Variance and standard deviation are both statistical measures that describe the variability or dispersion of a dataset.

Variance quantifies how far each number in the dataset is from the mean (average) and is calculated as the average of the squared differences from the mean. The formula for variance is given by:

[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} ]

where ( x_i ) represents each value in the dataset, ( \mu ) is the mean, and ( n ) is the number of values. For sample variance, the denominator is adjusted to ( n-1 ) to account for the degrees of freedom ⁴.

Standard deviation, on the other hand, is simply the square root of the variance. It provides a measure of the average distance of each data point from the mean, making it more interpretable than variance. The formula for standard deviation is:

[ \sigma = \sqrt{\sigma^2} ]

Both measures are essential in statistics for understanding data distribution and variability ²³.