1

I have a test which contains 24 items, so the maximum result is 24 and the minimum result is 0. How can I calculate the maximum standard deviation possible? Thank you!

user45118
  • 11
  • 1
  • 1
  • 2

2 Answers2

1

The maximum standard deviation would arise from an outcome where half of the observations are at one extreme and the other half are at the other extreme.

nickflees
  • 191
  • 6
1

As @nickflees says, the maximum SD is when the observations are equally split across the extremes. You don't have to trust him or me, if you use R you can see it for yourself with the following R code:

#Check all possible extents of extreme values
atbottom <- 23:12
mat <- sapply(atbottom,function(x) {c(rep(0,x),rep(24,24-x))})
sds <- apply(mat,2,sd)
sds
#simulate a bunch of possible response distributions and see if any produce an sd higher than the ones with half at each extreme
any(replicate(10000,sd(sample(0:24,24,replace=TRUE))) > sds[12])
russellpierce
  • 17,079
  • 16
  • 67
  • 98
  • This solution assumes without proof (but correctly) that the SD is maximized by choosing values only at the extremes. Proofs are given in the duplicate thread. (Although this seems obvious, contemplating the situation with an odd number of items helps one to realize that it's not completely trivial.) – whuber May 07 '14 at 14:12
  • Good point @whuber. All possible counter-examples would be computationally intensive... and the closed form mathematical solution is daunting. – russellpierce May 07 '14 at 14:15
  • I'm not quite sure what you mean by the "closed-form mathematical solution," but in this regard you might like to look at the duplicate thread. The answer by @Zen is particularly nice as well as being completely general. – whuber May 07 '14 at 14:20
  • @whuber: Not everybody who interacts with statistics has the mathematical background to understand his answer. What I see on Zen's answer is a mathematical proof which is great. Much better than my poor person's approximation. However, it is also an answer that I personally wouldn't expect to understand even in my wildest dreams, because the explanation is based on equations rather than a narrative in a language I can understand. – russellpierce May 07 '14 at 14:26
  • The limitation is entirely mine; but it makes me glad that SE's policy is to link to the duplicate and then close rather than to just delete. I can't imagine wording this question asker's comment in the way provided in the referenced question. – russellpierce May 07 '14 at 14:32