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I'll start by declaring I am no mathematician/statistician whatsoever. So be gentle.

I have a data set of 400 observations against discrete values from 1 to 10. Using excel, the median returned a value of 7. I created a histogram with this data set and it returned this:

enter image description here

I wanted to find the 50% value of the cumulative line and used the interpolation formula between 6 and 7, which resulted in a value of 6.42.

Is it correct to say that both 7 (median) and 6.42 are the 50th percentiles for this distribution? Is it accurate to say there can be two 50th percentiles? The data are discrete while the cumulative line is continuous on the graph. Is this the reason for the two different values?

Thank you for the help!

David E PMP
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  • https://en.wikipedia.org/wiki/Median the wikipedia entry may help – Barry Carter Aug 18 '22 at 15:21
  • If the observations are discrete values, the median - the value of the middle ranked observation - will usually be one of those discrete values (or possibly the average of two if either could be the median). Here, fewer than half are less than $7$ and fewer than half are greater then $7$, so $7$ looks like the median – Henry Aug 18 '22 at 15:26

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If this were from continuous date binned to create a histogram the sensible thing to say is that the median (50th percentile) for the underlying population is "between 6 and 7", or perhaps "roughly halfway between 6 and 7".

For discrete data there is no well defined 50th percentile. You could say "somewhat more than half the responses were 7 or less" or "somewhat more than half were 6 or more", with slightly different psychological effect.

In either case the two decimal places in 6.42 are a consequence of the interpolation formula you chose, but that level of precision is not supported by the data.

Ethan Bolker
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