The median of a group of observations is usually, and somewhat casually, thought of as the middle observation when they are in sorted order. And that’s usually a good way to think of it, even if it’s a little imprecise.

It’s often said, for example, that half the observations lie below the median while half lie above it. The Excel documentation says so. So does my old college stats text. But no. Suppose that your observations consist of the numbers 1, 2, 3, 4, and 5. The middlemost number in that set is 3. But it is not true that half the numbers lie above it or below it. It is accurate to state that the same number of observations lie below the median as lie above it. In the prior example, two observations lie below 3 and two lie above 3.

If there is an even number of observations in the data set, then it’s accurate to say that half lie below the median and half above it. But with an even number of observations there is no specific, middle record, and therefore there is no identifiable median record. Add one observation to the prior set, so that it consists of 1, 2, 3, 4, 5, and 6. There is no record in the middle of that set. Or make it 1, 2, 3, 3, 3, and 4. Although one of the 3’s is the median, there is no specific, identifiable record in the middle of the set.

One way, used by Excel, to calculate the median with an even number of records is to take the mean of the two middle numbers. In this example, the mean of 3 and 4 is 3.5, which Excel calculates as the median of 1, 2, 3, 4, 5, and 6. And then, with an even number of observations, exactly half the observations lie below and half above the median.

Note

Other ways to calculate the median are available when there are tied values or an even number of values: One method is interpolation into a group of tied values. But the method used by Excel has the virtue of simplicity: It’s easy to calculate, understand, and explain. And you won’t go far wrong when Excel calculates a median value of 65.5 when interpolation would have given you 65.7.

The syntax for the MEDIAN() function echoes the syntax of the AVERAGE() function. For the data shown in Figure 1, you just enter =MEDIAN(A2:A61).

##### Figure 1. The mean and the median are always different in asymmetric distributions.

#### Choosing to Use the Median

The median is sometimes a more descriptive measure of central tendency than the mean. For example, Figure 1 shows what’s called a skewed distribution—that is, the distribution isn’t symmetric. Most of the values bunch up on the left side, and a few are located off to the right (of course, a distribution can skew either direction—this one happens to skew right). This sort of distribution is typical of home prices and it’s the reason that the real-estate industry reports medians instead of means.

In Figure 1, notice that the median home price reported is $193,000 and the mean home price is $232,000. The median responds only to the number of ranked observations, but the mean also responds to the size of the observations’ values.

Suppose that in the course of a week the price of the most expensive house increases by $100,000 and there are no other changes in housing prices. The median remains where it was, because it’s still at the 50th percentile in the distribution of home prices. It’s that 50% rank that matters, not the dollars associated with the most expensive house—or, for that matter, the cheapest.

In contrast, the mean would react if the most expensive house increased in price. In the situation shown in Figure 1, an increase of $120,000 in just one house’s price would increase the mean by $2,000—but the median would remain where it is.

The median’s relatively static quality is one reason that it’s the preferred measure of central tendency for housing prices and similar data. Another reason is that when distributions are skewed, the median provides a better measure of how things tend centrally. Have another look at Figure 1. Which statistic seems to you to better represent the typical home price in that figure: the mean of $232,000 or the median of $193,000? It’s a subjective judgment, of course, but many people would judge that $193,000 is a better summary of the prices of these houses than is $232,000.