This page explains why variance can't be negative.
Average Squared Deviation
The reason is that the way variance is calculated makes a negative result mathematically impossible.
Variance is the average squared deviation from the mean. Notice the word "squared".
Why Variance Calculation Can't Give Negative Result
To calculate variance, you need to take the following steps:
- Take each observation (number) in the data set.
- Calculate the differences between the individual numbers and the mean of the data set. Some of these differences can be and – unless all the numbers are exactly the same – will be negative.
- Then you square each of the differences (multiply it by itself). This is precisely the moment which eliminates the possibility of variance (and therefore also standard deviation) being negative. Something (negative or positive number) squared is always a positive number, except zero squared which is still zero. You can't get a negative number when squaring something.
- Then you find the average (mean) of all the squared numbers from the previous step. Because the squared deviations are all positive numbers or zeroes, their smallest possible mean is zero. It can't be negative. This average of the squared deviations is in fact variance. Therefore variance can't be negative.
Smallest Possible Variance Value
The smallest value variance can reach is exactly zero. This is when all the numbers in the data set are the same, therefore all the deviations from the mean are zero, all squared deviations are zero and their average (variance) is also zero.
If there are at least two numbers in a data set which are not equal, variance must be greater than zero.