Created By : Vaibhavi Kumari
Reviewed By : Phani Ponnapalli
Last Updated : Apr 10, 2023

Variance Calculator Probability or probability variance calculator helps to determine the variance of random variable x of discrete probability distribution or discrete random variable variance calculator & probability density function(PDF) or variance calculator from pdf. Just enter the required input values in the given fields then click on the calculate button to get the output with detailed steps in no time.

What is Variance in Probability ?

In probability theory, the expectation of the squared deviation of a random variable from its population means or sample mean is called variance. Also, the second central moment of a distribution, the square of SD, and the covariance of the random variable with itself are known as Variance in probability. It is denoted by σ², s², Var(X), or V(X).

Variance Probability Formulas

The variance of a random variable X is defined by

var(X) = E [ (X−μ)² ], where μ = E(X).

For a discrete random variable X, the variance of X is acquired as follows:

var(X) = ∑(x−μ)² pX(x)

For, Absolutely continuous random variable, the variance formula of the probability density function is defined as

For a population, the variance is computed as

σ² = ( Σ (x-μ)² ) / N or σ² = ( (Σ x²) / N ) - μ².

The variance of joint probability formula for the random variables X and Y are given as

Var(X) = ∑ x^2 PX(x)−(E(X))^2
Var(Y) = ∑ y^2 PY(y)−(E(Y))^2

How do you find Probability with variance ?

In order to calculate the variance of a discrete probability distribution, all you need to do is follow the below simple steps and find the results manually with ease.

  • Firstly, calculate each deviation from its mean or expected value.
  • Now, square each value obtained in the first step.
  • Next, product it by its probability and finally sum up the products.
  • Thus, you can get the desired variance probability output.
  • If you want to find the standard deviation(σ) of a probability distribution, simply calculate the square root of the variance(σ²).


Consider the rolling of a fair six-sided die, with X the number on the uppermost face. We know that the px of X is 16, x=1,2,3,4,5,6, and that mean is 3.5. What is the variance of random variable X?


Given x=1,2,3,4,5,6,

px(X) = 16, μX = 3.5

To find the variance of random vaiable (X) of discrete probability function, use the formula of var(X) i.e.,

var(X) = E [(X−μX)^2]
= ∑(x−μX)^2*pX(x)
= ∑(x−μX)^2*1/6
= 1/6((1−3.5)^2+(2−3.5)^2+(3−3.5)^2+(4−3.5)^2+(5−3.5)^2+(6−3.5)^2)
= 1/6×17.5
= 35/12 ≈ 2.9167.

Therefore, the variance of probability distribution of X is approx 2.9167.

The standard deviation probability distribution of X is σX = √35/12 ≈ 1.7078.

FAQs on Free Online Variance Calculator Probability with Steps

1. How do you find the variance of joint probability?

2. What is the formula for the variance and standard deviation of a discrete probability distribution?

The formula for the variance of a discrete probability distribution is σx^2 = Var (X) = ∑i (xi − μ)^2 p(xi) = E(X − μ)^2 and standard deviation (σx) of a discrete probability distribution is Square root of Var(X).

3. What is the mean and variance formula in probability?

The mean formula in probability is μ = n *p and the variance of the probability distribution formula is Var(X) = E(X^2) − [E(X)]^2.

4. Where can I discover the free online & handy variance & standard deviation probability calculator?

You can discover the handy & free online variance calculator probability on a reliable & trustworthy web portal that provides concept-wise probability calculators.