Introduction

In statistics, variance is a measure of how spread out numbers are. It is the average of the squared differences from the mean. A distribution with zero variance has all its values equal to the mean. The following is true if a distribution has zero variance:

-The mean, median, and mode are all equal.
-All the values are identical.
-There is no dispersion or spread in the data.

Theoretical Results

The results of this section derive from several theoretical assumptions. First, we assume that the random variables are independent and identically distributed (i.i.d.). Second, we assume that the distribution of each random variable has zero variance. Third, we assume that the random variables are distributed according to a normal distribution. fourth, we assume that the random variables are independent of each other.

Theorem 1

If a distribution has zero variance, then it is a point mass (i.e. all the probability is concentrated at a single point).

Theorem 2

Let X be a random variable with mean μX and variance σ2X. If X has a Normal distribution, then

(1) μX = E(X);

(2) σ2X = Var(X); and

(3) Cov(X,Y) = E[XY] − μXμY.

Corollary

In probability theory and statistics, the variance is the expectation of the squared deviation of a random variable from its mean. Informally, it measures how far a set of (random) numbers are spread out from their average value.

The variance of a random variable X is usually denoted by Var(X), sigma^2 (X), or σ^2 (X).

If a distribution has zero variance, then all values in the distribution are equal to the mean.

Examples

If a distribution has zero variance, it means that all the values in the distribution are the same. This can be useful if you want to create a distribution that is perfectly uniform.

Example 1

If a distribution has zero variance, it means that all of the values in the distribution are exactly the same. In this case, the mean and median would also be equal to each other.

Example 2

Example 2: If a distribution has zero variance, then all of the following are true EXCEPT

A) The distribution is always centered at its mean.
B) The distribution is always symmetric about its mean.
C) Every value in the distribution is equal to the mean.
D) The distribution has no spread.

Conclusion

A distribution with zero variance has all values equal to the mean. This is a very specific case, and in most cases when people say “zero variance” they simply mean “very small variance.”