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I work in a health service and have a PhD student who was told by my boss (her primary supervisor) to measure clinicians' responses concerning what proportion of their clients had a certain condition via a discrete outcome variable expressing the proportion as either 0% or a range: 0%, 1-20%, 21-40%, 41-60%, 81-100% (and 'do not know').

My boss suggested this then went on long service leave for six months. While he was away my student collected the data and then started to analyse it. The form we chose for this outcome has presented us with considerable difficulties in reporting descriptive statistics. Histograms work quite well but I have found it very difficult to express the descriptive statistics verbally in a meaningful or comprehensible way (e.g. "40% of clinicians reported that 21-40% of their clients had the condition, 25% reported that 61-80% of their client had the condition" Huh?).

I expressed these difficulties to my boss who wondered what all the fuss was about and told my student to just transform the variable to a numeric variable, with the midpoint of each range as the value, then report the mean and sd of those values. i.e. '1-20%' becomes 10%, '21-40%' becomes 30%, '41-60%' becomes 50% etc. My boss is very smart but not a statistician and instinctively his workaround just seems wrong to me; however, I wouldn't know where to begin looking to find out whether this is the case or not.

Can anyone give me advice about whether or not his approach - to take the midpoint of a range response in order perform mathematical operations on that midpoint - is valid? Or alternatively point me to some literature?

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    $\begingroup$ Your data are binned (interval censored). en.wikipedia.org/wiki/Censoring_(statistics)#Types ... this doesn't solve your problem but may help when searching the site. $\endgroup$
    – Glen_b
    Commented Oct 5, 2023 at 22:52
  • $\begingroup$ Another solution is to use interval arithmetic. $\endgroup$
    – whuber ♦
    Commented Oct 6, 2023 at 12:16
  • $\begingroup$ Thank you @whuber. What is interval arithmetic. $\endgroup$
    – llewmills
    Commented Oct 7, 2023 at 20:09
  • $\begingroup$ en.wikipedia.org/wiki/Interval_arithmetic $\endgroup$
    – whuber ♦
    Commented Oct 8, 2023 at 13:27

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This approach, as implemented, assumes that, within each category, the mean is actually at the midpoint. This is not necessarily the case.

Far better to get the median, which only assumes things about the category that the median is in. Then you can make different guesses as to how that is distributed, and see how much it matters. But the simplest way is to assume that the distribution within that category is uniform. You can, if you wish, also calculate the quartiles in the same way. Just know that they will all be based on certain assumptions, but that those are reasonable.

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  • $\begingroup$ Thanks so much @Peter Flom. So, if I understand you, you are saying it is valid to transform each categorical 'range response' (e.g. '1-20%', '61-80%') into the midpoint (10% and 20% respectively) then take the median and interquartile range (for example) of those midpoints? Is that right? $\endgroup$
    – llewmills
    Commented Oct 5, 2023 at 22:27
  • $\begingroup$ "Valid" isn't really a "yes/no" thing in this context. It is a reasonable thing to do, and many people do it. But it won't give you exact answers. $\endgroup$
    – Peter Flom
    Commented Oct 5, 2023 at 22:48
  • $\begingroup$ Ok thank you, 'it is a reasonable thing to do' was exactly what I needed to hear. Will make life much easier. $\endgroup$
    – llewmills
    Commented Oct 5, 2023 at 23:14
  • $\begingroup$ Typo above. I meant ' (e.g. '1-20%', '61-80%') into the midpoint (10 and 70% respectively)' $\endgroup$
    – llewmills
    Commented Oct 6, 2023 at 2:30
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you can take a weighted average of the ranges given by the doctors to give an overall range. If half the doctor say 40-60% and the other half say 60-80%, you can be sure that 50-70% of the patients overall have the condition. I hope the math is obvious enough with this simple example.

you can use middle values of the range if you declare so, but this may just as well be reason for more confusion.

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