Mean, Median, and Mode in Skewed Families

Why These Three Summaries Can Disagree

For a symmetric distribution, the mean, median, and mode often coincide. In a skewed distribution, they can separate: a long right tail can pull the mean above the median, while the mode still marks where the density is highest. The amount and even the ordering of that separation depend on the distribution family and its parameters.

This tutorial compares these summaries for lognormal, Weibull, and Pareto distributions. The mean is an expectation, so it need not be finite. A median is a central quantile, while a mode describes the density's peak; neither should be treated as interchangeable with the mean.

Lognormal: Three Different Locations on the Original Scale

Use the convention \(X\sim\operatorname{Lognormal}(\mu,\sigma^2)\) when \(Y=\log X\sim N(\mu,\sigma^2)\), with \(\sigma>0\). Thus \(X>0\), and the normal parameters describe the logarithm of \(X\), not \(X\) itself.

Because \(P(X\leq x)=P(Y\leq\log x)\) for \(x>0\), the median is \(\exp(\mu)\): the normal variable \(Y\) has median \(\mu\). The mean follows by completing the square in the normal integral:

For the mode, the lognormal density for \(x>0\) is \(f(x)=\frac{1}{x\sigma\sqrt{2\pi}}\exp\left(-\frac{(\log x-\mu)^2}{2\sigma^2}\right)\). Differentiating its log-density gives \(\frac{d}{dx}\log f(x)=-\frac{1}{x}-\frac{\log x-\mu}{\sigma^2x}\). The derivative vanishes at \(\log x=\mu-\sigma^2\); it is positive below this point and negative above it. Hence this point is the density's maximum.

Weibull: Shape Can Change the Ordering

Use the scale-shape convention \(X\sim\operatorname{Weibull}(k,\lambda)\), where \(k>0\) is the shape and \(\lambda>0\) is the scale. Its CDF and density for \(x>0\) are \(F(x)=1-\exp[-(x/\lambda)^k]\) and \(f(x)=\frac{k}{\lambda}(x/\lambda)^{k-1}\exp[-(x/\lambda)^k]\).

Solving \(F(m)=1/2\) gives the median. For the mean, substitute \(u=(x/\lambda)^k\) in the density integral; then \(x=\lambda u^{1/k}\) and the resulting integral is the defining gamma integral. For the mode, differentiating the log-density locates its interior stationary point when one exists.

where \(\Gamma(a)=\int_0^\infty u^{a-1}e^{-u}\,du\) for \(a>0\). To determine the mode, write \(\log f(x)=\log(k/\lambda)+(k-1)\log(x/\lambda)-(x/\lambda)^k\). For \(k>1\), setting its derivative to zero yields \((x/\lambda)^k=(k-1)/k\), so the mode is \(\lambda((k-1)/k)^{1/k}\). When \(k=1\), the density is decreasing from its largest value at zero. When \(0<k<1\), the density diverges as \(x\) decreases to zero; zero is the limiting boundary mode, but the density has no finite maximum value there.

Pareto: A Finite Median Does Not Guarantee a Finite Mean

For a Type I Pareto distribution, let \(X\) have support \(x\geq x_m\), where \(x_m>0\) is the minimum value, and shape \(\alpha>0\). Its CDF is \(F(x)=1-(x_m/x)^\alpha\) on the support. Inverting \(F(m)=1/2\) gives a finite median for every \(\alpha>0\). The density decreases on the support, so its maximum is at the endpoint \(x_m\).

The mean requires more care: its density integral is \(\alpha x_m^\alpha x^{-\alpha-1}\), and the expectation is finite only when the corresponding integral of \(x^{-\alpha}\) over \([x_m,\infty)\) converges. When \(\alpha>1\), direct integration gives

For \(0<\alpha\leq1\), that integral diverges and the mean is infinite. The median and mode remain finite: \(\operatorname{median}(X)=x_m2^{1/\alpha}\), and the mode is \(x_m\).

Interpreting the Comparisons

The mean responds to the magnitude of all observations, including rare extreme values; the median depends only on which side of it contains half the probability; the mode concerns the density peak. The examples illustrate three distinct patterns: the lognormal has mean \(>\) median \(>\) mode for positive log-scale variance; the Pareto has mean \(>\) median \(>\) mode when its mean exists; and the Weibull ordering can change with its shape.

A common mistake is to claim that every right-skewed distribution must have mean \(>\) median \(>\) mode. That is a useful pattern in some familiar cases, not a general theorem about skewness. The Weibull calculations with \(k=4\) directly contradict it. Another mistake is to report a mean formula without checking its existence condition, as the Pareto example shows. Always identify the parameter regime first, and distinguish an infinite or undefined mean from a large but finite one.