Question

What does variance measure?

Answer 1

We know that the variance of a random variable X is the mean or expected value of the squared deviation from the mean of X. Using this definition, we can arrive at a simpler formula for variance. It measures how wide the data is spread from the mean value.

Complete step-by-step answer:
We know very well that the variance measures how far a set of numbers is spread out, from their average value. Also, it is clear that variance is the square of standard deviation. The variance of a random variable X is mathematically represented by ${{\sigma }^{2}},{{s}^{2}}$ or Var(X).
We know that the variance measures the distance between the mean of a data set, and how far a data point has dispersed. Variance actually measures the variability, which we can also define as the degree of spread.
Using the definition of variance, for a random variable X, we can write the variance as
$Var\left( X \right)=E\left[ {{\left( X-E\left[ X \right] \right)}^{2}} \right]$
where, E(X) represents the expected value or mean for the random variable X.
We can expand the above equation as
$Var\left( X \right)=E\left[ {{X}^{2}}-2X\cdot E\left[ X \right]+E{{\left[ X \right]}^{2}} \right]$
Hence, we can write
$Var\left( X \right)=E\left[ {{X}^{2}} \right]-2E\left[ X \right]\cdot E\left[ X \right]+E{{\left[ X \right]}^{2}}$
Thus, we can simplify this to get
$Var\left( X \right)=E\left[ {{X}^{2}} \right]-E{{\left[ X \right]}^{2}}...\left( i \right)$
Hence, we can also say that the variance of X is equal to the difference of the mean of the square of X and the square of the mean of X.

Note: We must note that variance is similar to deviation, but variance is squared and hence gives better information about the spread of data in a data set. Since variance is squared value, hence we can say that variance is always non-negative.