# Nonintegral (and not half-integral) median

I was recently looking at the National Survey of Family Growth (US Census Bureau) and noticed that the numbers on the median column were not whole numbers, but rather numbers like 5.1 and 4.6. (Not 5.1 thousand or 5.1 million, just 5.1.)

This seemed unusual to me, so I looked to see if there was a note. There was:

For definition of median, see Guide to Tabular Presentation.

Eventually I found the Guide to Tabular Presentation, which says

The median of a group of numbers is the middle number or value when each item in the group is arranged according to size (lowest to highest or visa versa); it generally has the same number of items above it as well as below it. If there is an even number of items in the group, the median is taken to be the average of the two middle numbers.

This does not seem to allow results other than whole numbers, unless (very unlikely in the case of large data sets) the middle numbers differ, in which case the result is half of a whole number. So what's going on?

• Exactly what "median column" were you looking at? Family size? Mean family size per Census tract? Growth rates? Percent below poverty line? Something else? – whuber Jun 1 '12 at 16:04
• @whuber: There are a number of examples, but say Table 95, "Number of opposite-sex partners in lifetime". – Charles Jun 1 '12 at 17:38
• I don't have an answer for you, but I do notice this is a survey and not a full census tabulation. Census surveys are typically complexly stratified samples, not simple random ones. Therefore, estimates--including medians--are based on weighted combinations of data. This could explain the apparent anomalies. You would need to refer to the survey methodology documents (usually way in back the appendices where the uses of the weights are described) to check up on this. – whuber Jun 1 '12 at 17:42
• @whuber So I notice sometimes you give a good answer like you did here with two comments rather than make it an answer and collect reputation points. Are you so high up in reputation ponts that it doesn't matter at all to you. Even if that is the case i think answers get much more attention than comments. So you probably should post these anyway, – Michael Chernick Jun 1 '12 at 23:32
• @Charles Certainly Bill's point that counts per unit of something are not necessarily integers does explain how items that start out as integer could become fractions and then have factional medians. Also I think number like 4.5 coming up as sample medians for even sized samples can and do occur often. – Michael Chernick Jun 1 '12 at 23:40