# What significance does variance hold?

Feb 6, 2018

As below

#### Explanation:

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean.

The technical definition is “The average of the squared differences from the mean”.

The Variance is defined as the average of the squared differences from the Mean and the symbol is color(green)(sigma^2. It is also called mean square deviation

The variance is a numerical value used to indicate how widely individuals in a group vary. If individual observations vary greatly from the group mean, the variance is big; and vice versa.

In short, Variance measures how far a data set is spread out.

A value of zero means that there is no variability; All the numbers in the data set are the same.

It is important to distinguish between the variance of a population and the variance of a sample . They have different notation, and they are computed differently. The variance of a population is denoted by σ2; and the variance of a sample, by s2.

The variance of a population is defined by the following formula:
color(green)(σ^2 = Σ ( X_i - X )^2 / N
where color(green)(σ^2) is the population variance, X is the population mean, ${X}_{i}$ is the ${i}_{t h}$ element from the population, and N is the number of elements in the population.

The variance of a sample is defined by slightly different formula:
color(blue)(s^2 = Σ ( X_i- X )^2 / ( n - 1 )
where $\textcolor{b l u e}{{s}^{2}}$ is the sample variance, X is the sample mean, ${X}_{i}$ is the ${i}_{t h}$ element from the sample, and n is the number of elements in the sample. Using this formula, the variance of the sample is an unbiased estimate of the variance of the population.

The Standard Deviation is just the square root of Variance and the symbol is color(brown)(sigma