Understanding Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

Formulas

Population Standard Deviation ( $\sigma$ ): $\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$

Sample Standard Deviation ( $s$ ): $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

Where:

$x_i$ : Each value in the data set
$\mu$ : Population mean
$\bar{x}$ : Sample mean
$N$ : Size of the population
$n$ : Size of the sample

Key Concepts

Variance: The average of the squared differences from the Mean. Standard deviation is the square root of the variance.
Normal Distribution: In a normal distribution, approximately 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.