Household income in most countries has roughly the same mean as a symmetric bell-shaped dataset might, but plot it and the picture looks nothing like a bell curve: a large cluster of households earns modest incomes, with a long thin tail stretching out toward a small number of extremely high earners. The mean gets pulled upward by that tail, while the bulk of households sit below it. Mean and standard deviation, by themselves, cannot tell you this is happening — you need a measure of shape, and that’s what skewness and kurtosis provide.
What Is Skewness?
Skewness measures the asymmetry of a distribution — whether one tail is longer or heavier than the other. A distribution is symmetric (zero skew) when its left and right sides are mirror images, like a perfect normal distribution. It is positively skewed (right-skewed) when it has a long tail stretching to the right — the household income example is the classic case, where most values cluster at the low-to-moderate end and a few extreme high values pull the tail rightward. It is negatively skewed (left-skewed) when the long tail stretches to the left instead — exam scores on an easy test often show this pattern, with most students clustering near the top and a small number of very low scores dragging a tail leftward.
A crucial and frequently tested consequence of skewness: in a right-skewed distribution, the mean is pulled toward the tail and typically exceeds the median, while in a left-skewed distribution, the mean is typically less than the median. In a perfectly symmetric distribution, mean and median coincide. This relationship is often the fastest way to spot skewness without plotting anything at all — just compare mean to median.
The Skewness Formula
One common formula for sample skewness (Pearson’s moment coefficient of skewness) is:
Skewness = [∑(xi − x̄)³ / n] / s³
where x̄ is the sample mean, s is the sample standard deviation, and n is the sample size. This is the third standardised moment of the distribution — “standardised” because dividing by s³ makes the result independent of the units the data is measured in, allowing skewness to be compared across completely different datasets, whether measured in dollars, centimetres, or test scores.
Cubing the deviations (rather than squaring, as with variance) preserves their sign: large positive deviations (values well above the mean) contribute positive terms, and large negative deviations contribute negative terms. If positive deviations dominate the sum — a few very large values pulling the total upward — skewness comes out positive, matching a right-skewed shape.
Worked Example: Calculating Skewness
Take a small dataset of 5 household incomes ($000s): 30, 32, 35, 38, 120. Clearly one extreme value (120) will pull this distribution’s shape.
Mean x̄ = (30+32+35+38+120)/5 = 255/5 = 51. Deviations from the mean: −21, −19, −16, −13, +69.
Standard deviation: squared deviations are 441, 361, 256, 169, 4761, summing to 5988. Sample variance = 5988/4 = 1497, so s = √1497 ≈ 38.7.
Cubed deviations: (−21)³=−9261, (−19)³=−6859, (−16)³=−4096, (−13)³=−2197, (69)³=328,509. Sum = 306,096.
Skewness = [306,096 / 5] / (38.7)³ = 61,219.2 / 57,960.6 ≈ 1.06
A skewness of about +1.06 confirms a moderate-to-strong right skew — consistent with the visual picture of four modest incomes and one extreme outlier pulling the distribution’s tail rightward. As a rough guide, skewness between −0.5 and +0.5 is often described as approximately symmetric, between ±0.5 and ±1 as moderately skewed, and beyond ±1 as highly skewed — though these thresholds are conventions, not hard rules.
What Is Kurtosis?
While skewness measures asymmetry, kurtosis measures the “tailedness” of a distribution — how much of the data sits in the extreme tails versus clustered near the centre, compared to a normal distribution. A distribution with high kurtosis (leptokurtic) has fatter tails and a sharper peak than normal — more extreme values than you’d expect, alongside more values tightly clustered near the mean. A distribution with low kurtosis (platykurtic) has thinner tails and a flatter peak — fewer extreme outliers, with values more evenly spread. A distribution matching the normal distribution’s tail behaviour exactly (mesokurtic) has kurtosis of 3 in the raw formula, which is why excess kurtosis (raw kurtosis minus 3) is commonly reported instead, so that a normal distribution reads as 0, positive values indicate fatter-than-normal tails, and negative values indicate thinner-than-normal tails.
The Kurtosis Formula
The formula for kurtosis (fourth standardised moment) is:
Kurtosis = [∑(xi − x̄)⁴ / n] / s⁴
with excess kurtosis calculated as this value minus 3. Raising deviations to the fourth power (an even power, so all terms are positive) heavily weights extreme values — a deviation twice as large contributes sixteen times as much to the sum, which is exactly why kurtosis is so sensitive to outliers in the tails specifically, rather than to the general spread that variance already captures.
Why This Matters: Fat Tails in Finance
Kurtosis is not just a textbook curiosity — it played a direct role in the 2008 financial crisis. Many risk models assumed asset returns followed a normal distribution, which implies extreme market moves should be astronomically rare. In reality, financial returns exhibit significant excess kurtosis — fat tails meaning extreme gains and losses happen far more often than a normal-distribution model would predict. Models built on the false assumption of normality systematically underestimated the probability of severe losses, contributing to inadequate risk buffers across the financial system when several “impossible” events occurred close together.
Skewness, Kurtosis and the Choice of Statistical Method
Both measures directly affect which statistical tools are appropriate. Highly skewed data can make the mean a misleading measure of “typical” value — which is exactly why median household income, not mean household income, is the standard figure reported in economic statistics. High kurtosis warns that methods assuming normality (many t-tests and confidence intervals, at small sample sizes) may understate the true probability of extreme outcomes, which is why analysts routinely check skewness and kurtosis as a first diagnostic step before trusting a normal-based model.
Common Mistakes
Assuming kurtosis measures “peakedness” alone. While high kurtosis often does correspond to a sharper peak, the defining feature is tail weight — two distributions can have similar peak shapes but very different tail behaviour, and it’s the tails that kurtosis is fundamentally measuring.
Confusing raw kurtosis with excess kurtosis. A raw kurtosis value of 3 corresponds to a normal distribution, not zero — always check whether a reported figure is raw or excess kurtosis before interpreting it, since software packages differ in which convention they report by default.
Ignoring skewness when choosing between mean and median. Reporting the mean of a heavily skewed dataset (like income or house prices) without checking for skewness can badly mislead readers about what a “typical” value actually looks like.
Practice Question
A dataset has mean = 50 and median = 42. What does this suggest about the distribution’s skewness, and why?
Answer: Since the mean (50) is greater than the median (42), this suggests the distribution is positively (right) skewed. In a right-skewed distribution, a small number of unusually large values pull the mean upward while the median, which is resistant to extreme values, stays closer to where most of the data actually clusters. This pattern is exactly what appears in datasets like income or house prices, where a few very high values inflate the mean relative to the bulk of the data.
References
1. Moore, D.S., McCabe, G.P. and Craig, B.A. (2021) Introduction to the Practice of Statistics. W.H. Freeman.
2. Taleb, N.N. (2007) The Black Swan: The Impact of the Highly Improbable. Random House.
3. Triola, M.F. (2022) Elementary Statistics. Pearson.
4. Field, A. (2018) Discovering Statistics Using IBM SPSS Statistics. SAGE.
5. Groeneveld, R.A. and Meeden, G. (1984) ‘Measuring Skewness and Kurtosis’, The Statistician, 33(4), pp. 391–399.