What does Zero Variance mean for a set of data?

If a given set of data values has zero variance, then it means that the data values are constant. The data values consist of the same number repeated a certain number of times. Let us try to understand why this is true.

Let x₁,…,x_n be the given set of data values that has zero variance. The variance σ² is given by the formula,

σ²  = Σ(x_i–x̄)²/n.

Setting σ²=0 we get that,

Σ(x_i–x̄)²/n =0 ⇒ Σ(x_i–x̄)² = 0.

We know that squares of real numbers are always non-negative. Since the quantities Σ(x_i–x̄)² on the left side of the above equation are all nonnegative and add up to 0 this means all these quantities must be equal to zero. Therefore we have,

(x_i–x̄)² = 0 ⇒ (x_i–x̄) = 0 for all i=1,2,…,n.

So we conclude that

x_i = x̄ for all i=1,2,…,n.

This means that all the data values x₁,…,x_n are equal to each other. That is the data consists of a single value repeated some number of times.

For example, the set of data values 3, 3, 3, 3, 3 has a variance equal to 0 and the mean of this data set is equal to 3 itself.

Can a nonconstant random variable have zero variance?

It is not possible for a nonconstant random variable to have zero variance. As we saw above if a distribution has zero variance then it means that the distribution is constant.

Give an example of a distribution that has zero variance.

The uniform distribution is an example of a distribution that has zero variance. Any constant set of discrete data values such as 3, 3, 3, 3, 3 also has zero variance.

Give an example of two random variables that have positive variance whose sum has zero variance.

Such a distribution does not exist. This is because variance is additive therefore the variance of the sum of the two random variables will equal the sum of the individual variances. Since the individual variances are positive therefore their sum will also be strictly positive.

Find a random variable with a finite mean and zero variance.

Let X be a random variable following uniform distribution with parameters a and b. Then the mean of the random variable is finite and equals (a+b)/2 whereas, the variance of X is 0.