Statistics Question

Question

Suppose I conduct a survey of 10 people asking whether to rank a movie as 0 to 4 stars. Allowable answers are 0, 1, 2, 3, and 4.

The mean is 2.0 stars.

How do I calculate the certainty (or uncertainty) about this 2.0 star rating? Ideally, I would like a number between 0 and 1, where 0 represents complete uncertainty and 1 represents complete certainty.

It seems clear that the case where the 10 people choose ( 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ) would be the most certain, while the case where the 10 people choose ( 0, 0, 0, 0, 0, 4, 4, 4, 4, 4 ) would be the least certain. ( 0, 1, 1, 2, 2, 2, 2, 3, 3, 4 ) would be somewhere in the middle.

Answer 1

The standard deviation does not have the properties requested. It is zero when everyone chooses the same answer, and can be as great as sqrt(40/9) = 2.11 when there are five 0s and five 4s.

I suggest you use 1-stdev(x)/sqrt(40/9) which will take value 1 when everyone agrees, and value 0 when there are five 0s and five 4s.