Typical Value, Median, and Arithmetic Mean Under Lognormality

Why Three “Typical” Summaries Can Differ

In a population model, a parameter value assigned to an individual is not generally the same for everyone. With an exponential random effect, the population typical value is the parameter value for which the random effect is zero. That value is a useful reference, but it is not automatically the arithmetic average across individuals. Under lognormality, the distribution is right-skewed: the median equals the typical value, while the arithmetic mean is larger whenever there is nonzero random-effect variance.

This distinction matters when a model-based parameter is reported or used in a calculation. “Typical value,” “median,” and “arithmetic mean” answer different questions, even when they are all expressed in the same units. The results below concern a single individual parameter under the exponential random-effects model; they describe the model-implied population distribution, not necessarily the summaries of a finite observed sample.

The assumptions are important: the typical value is positive; the random effect has a normal distribution with mean zero and variance \(\omega^2\); and the individual parameter is obtained by multiplying \(\theta\) by \(\exp(\eta_i)\). The zero mean is on the log scale, not the natural parameter scale. If \(\omega^2=0\), every modeled individual has \(P_i=\theta\), so the median and arithmetic mean both equal \(\theta\). If \(\omega^2>0\), the conclusions below apply.

Why the Typical Value Is the Median

Because the exponential function is increasing, \(P_i\leq\theta\) holds exactly when \(\eta_i\leq 0\):

The second equivalence uses \(\theta>0\). A normal random effect centered at zero has \(P(\eta_i\leq0)=0.5\). Therefore \(P(P_i\leq\theta)=0.5\), which establishes that \(\theta\) is a median of the individual-parameter distribution. Since the lognormal distribution is continuous, its median is unique. The same result follows from the quantile function: for a probability \(q\) between zero and one, the \(q\)-quantile is \(\theta\exp(z_q\omega)\), where \(z_q\) is the \(q\)-quantile of the standard normal distribution and \(\omega=\sqrt{\omega^2}\). At \(q=0.5\), \(z_q=0\), giving the median \(\theta\).

This is also why \(\theta\) is called the typical value in this model: it is the value obtained by setting \(\eta_i=0\), and it is the midpoint of the modeled population by probability. It does not mean that the parameter values are symmetrically distributed around \(\theta\) on the natural scale.

Deriving the Arithmetic Mean

The natural-scale arithmetic mean is an expectation of \(P_i\), not an expectation of \(\eta_i\). Starting from the model,

For a normal random effect with mean zero and variance \(\omega^2>0\), its density is \(\frac{1}{\sqrt{2\pi\omega^2}}\exp[-\eta^2/(2\omega^2)]\). Integrating against this density gives

The second line follows by completing the square: \(\eta-\eta^2/(2\omega^2)=-(\eta-\omega^2)^2/(2\omega^2)+\omega^2/2\). The remaining integral is the total area of a normal density kernel with mean \(\omega^2\) and variance \(\omega^2\), so it equals \(\sqrt{2\pi\omega^2}\) and cancels the denominator. When \(\omega^2=0\), the model is degenerate at \(\theta\), and the same mean formula gives \(\theta\).

A common source of numerical error is to use the standard deviation \(\omega\) in place of the variance \(\omega^2\). The mean formula uses \(\omega^2/2\) in the exponent. If a model reports \(\omega=0.6\), for example, the variance is \(0.36\), and the mean multiplier is \(\exp(0.36/2)\), not \(\exp(0.6/2)\).

Reading the Difference in Context

The mean-median difference grows with the random-effect variance. Its ratio has a particularly simple form:

Thus, the ratio is determined by \(\omega^2\), not by the units or magnitude of \(\theta\). Increasing \(\theta\) scales both summaries by the same factor, whereas increasing \(\omega^2\) raises the mean relative to the median. This ratio is a model implication; it does not by itself establish that the model fits a particular data set.

Comparisons, Diagnostics, and Limitations

Under an additive normal parameter model, the natural-scale distribution is symmetric around its mean, subject to the model’s assumptions. The exponential model instead ensures positive parameter values and produces a right-skewed natural-scale distribution. In the exponential model, \(\theta\) is the median, not the arithmetic mean. Which summary is useful depends on the question: the median describes the central point by probability, while the arithmetic mean describes the expected parameter value on the natural scale.

Several checks help prevent misinterpretation. Confirm that the parameter model is exponential and that the log-scale random effect is centered at zero; confirm whether a reported quantity is a variance or a standard deviation; and check that the arithmetic mean is calculated in the parameter’s units. If \(\omega^2>0\), a computed mean below \(\theta\) signals an arithmetic or coding error under this model. A sample mean from a small data set may nevertheless fall below the model-implied mean, because random samples vary.

The formulas are exact for the stated lognormal distribution, but they do not guarantee that real individual parameters follow that distribution. A poor fit, influential high values, or substantial departures from normality on the log scale can make these summaries unsuitable. Estimated values of \(\theta\) and \(\omega^2\) also carry uncertainty, which these formulas do not quantify. Finally, the arithmetic mean is sensitive to the upper tail: when variability is large, a relatively small fraction of high parameter values can raise it substantially above the median.