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: We use kurtosis as a measure of peakedness (or flatness). Positive kurtosis indicates a relatively peaked distribution. Negative kurtosis indicates a relatively flat distribution.

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.

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. Since this value is negative, the curve representing the distribution is relatively flat. 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

The red curve (df = 10) is flatter than the blue curve (df = 5), which is reflected in the fact that the kurtosis value of the red curve is lower.

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.

20 Responses to Symmetry, Skewness and Kurtosis

  1. rose says:

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

  2. 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?

  3. 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

  4. adekola says:

    What the differences and similarities between skewness and kurtosis?

  5. 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

  6. 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

  7. soniya says:

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

  8. 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

  9. 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 *