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## Tuesday, December 1st, 2020

### Unpopular Ratings

Categories: [ Science ]

For quite a while, I've been wondering what to do with movies or games which have been rated with a small number of people, such as the ones found on IMDb or BoardGameGeek: users can give a 0 – 10 rating to items, and the users' average is then shown. Is an obscure movie rated 8 by 42 people actually worth an 8?

Putting aside the actual usefulness of those ratings, I wanted to know how much uncertainty there is on those values, knowing only the arithmetic mean and the number of people who have voted.

Given population (i.e., all the users who have seen the film or played the game and who could vote), when we take a sample (i.e., the users who have rated the item) of size n, we get a sample mean m. But if another sample had been taken, we would have gotten a slightly different sample mean. How much much different would these two values be? In other words, how close is this value from the value we would get if the whole population would have rated the item?

The standard error of the mean indicates how much the sample mean m can vary by calculating its standard deviation sm = s/√n. The problem is that it is based on the standard deviation s of the population, which is unknown. Given that we do not know the users' individual ratings, we cannot either calculate the standard deviation of the sample.

All is not lost, however: since the ratings are always between 0 and 10, we can estimate the maximum value of the standard deviation. My best guess is that this value is maximal when half the ratings are 0 and the other half are 10, leading to a standard deviation of 5.

Given that the means of 96% of the samples of size n falls within ±2sm of the population's mean, we know that the mean of the population is very likely to be between m−10√n (but no less than 0) and m+10√n (but no more than 10). So this obscure movie rated 8 by 42 users is worth between 6.5 and 9.5. Given that on IMDb movies rated above about 7 are reasonably good, the obscure movie may be worth watching.

[ Posted on December 1st, 2020 at 22:59 | no comment | ]