Symmetry, Skewness and Kurtosis

We consider a random variable x and a data set S = {x1, x2, …, xn} of size n which contains possible values of x. The data set can represent either the population being studied or a sample drawn from the population.

Looking at S as representing a distribution, the skewness of S is a measure of symmetry while kurtosis is a measure of peakedness of the data in S.

Symmetry and Skewness

Definition 1: We use skewness as a measure of symmetry. If the skewness of S is zero then the distribution represented by S is perfectly symmetric. If the skewness is negative, then the distribution is skewed to the left, while if the skew is positive then the distribution is skewed to the right (see Figure 1 below for an example).

Excel calculates the skewness of a sample S as follows:

Skewness Formula in Excel

where is the mean and s is the standard deviation of S. To avoid division by zero, this formula requires that n > 2.

Observation: When a distribution is symmetric, the mean = median, when the distribution is positively skewed the mean > median and when the distribution is negatively skewed the mean < median.

Excel Function: Excel provides the SKEW function as a way to calculate the skewness of S, i.e. if R is a range in Excel containing the data elements in S then SKEW(R) = the skewness of S.

Excel 2013 Function: There is also a population version of the skewness given by the formula

Skewness population

This version has been implemented in Excel 2013 using the function, SKEW.P.

It turns out that for range R consisting of the data in S = {x1, …, xn}, SKEW.P(R) = SKEW(R)*(n–2)/SQRT(n(n–1)) where n = COUNT(R).

Real Statistics Function: Alternatively, you can calculate the population skewness using the SKEWP(R) function, which is contained in the Real Statistics Resource Pack.

Example 1: Suppose S = {2, 5, -1, 3, 4, 5, 0, 2}. The skewness of S = -0.43, i.e. SKEW(R) = -0.43 where R is a range in an Excel worksheet containing the data in S. Since this value is negative, the curve representing the distribution is skewed to the left (i.e. the fatter part of the curve is on the right). Also SKEW.P(R) = -0.34. See Figure 1.

Shape: skewness and kurtosis

Figure 1 – Examples of skewness and kurtosis

Observation: SKEW(R) and SKEW.P(R) ignore any empty cells or cells with non-numeric values.

Kurtosis

Definition 2: Kurtosis provides a measurement about the extremities (i.e. tails) of the distribution of data, and therefore provides an indication of the presence of outliers.

Excel calculates the kurtosis of a sample S as follows:

Kurtosis Formula in Excel

where is the mean and s is the standard deviation of S. To avoid division by zero, this formula requires that n > 3.

Observation: It is commonly thought that kurtosis provides a measure of peakedness (or flatness), but this is not true. Kurtosis pertains to the extremities and not to the center of a distribution.

Excel Function: Excel provides the KURT function as a way to calculate the kurtosis of S, i.e. if R is a range in Excel containing the data elements in S then KURT(R) = the kurtosis of S.

Observation: The population kurtosis is calculated via the formula

image089x

which can be calculated in Excel via the formula

=(KURT(R)*(n-2)*(n-3)/(n-1)-6)/(n+1)

Real Statistics Function: Excel does not provide a population kurtosis function, but you can use the following Real Statistics function for this purpose:

KURTP(R, excess) = kurtosis of the distribution for the population in range R1. If excess = TRUE (default) then 3 is subtracted from the result (the usual approach so that a normal distribution has kurtosis of zero).

Example 2: Suppose S = {2, 5, -1, 3, 4, 5, 0, 2}. The kurtosis of S = -0.94, i.e. KURT(R) = -0.94 where R is a range in an Excel worksheet containing the data in S. The population kurtosis is -1.114. See Figure 1.

Observation: KURT(R) ignores any empty cells or cells with non-numeric values.

Graphical Illustration

We now look at an example of these concepts using the chi-square distribution.

Chi-square distribution

Figure 2 – Example of skewness and kurtosis

Figure 2 contains the graphs of two chi-square distributions (with different degrees of freedom df). We study the chi-square distribution elsewhere, but for now note the following values for the kurtosis and skewness:

Comparison of skewness and kurtosis

Figure 3 – Comparison of skewness and kurtosis

Both curves are asymmetric, and skewed to the right (i.e. the fat part of the curve is on the left). This is consistent with the fact that the skewness for both is positive. But the blue curve is more skewed to the right, which is consistent with the fact that the skewness of the blue curve is larger.

26 Responses to Symmetry, Skewness and Kurtosis

  1. Emmanuel says:

    I think the Kurtosis formula is too long to be crammed, can I get assistance on how go understand if?

  2. Mina says:

    hello,
    the Kurtosis value on my data is above 2 (+3). i think it should be between negative and positive 2. how can I change it to obtain normality??

  3. Your description of kurtosis is incorrect. Kurtosis measures nothing about the peak of the distribution. It only measures tails (outliers).

    The logic is simple: The average of the Z^4 values (which is the kurtosis) gets virtually no contribution from |Z| values that are less than 1.0, where any “peak” would be. It’s only the large |Z| values (the outliers) that contribute to kurtosis.

    Here is an article that elaborates : http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4321753/pdf/nihms-599845.pdf

    The “peakedness” description is an unfortunate historical error, promoted for ages, apparently by inertia. People just parroted what others said.

    • Charles says:

      Peter,
      Thank you very much for sharing this and setting the record straight. I will change the website accordingly.
      I will also add your article to the Bibliography.
      Charles

  4. rose says:

    hi;
    I want to make sure by ” n ”
    did you mean the sample size ?

  5. Steven says:

    Hey Charles

    Say you had a bunch of returns data and wished to check the skewness of that data. In this instance, which would be appropriate – Skew() or Skew.P()

    I would imagine Skew() because Skew.P() refers to a population and you don’t have the population here, you merely have a bunch of return data don’t you. OR when dealing with financial returns do you assume that the data you have is the population?

  6. sandeep says:

    I want two suggestion
    1. I have 1000 dollar money i wants to distribute it in 12 month in such a way that peak is 1.6 time the average ( using normal distribution curve)
    2. As per my knowledge the peak in bell curve is attended in mean (i.e by 6.5 month) but if i want peak at 40% month (i.e 12*40/100 time ) and peak will still remain 1.6 time the average( i.e peak= 1.6*100/12) than what will be the distribution

    • Charles says:

      The peak is usually considered to be the high point in the curve, which for a normal distribution occurs at the mean. Thus, I don’t know what it means for the peak to be 1.6 times the average (which is the mean). Please explain what you mean by the peak?
      Charles

    • very dificult to compute a curtosis how to be know a sample is group or ungrouped data

      • Charles says:

        Jessa,
        You can compute kurtosis using the KURT function. I don-t understand teh part about group or ungrouped data.
        Charles

  7. adekola says:

    What the differences and similarities between skewness and kurtosis?

  8. Professor Amir says:

    Based on my experience of teaching the statistics, you can use pearson coefficient of skewness which is = mean – mode divide by standard deviation or use this = 3(mean – median) divide by standard deviation. mostly book covered use the first formula for ungrouped data and second formula for grouped data

  9. Gaylord Lussac says:

    “the kurtosis value of the blue curve is lower” should read “the kurtosis value of the blue curve is higher”.
    In fact, zero skew is seldom observed. See for example http://www.aip.de/groups/soe/local/numres/bookcpdf/c14-1.pdf

    • Charles says:

      Gaylord,
      Thanks for catching this typo. I have now corrected the webpage. I appreciate your help in making the website better.
      Charles

  10. soniya says:

    Hi Charles. I want to know ‘what is the typical sort of skew?’

  11. Namo Ghate says:

    Using the scores I have, how can I do the GRAPHIC ILLUSTRATION of skewness and kurtosis on the excel?

    • Charles says:

      Namo,
      I am not sure what you mean by a graphic illustration. I have tried to do this with the graph of the chi-square distribution, which was done using Excel (see the details in the Examples Workbook, which you can download for free).
      Charles

  12. Karim says:

    Thanks for helping us understanding those basics of stat.

Leave a Reply

Your email address will not be published. Required fields are marked *