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Pearson’s Coefficient of Skewness

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What is Pearson’s Coefficient of Skewness?

Karl Pearson developed two methods to find skewness in a sample.

  1. Pearson’s Coefficient of Skewness #1 uses the mode. The formula is:
    pearson skewness
    Where xbar = the mean, Mo = the mode and s = the standard deviation for the sample.
    See: Pearson Mode Skewness.
  2. Pearson’s Coefficient of Skewness #2 uses the median. The formula is:
    Pearson's Coefficient of Skewness
    Where xbar = the mean, Mo = the mode and s = the standard deviation for the sample.
    It is generally used when you don’t know the mode.

Sample problem: Use Pearson’s Coefficient #1 and #2 to find the skewness for data with the following characteristics:

  • Mean = 70.5.
  • Median = 80.
  • Mode = 85.
  • Standard deviation = 19.33.

Pearson’s Coefficient of Skewness #1 (Mode):
Step 1: Subtract the mode from the mean: 70.5 – 85 = -14.5.
Step 2: Divide by the standard deviation: -14.5 / 19.33 = -0.75.

Pearson’s Coefficient of Skewness #2 (Median):
Step 1: Subtract the median from the mean: 70.5 – 80 = -9.5.
Step 2: Multiply Step 1 by 3: -9.5(3) = -28.5
Step 2: Divide by the standard deviation: -28.5 / 19.33 = -1.47.

Caution: Pearson’s first coefficient of skewness uses the mode. Therefore, if the mode is made up of too few pieces of data it won’t be a stable measure of central tendency. For example, the mode in both these sets of data is 9:
1 2 3 4 5 6 7 8 9 9.
1 2 3 4 5 6 7 8 9 9 9 9 9 9 9 9 9 9 9 9 10 12 12 13.
In the first set of data, the mode only appears twice. This isn’t a good measure of central tendency so you would be cautioned not to use Pearson’s coefficient of skewness. The second set of data has a more stable set (the mode appears 12 times). Therefore, Pearson’s coefficient of skewness will likely give you a reasonable result.

Interpretation

In general:

  • The direction of skewness is given by the sign.
  • The coefficient compares the sample distribution with a normal distribution. The larger the value, the larger the distribution differs from a normal distribution.
  • A value of zero means no skewness at all.
  • A large negative value means the distribution is negatively skewed.
  • A large positive value means the distribution is positively skewed.

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Pearson’s Coefficient of Skewness was last modified: October 15th, 2017 by Andale

29 thoughts on “Pearson’s Coefficient of Skewness

  1. Abhishek mittal

    sir
    can i calculate coefficient of skewness from
    mean , Q1 and Q3
    if yes, then pls tell me the formula……
    if no, then how i calculate coefficient of skewness……

  2. Omar

    Pearson’s Coefficient of Skewness #2 :
    what about excel function of this coefficient … can you tell me what is it please ?

    thank you

  3. Andale Post author

    As long as you can find the mode, median etc. you should be able to use one of these formulas.

  4. Varun Marlecha

    Sir
    Karl pearson’s coefficient of skweness of a distributions+0.32.it’s standard deviation is 6.5 and mean is 29.6. Find the mode and median of the distribution.

  5. Andale Post author

    The results of the calculation tell you:

    The direction direction of the skew (positive or negative).
    How the sample compares with a normal (symmetric) distribution. The further the skew result is from zero, the greater the skewness.

  6. Barry

    You repeated the same sentence (explanation) from the 1st formula for the 2nd formula. The 2nd formula defines everything in the first formula not the 2nd.

  7. Andale Post author

    Hi, Barry,
    Thanks for your comment. I’m always happy to make a correction. However, I don’t see what part you’re referring to. The explanation for the first formula includes “mode,” the second, the “median.”
    “Pearson’s Coefficient of Skewness #1 uses the mode” and “Pearson’s Coefficient of Skewness #2 uses the median.
    If this isn’t the part you mean, could you quote the part where you see a correction is needed?
    Thanks,
    Stephanie

  8. Rezaul

    if mode is 50 and median is 55,standard deviation 625.then what is the skewness of the distribution?

  9. Phelly Okoth

    Sir, how do i calculate median weight and standard deviation given data in class intervals and frequency, then compute Kearl Pearson’s coefficient of skewness and interprete?

  10. Andale Post author

    Hi, Phelly,
    That’s a complicated question. I’d start by finding the median weight for class intervals, which you can find instructions for here. You would use a similar process to find the SD. After that, you should be able to plug in the numbers for the coefficient of skew.

  11. Camille

    Hello,
    when can you tell that you serie is asymetric? From which value?
    For example, if I have 0.2, can I say that my serie isn’t symetric?
    I am working with the coefficient of variation and have to verify that my serie are symetric to be able to use this coefficient.
    Thank you

  12. Andale Post author

    0.2 is pretty low. It’s not perfectly symmetric (a “0” would be perfect symmetry), so I would say you have a slight positive skew, compared to a normal distribution.
    There are other tests to figure out if your data comes from a normal distribution (which would make it symmetric). I’d consider running one. See: assumption of normality for all the different types of tests you can run.

  13. Patrick

    I have a data set with the following properties:
    Mean = 45,452
    Median = 40,003
    Std. Dev. = 34,988
    (45,452-40,003)/34,988=0.156 This is the difference between the mean and median as a fraction of the standard deviation. Why is this multiplied by 3 in the Pearson Coefficient?

  14. Andale Post author

    Hi, Patrick,
    It looks like it’s just a relationship that Pearson noticed (as opposed to something he derived). They give a pretty good explanation over at StatsExchange.
    Also check out the explanation given in this book.

  15. Yiran Michael Yinmaheme Yidaan

    My purpose for the research is accomplished by your materials, and more to the point my understanding of the topic has been enlighten by your materials.
    Thanks

  16. Andale Post author

    Where are you seeing this? If it’s in a book, then different authors sometimes use different symbols.