9 Measures of Kurtosis
Skewness, in the last chapter, described which way a distribution leans. Kurtosis describes something different: how peaked or flat a distribution is, and specifically how heavy or light its tails are, compared to a normal (bell-curve) distribution with the same variance. Two distributions can be perfectly symmetric, sharing the same mean, spread, and zero skewness, and still differ sharply in kurtosis, one with a sharp peak and fat tails prone to extreme values, the other flatter and more evenly spread. This chapter covers the moment-based coefficient of kurtosis and its classification into leptokurtic, mesokurtic, and platykurtic shapes, for both raw and grouped data.
9.1 What Is Kurtosis?
Kurtosis is often described informally as “peakedness,” but it is more accurately about the tails: a distribution has high kurtosis when it produces extreme values (far from the mean) more often than a normal distribution would, and low kurtosis when extreme values are rarer than normal. The three standard classifications, all defined relative to the normal distribution, are:
- Mesokurtic: Kurtosis matches the normal distribution. The baseline case.
- Leptokurtic: Higher kurtosis than normal, a sharper peak and fatter tails, meaning more extreme values than a normal distribution would produce.
- Platykurtic: Lower kurtosis than normal, a flatter peak and thinner tails, meaning fewer extreme values than a normal distribution would produce.
A useful business intuition: two investment portfolios can have identical average returns and identical standard deviations, yet the leptokurtic one is riskier, because it is more prone to rare but extreme swings (“fat tails”) than its bell-curve-shaped standard deviation alone would suggest. Kurtosis is exactly the number that exposes this hidden risk.
9.2 Moment-Based Coefficient of Kurtosis
Kurtosis is computed from the fourth moment about the mean, \(m_4 = \frac{\sum (x_i - \bar{x})^4}{n}\), standardised by the square of the variance.
\[\beta_2 = \frac{m_4}{SD^4} = \frac{m_4}{(\text{Variance})^2}\]
A normal distribution has \(\beta_2 = 3\) exactly, which is why statisticians usually report excess kurtosis instead, simply \(\beta_2\) minus 3, so that a normal distribution scores 0 and the classifications become easy to read directly off the sign:
\[\text{Excess Kurtosis} = \gamma_2 = \beta_2 - 3\]
\[\gamma_2 > 0 \Rightarrow \text{Leptokurtic} \qquad \gamma_2 = 0 \Rightarrow \text{Mesokurtic} \qquad \gamma_2 < 0 \Rightarrow \text{Platykurtic}\]
Example
For the seven test marks 58, 62, 58, 74, 65, 58, 70 (mean \(\approx 63.57\), population \(SD \approx 5.95\)), the fourth moment is \(m_4 \approx 2348.06\).
\[\beta_2 = \frac{2348.06}{5.95^4} \approx \frac{2348.06}{1252.29} \approx 1.88 \qquad \gamma_2 = 1.88 - 3 = -1.12\]
The excess kurtosis is negative, so this small dataset is platykurtic, flatter than a normal distribution. That is consistent with the data itself: three of the seven marks are tied at exactly 58, clustering the middle rather than spreading into extreme tails.
9.3 Kurtosis for Grouped Data
As with skewness, kurtosis extends to grouped data by substituting class marks for individual observations, using the grouped mean and standard deviation already computed in Chapters 5 and 6. Continuing with the same 30-student marks distribution:
| Class Interval | Class Mark (\(x\)) | Frequency (\(f\)) |
|---|---|---|
| 30–40 | 35 | 2 |
| 40–50 | 45 | 4 |
| 50–60 | 55 | 6 |
| 60–70 | 65 | 6 |
| 70–80 | 75 | 5 |
| 80–90 | 85 | 4 |
| 90–100 | 95 | 3 |
\[m_4 = \frac{\sum f_i (x_i - \bar{x})^4}{N} \qquad \beta_2 = \frac{m_4}{SD^4} \qquad \gamma_2 = \beta_2 - 3\]
Example
Using the grouped mean of 65.67 and grouped \(SD\) of 17.11 from Chapters 5 and 6, the grouped fourth moment works out to \(\beta_2 \approx 2.10\), giving excess kurtosis \(\gamma_2 \approx -0.90\). Once again platykurtic, and close to the excess kurtosis of the raw, ungrouped 30 marks (\(\approx -0.91\)), confirming that the 30 exam marks are genuinely flatter and more evenly spread than a normal distribution, not bunched into a sharp central peak with occasional extreme scores.
9.4 Skewness and Kurtosis Together
Skewness and kurtosis are the third and fourth “moments” of a distribution’s shape, following the mean (first moment, location) and variance (second moment, spread). A distribution with skewness \(\approx 0\) and excess kurtosis \(\approx 0\) is a reasonable match for the normal distribution, the single most important distribution in statistics and the foundation for most of the inferential techniques (confidence intervals, hypothesis tests) that this book covers in its next module. Checking skewness and kurtosis together is, in effect, an informal first check of whether that assumption is reasonable for a given dataset.
Neither skewness nor kurtosis is “good” or “bad” on its own, they simply describe shape. But both matter enormously downstream: many statistical tests and models formally assume approximately normal data (skewness near 0, excess kurtosis near 0), and a highly skewed or heavily leptokurtic dataset can silently invalidate their conclusions if the assumption is not checked first.
Recap
Chapters 4 through 8 have now built a complete descriptive toolkit from a single 30-student worked example: organizing raw marks into a frequency distribution (Chapter 4), locating its center with mean, weighted mean, geometric mean, median, and mode (Chapter 5), measuring its spread with range, quartile deviation, mean deviation, variance, standard deviation, and coefficient of variation (Chapter 6), and now describing its shape with skewness and kurtosis (Chapters 7 and 8). Together these numbers can describe almost any dataset without ever needing to see every individual value. The next modules in this book build on this foundation toward probability, inferential statistics, and data visualization.
Summary
| Concept | Description |
|---|---|
| Core Idea | |
| Kurtosis | A measure of how peaked or flat a distribution is, and how heavy or light its tails are, relative to normal |
| Mesokurtic | Kurtosis matches the normal distribution; the baseline case, excess kurtosis of zero |
| Leptokurtic | Higher kurtosis than normal: a sharper peak and fatter tails, more prone to extreme values |
| Platykurtic | Lower kurtosis than normal: a flatter peak and thinner tails, fewer extreme values than normal |
| The Coefficient | |
| Fourth Moment | Average of each deviation from the mean raised to the fourth power |
| Moment-Based Coefficient (beta2) | Fourth moment divided by the square of the variance; equals 3 for a normal distribution |
| Excess Kurtosis | Beta2 minus 3, so a normal distribution scores 0 and the sign gives the classification directly |
| Grouped Data | |
| Grouped Kurtosis | Kurtosis computed from class marks and frequencies, using the grouped mean and standard deviation |
| Putting It Together | |
| Skewness and Kurtosis Together | Skewness (3rd moment, shape lean) and kurtosis (4th moment, tail weight) together describe distribution shape |
| Normal Distribution Baseline | Skewness near 0 and excess kurtosis near 0 together suggest data reasonably close to a normal distribution |