In statistics, sometimes we use mean sum of error, and sometimes variance. I just want to know what is the difference between mean sum of error and variance?
They measure two different things.
Variance measures the spread of a variable. While MSE measures the deviation from a predictor. MSE is usually associated with a function:
If you are trying to measure how well a function, say Y=mX+b, predicts Y then you would use MSE. Note that here the MSE <> Variance.
I could guess the confusion mostly comes from when you have a target value in manufacturing. Say you have a target value of 4 on a measurement. MSE would take into account the mean's shift from the target as well as the variance. So values of 2, 3, 4, 5 would yield MSE of 1.5 and variance of 1.3