There are three types of skewness:

• Positive skewness: The right tail is longer; the mass of the distribution is concentrated on the left.
• Negative skewness: The left tail is longer; the mass of the distribution is concentrated on the right.
• Zero skewness: The distribution is perfectly symmetrical.

Skewness is important in statistics because it affects the interpretation of data and the choice of appropriate statistical tests.

Karl Pearson's Formula

Karl Pearson developed a formula to measure skewness based on the third moment of a distribution. The formula is:

Where:

• n = number of observations
• xᵢ = each individual value
• x̄ = mean of the values
• s = standard deviation

This formula calculates the coefficient of skewness, which can range from negative infinity to positive infinity. A value of zero indicates no skewness, while positive and negative values indicate right and left skewness, respectively.

How to Calculate

To calculate Karl Pearson's coefficient of skewness:

1. Collect your dataset of at least 3 values.
2. Calculate the mean (average) of your data.
3. Calculate the standard deviation of your data.
4. For each data point, subtract the mean and divide by the standard deviation to get the z-scores.
5. Cube each z-score.
6. Sum all the cubed z-scores.
7. Multiply the sum by the skewness coefficient [n / (n-1)(n-2)].

The result is your coefficient of skewness. Use the calculator on this page to perform these calculations automatically.

Interpreting Results

The coefficient of skewness can be interpreted as follows:

• Positive value: The distribution is skewed to the right (positive skewness).
• Negative value: The distribution is skewed to the left (negative skewness).
• Zero value: The distribution is symmetrical.

The absolute value of the coefficient indicates the degree of skewness. A value between -0.5 and 0.5 is considered approximately symmetrical, while values outside this range indicate significant skewness.

Worked Example

Let's calculate the coefficient of skewness for the following dataset: 2, 3, 4, 5, 6.

1. Mean (x̄) = (2 + 3 + 4 + 5 + 6) / 5 = 4
2. Standard deviation (s) ≈ 1.414
3. Calculate z-scores:
   • (2-4)/1.414 ≈ -1.414
   • (3-4)/1.414 ≈ -0.707
   • (4-4)/1.414 = 0
   • (5-4)/1.414 ≈ 0.707
   • (6-4)/1.414 ≈ 1.414
4. Cube each z-score:
   • (-1.414)³ ≈ -2.828
   • (-0.707)³ ≈ -0.353
   • 0³ = 0
   • (0.707)³ ≈ 0.353
   • (1.414)³ ≈ 2.828
5. Sum of cubed z-scores ≈ -2.828 - 0.353 + 0 + 0.353 + 2.828 = 0
6. Skewness coefficient = [5 / (5-1)(5-2)] * 0 = 0

The result is 0, indicating the distribution is symmetrical. This matches our expectation since the data is evenly distributed around the mean.