What Is an Average? Three Different Answers to the Same Question

The mean adds every value and divides by how many there are, which uses all the information and is the basis of most further statistics. The median is the middle value when everything is sorted, so half the data lie above and half below, and it ignores how extreme the extremes are. The mode is the most frequent value, which is the only one that works for data with no numerical order and which can be absent or multiple. For a symmetric distribution the three coincide and the choice does not matter. For a skewed one they separate, and the more skewed the data the further apart they are, which is exactly the situation where a single figure is most likely to mislead and most likely to be quoted.

The appropriate choice follows from the data and the purpose:

• •The mean where the data are roughly symmetric and where totals matter, since the mean multiplied by the count gives the total
• •The median where the data are skewed or contain extreme values, which is why incomes and house prices are reported this way
• •The mode for categories with no order, and for questions about the most common case
• •The mean for further calculation, since medians do not combine and means do
• •The median where robustness matters, since one extreme value can move a mean arbitrarily far and cannot move a median much
• •All three together where the shape of the data matters, since the gap between them describes the skew

Reporting an average without any indication of spread conceals most of what the data contain. Two distributions with identical means can differ enormously in whether values cluster tightly or scatter widely, and the practical meaning differs completely, since an average journey time of thirty minutes with everything between twenty-eight and thirty-two is a different situation from the same average with journeys between five minutes and two hours. The standard companion figures are the range, the interquartile range and the standard deviation, and a report giving a central value with none of them is incomplete. The stronger recommendation is to look at the distribution itself, since summaries hide multiple peaks, gaps and clusters, and several famous examples exist of quite different datasets sharing identical means, variances and correlations while looking nothing alike when plotted.

The gap between the mean and the median is exploited routinely and recognising the pattern is a practical skill. Income and wealth are strongly skewed, so the mean sits well above the median and quoting the mean overstates the typical case substantially, which is why claims about average household wealth and average earnings should always prompt a question about which measure was used. Averages over groups that differ can produce a figure describing nobody, which is the situation where a bimodal distribution has two clusters and the mean falls in the empty space between them. Averaging averages without weighting by group size gives a wrong answer and is a common error in summarising survey data. And an average computed over a period that includes an unusual event carries that event permanently, which is why long-run averages are quoted selectively by whoever benefits.

Further averages exist for situations where the ordinary mean gives a wrong answer. The geometric mean multiplies the values and takes the appropriate root, and it is the correct average for quantities that compound, including growth rates and investment returns, where averaging percentages arithmetically overstates the result. The harmonic mean divides the count by the sum of the reciprocals, and it is correct for rates over a fixed distance, which is why the average speed of a journey out at one speed and back at another is not the arithmetic mean of the two. Weighted means assign different importance to different values and are used whenever the observations are not equally representative. Knowing that these exist is more useful than knowing the formulas, since the common error is applying the arithmetic mean to a situation that calls for one of the others.

Mean, median and mode coincide for symmetric data and separate for skewed data, which is exactly when a single figure misleads. The median suits incomes and prices because one extreme value cannot move it. An average without a measure of spread conceals most of what the data contain. Compounding quantities need the geometric mean and rates over a fixed distance need the harmonic one.