Question

Assume that a gallon of milk bought at the store goes bad some time in the interval from 5 days to 10 days after purchase. What is the standard deviation of this distribution?

Answer 1

There is not enough information in this question to answer it.

One would need to know the entire distribution and this could be accomplished by either describing the probability on each day (with all of the probabilities adding to 1) or by defining a particular distribution.

For example, if this is a continuous uniform distribution from 5 to 10, then the standard deviation (SD) is calculated to be `5/sqrt(12)` based on the formula that variance = `1/12*(a-b)^2` and that SD = `sqrt(variance)`

But to illustrate that there is not enough information, take for example the fact that one can instead say that the milk will spoil either on day 5 or day 10 with equal probability. In this case, the SD = `5/2` based on the fact that the variance = `p*(1-p)*(a-b)` (This, by the way, is the highest the SD can be in this case).

You can also make the SD as arbitrarily small as you want by putting as much of the distribution on one day as possible. If 0.99999... is on day 5 and a very small amount is on each of the other days, then the SD approaches 0.