Variance
![Distribution and cumulative distribution of s2/σ2, for various values of ν = n − 1, when the yi are independent normally distributed.](/Images/godic/202502/06/Scaled_chi_squared.svg1156.png")
![](/Images/godic/202502/06/Scaled_chi_squared_cdf.svg1156.png")
![Geometric visualisation of the variance of an arbitrary distribution (2, 4, 4, 4, 5, 5, 7, 9):
1. A frequency distribution is constructed.
2. The centroid of the distribution gives its mean.
3. A square with sides equal to the difference of each value from the mean is formed for each value.
4. Arranging the squares into a rectangle with one side equal to the number of values, n, results in the other side being the distribution's variance, σ².](/Images/godic/202502/06/Variance_visualisation.svg1156.png")
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean, and it informally measures how far a set of (random) numbers are spread out from their mean. The variance has a central role in statistics. It is used in descriptive statistics, Statistical inference, hypothesis testing, goodness of fit, Monte Carlo sampling, amongst many others. This makes it a central quantity in numerous fields such as Physics, Biology, Chemistry, Economics, and Finance. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by σ² or Var(X).