# Variance Calculator

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Lets first understand, what is variance and how we can calculate variance?

Variance is the average of the squared deviation of a random variable from its mean and it measures how far a set of numbers is spread out from their average value and variance has a central role in statistics.

Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation and
it is often represented by \(\sigma ^2\), s^{2} or Var(X).

Calculation of Variance

$$Mean(\mu)= \frac{1}{n}\sum_{i=1}^n x_n$$ $$\mu= \frac{1}{n}(x_1 + x_2 + x_3 + ... + x_n)$$ $$Variance(\sigma^2)=\frac{1}{n}\sum_{i=1}^n (x_i\hspace{0.1cm}-\hspace{0.1cm}\mu)^2 $$Example : Calculate the variation for the data set 20, 22, 28, 32, 56.

Solution:

$$\mu=\frac{20+22+28+32+56}{5}$$ $$\mu=\frac{158}{5}$$ $$\mu=31.6$$ $$(31.6-20)^2=134.56$$ $$(31.6-22)^2=92.16$$ $$(31.6-28)^2=12.96$$ $$(31.6-32)^2=0.16$$ $$(31.6-56)^2=595.36$$ $$Variance(\sigma ^2)=\frac{835.2}{5}$$ $$Variance(\sigma ^2)=167.04$$Basic properties

1. Variance is non-negative.

2. The variance of a constant is zero.

3. Conversely, if the variance of a random variable is 0, then it is almost surely a constant.

4. If a constant is added to all values of the variable, the variance is unchanged.

5. If all values are scaled by a constant, the variance is scaled by the square of that constant.

For variance calculation using above variance calculator, you have to just write the comma separated values in the given input box and press the calculate button, you will get the result as the variance of the given data set.