# Is it possible to estimate the frequency distribution given only the min, max and mode?

I have access to data containing min, max and mode. Is it possible to estimate a frequency distribution only with this data? If yes, how?

• In some instances you can exclude certain distributions: minimum below 0 excludes exponential and other gamma distributions. If you have min/mode/max info for many subjects from the same dist'n family, you might be able to make some reasonable inferences: if mode-min = max-mode is aprx true for most, then population seems symmetrical. // If many subjects from exactly same normal dist'n then avg mode should estimate $\mu.$ Then (if you know sample sizes) ranges (max-min) can be used to estimate normal population $\sigma.$ May 15 '20 at 15:37
• A similar question with some answers May 15 '20 at 16:10

You can give some very crude information about your distribution. Specifically, you can estimate the min, the max and the mode, and you know that the number of data points at the min and the max (and in between) is less than or equal to the mode. So, precisely what you already know.

Apart from that, your distribution can look very different. Here are three possible histograms with 101 data points, a min of 1, a max of 10 and a mode of 7:

R code:

breaks <- seq(0.5,10.5,1)
par(mfrow=c(1,3),las=1)
hist(c(rep(1:10,each=4),7),xlab="",ylab="",main="",breaks=breaks,ylim=c(0,40))
hist(c(1,10,rep(7,39)),xlab="",ylab="",main="",breaks=breaks,ylim=c(0,40))
hist(c(rep(c(1,10),each=13),rep(7,14)),xlab="",ylab="",main="",breaks=breaks,ylim=c(0,40))


# Using sample range to estimate normal $$\sigma.$$

In my Comment above, I mention the possibility of estimating the population standard deviation from a normal sample using the range.

Some elementary textbooks suggest dividing the range by 4 or 5 to estimate $$\sigma,$$ and that works fairly well for samples of moderate size. (Textbook examples sometimes seem contrived to make this idea seem to work better than it really does.)

For small normal samples $$(n \le 20)$$ the range $$D = \max - \min$$ can be used to get a good estimate of $$\sigma.$$ Quality management engineers often use ranges to estimate SD for making control charts, and tables to convert ranges to SDs are found in some engineering texts. (But for very small $$n,$$ it's a bad idea to divide by 4 or 5.)

The constant $$c$$ such that $$D/c \approx \sigma$$ can be readily estimated by simulation for various sample sizes; illustrated below for $$n=20, 35, 50, 100,$$ for which respective values of $$c$$ are about $$3.73, 4.2, 4.5, 5.0.$$

set.seed(2020)
c = replicate(10^5, diff(range(rnorm(20)))); mean(c)
[1] 3.732738
c = replicate(10^5, diff(range(rnorm(35)))); mean(c)
[1] 4.213065
c = replicate(10^5, diff(range(rnorm(50)))); mean(c)
[1] 4.498718
c = replicate(10^5, diff(range(rnorm(100)))); mean(c)
[1] 5.016006


Note: In R, range returns min and max, so you need to use diff to get the true range.

• Once I thus estimate the σ , would it make sense to use the mode in place of mean to draw the distribution using these values? May 18 '20 at 5:33