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Kurtosis – Quick Introduction

In statistics, kurtosis refers to the “peakedness”
of the distribution for a quantitative variable.
What's meant by “peakedness” is best understood from the example histograms shown below.

Kurtosis Examples

Kurtosis Examples In Histograms

Test 4 is almost perfectly normally distributed. Its excess kurtosis is therefore close to 0.

The distribution for test 3 is somewhat “flatter” than the normal curve: the histogram bars are lower than the middle of the curve and higher towards its tails. Test 3 therefore has a negative excess kurtosis.

Test 5 is more “peaked” than the normal curve: its bars are higher than the peak of the curve and lower towards its tails. Therefore, test 5 has a positive excess kurtosis.

Test 2 roughly follows a uniform distribution. Because it's even flatter than test 3, it has a stronger negative excess kurtosis.

The strongest negative excess kurtosis is seen for test 1, which has a bimodal distribution.

Positive excess kurtosis is often seen for variables having strong (positive) skewness such as test 6.

So now that we've an idea what (excess) kurtosis means, let's see how it's computed.

Kurtosis Formulas

If your data contain an entire population rather than just a sample, the population kurtosis \(K_p\) is computed as

$$K_p = \frac{M_4}{M_2^2}$$

where \(M_2\) and \(M_4\) denote the second and fourth moments around the mean:

$$M_2 = \frac{\sum\limits_{i = 1}^N(X_i - \overline{X})^2}{N}$$

and

$$M_4 = \frac{\sum\limits_{i = 1}^N(X_i - \overline{X})^4}{N}$$

Note that \(M_2\) is simply the population-variance formula.

Kurtosis or Excess Kurtosis?

A normally distributed variable has a kurtosis of 3.0. Since this is undesirable, population excess kurtosis \(EK_p\) is defined as

$$EK_p = K_p - 3$$

so that excess kurtosis is 0.0 for a normally distributed variable.

Now, that's all fine. But what's not fine is that “kurtosis” refers to either kurtosis or excess kurtosis in standard textbooks and software packages without clarifying which of these two is reported.

Anyway.

Our formulas thus far only apply to data containing an entire population. If your data only contain a sample from some population -usually the case- then you'll want to compute sample excess kurtosis \(EK_s\) as

$$EK_s = (N + 1) \cdot EK_p + 6 \cdot \frac{(N - 1)}{(N - 2) \cdot (N - 3)}$$

This formula results in “kurtosis” as reported by most software packages such as SPSS, Excel and Googlesheets. Finally, most text books suggest that the standard error for (excess) kurtosis \(SE_{eks}\) is computed as

$$SE_{eks} \approx \sqrt{\frac{24}{N}}$$

This approximation, however, is not accurate for small sample sizes. Therefore, software packages use a more complicated formula. I won't bother you with it.

Kurtosis Calculation Example

An example calculation for excess kurtosis is shown in this Googlesheet (read-only), partly shown below.

Kurtosis Example Calculation Googlesheets Data

We computed excess kurtosis for scores 1 through 10. First off, we added their mean, M = 5.50. Next,

\(D\) denotes the difference scores (score - mean);
\(D^2\) are squared difference scores that are used for computing a variance;
\(D^4\) are difference scores raised to the fourth power.

As shown below, the remaining computations are fairly simple too.

Kurtosis Example Calculation Googlesheets Formulas

Note that the excess kurtosis = 1.20. The SPSS output shown below confirms this result.

Kurtosis In SPSS Output Table

Platykurtic, Mesokurtic & Leptokurtic

Some terminology related to excess kurtosis is that

Our kurtosis examples illustrate what platykurtic, mesokurtic and leptokurtic distributions tend to look like.

Finding Kurtosis in Excel & SPSS

First off, “kurtosis” always refers to sample excess kurtosis in Excel, Googlesheets and SPSS. It's found in Excel and Googlesheets by using something like =KURT(A2:A11)

SPSS has many options for computing excess kurtosis but I personally prefer using Analyze SPSS Menu Arrow Compare Means SPSS Menu Arrow Means

As shown below, kurtosis can be selected from Options and dragged into the desired position.

Find Kurtosis In SPSS

Right, I think that's about it regarding (excess) kurtosis. If you've any questions or remarks, please throw me a comment below. Other than that:

thanks for reading!

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