Calculating the Median Age of a Group of People

Jan has a worksheet that has a list of ages in column A. The ages start at 1 and go through to 100. In column B she has the number of people in each of those ages. Jan needs a formula that will tell her the median age of this group of people.

At first blush you might think that you can use the MEDIAN function to calculate the median. That function works great if you have a simple list of values. For instance, if you were calculating the median for the ages alone, then MEDIAN would work fine. However, in Jan's case she needs the median value for the ages of the people, not for the ages themselves. In other words, the median needs to be weighted by the number of people that are each age. The MEDIAN function cannot handle such a requirement.

It should be pointed out that the median age is going to be different than the average age for a group of people. The average can be calculated most easily by multiplying the age by the number of people that is each age. For instance, in column C you could place a formula such as =A1*B1 and then copy it down the column. Add up the values in columns B and C, and then divide the sum in column C by the sum in column B. The result is the average age for the list of people.

The median age, on the other hand, is the age at which half of the people fall below that age and half above that age. The median age can best be calculated by using an array formula, such as the following:

=MATCH(SUM($B$1:$B$100)/2,SUMIF($A$1:$A$100,
"<="&$A$1:$A$100,$B$1:$B$100))

This is a single formula, entered by pressing Ctrl+Shift+Enter. The SUMIF function in the formula is used to generate an array of the cumulative number of people who are less than or equal to each age. The SUM portion of the formula gives the midpoint of the total frequency of ages. The MATCH function is then used to look up the midpoint value in the array of cumulative frequencies. This yields an "index number" in the initial array, and since the array consists of the all ages 1 through 100, this index number matches is equivalent to the median age.

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