Does mean/expectation affect the variance of a sample?

Question

Let's say I have an app $A$ that has a mean of $4$ stars and an app $B$ that has a mean of $3$ stars. Both have the same sample size. Would the variance stay the same? I thought it would because what's up top (of the unbiased variance formula) would stay the same, since it's $X$ minus the mean and the $X$ would change too along with the mean. I am kinda confused on it though.

Answer 1

Typically, when you have a bounded random variable, the closer the mean is to one of the extremes, the lower the variance. Consider an app with a 5-star mean. It will necessarily have 0 variance as if an outcome other than a 5-star rating were possible, this would imply the mean is less than 5. Examples of bounded families of random variables are the beta distribution and the binomial distribution, which exhibit this property. Though an app rating would not necessarily follow any family of distribution other than the general multinomial distribution.

Answer 2

The mean of a sample is not informative of the variance of that sample. For instance, both of your samples could have zero variance--all of your app A ratings are exactly $4$ stars, and all of your app B ratings are exactly $3$ stars. Then the mean ratings are $4$ and $3$ respectively, and each has variance $0$ because all the ratings are identical within each type of app.

On the other hand, app A ratings could have an average of $4$ stars, half of them are $3$ star ratings and half are $5$ star ratings. Now the variance of app A ratings is positive. Or, you can apply the same reasoning to app B ratings.

If you don't specify any other information about how individual ratings are distributed on the rating scale, then you cannot answer the question about how the variance of the rating might or might not change with a changing mean. And even if you did supply some underlying parametric model, it may not be possible to relate mean and variance because the answer depends on the choice of model and other assumptions.