Can a variance be negative?

Let's talk about a variance being negative. The other day I worked out a discrete probability distribution problem, and I got a negative variance. The variance is always positive. It would be zero if the data was all the same value. It should never be negative. 

In this post, I will show how I first did the problem. Then I’ll show why that the negative answer is incorrect, and how it happened.

Problem

Calls for a crisis hotline. The number of calls received per day at a crisis hotline is distributed as follows:

Number X	30	31	32	33	34
Probability P(x)	0.05	0.21	0.38	0.25	0.11

Find the mean, variance, and the standard deviation of the distribution.

(Problem #18 pg. 307 from Bluman, A. (2013). Elementary statistics (9th ed). New York, NY: McGraw-Hill)

Part 1: The work at first glance

Finding the mean

The mean is 32.2 using the rounding rule. This is correct.

Finding the variance

Obviously, you can’t get a negative variance, so I tried the exact mean, 32.16.

1.1 is the correct variance using the rounding rules.

Part 2: Finding the Error

The work is correct. It should not matter if you plug in the rounded or exact mean. The variance should still be positive. The issue comes up with the shortcut formula. When the regular formula is used for variance, it results in variance close to 1.0744 using the rounded mean, 32.2.

Note: the differences are always being squared resulting in either a zero or positive value. They are never negative. Because of this, variance is always positive. It should not matter if 32.16 or 32.2, but it does.

Using the regular formula with the rounded mean, it is pretty close to the variance using the exact mean. Here is the regular formula for variance using the exact mean:

It is the same as the shortcut formula as it should be. Notice with using the exact mean or the rounded mean, you still get 1.1 for your variance when using the rounding rules. But why is it when you use the rounded mean for the shortcut and regular formula you get two different answers with one being negative.

It is important here to look at how the shortcut formula is derived from the regular formula. Note that µ is a constant in the formula. 

The problem arises with the substitution. In order to make the substitution and combine like-terms, you are assuming that all µ’s are the same. If you use the rounded mean, the µ’s are not all the same. You will have unlike terms, so you will not able to combine them to simplify.

Notice the solution is the same one we found using the regular formula.

Conclusion

When you using the shortcut formula for calculating the variance for a discrete probability distribution, you have to be careful when you plug in a rounded mean. It is better to use the exact mean if possible, or to use a rounded mean closer to the exact. The question arises from this, “Does this happen with other shortcut formulas found in statistics?”

In case, if you worrying how to get the standard deviation, you just need to take the square root of the variance. By the way, variance should always be positive or equal to zero.