Standard Deviation
Statistics & ProbabilityStandard deviation measures how spread out the values in a dataset are from the mean, expressed in the same units as the data.
Formula
s = \sqrt{\dfrac{\sum (x_i - \bar{x})^2}{n-1}}
Definition
Standard deviation is a number that tells you how spread out data values are from the average. A small standard deviation means values cluster close to the mean; a large one means they are more spread out.
Example
Class A scores: $78$, $80$, $82$ (mean $80$, small spread). Class B scores: $50$, $80$, $110$ (mean $80$, huge spread). Class B has a much larger standard deviation even though both averages are $80$.
Key Insight
Standard deviation is like the "typical distance" each value is from the mean. It tells you whether most values are close together or scattered widely.
Definition
The sample standard deviation $s = \sqrt{\sum (x_i-\bar{x})^2/(n-1)}$ measures average distance from the mean. The denominator $n-1$ (Bessel's correction) makes it an unbiased estimator of the population standard deviation $\sigma$. Standard deviation is in the original units of measurement.
Example
Data: $4$, $7$, $13$, $16$. Mean $= 10$. Deviations: $-6$, $-3$, $3$, $6$. Squared deviations: $36$, $9$, $9$, $36$. Sum $= 90$. $s = \sqrt{90/3} = \sqrt{30}$ = approximately $5.48$.
Key Insight
The empirical rule (for bell-shaped distributions): about $68\%$ of data falls within $1$ standard deviation of the mean, $95\%$ within $2$, and $99.7\%$ within $3$.
Definition
The sample standard deviation $s$ is the square root of the sample variance $s^2 = \frac{1}{n-1}\sum (x_i-\bar{x})^2$. By the central limit theorem, $(\bar{x}-\mu)/(s/\sqrt{n})$ follows a t-distribution with $n-1$ degrees of freedom when population is normal, enabling t-tests and t-intervals. For non-normal data, the bootstrap provides distribution-free confidence intervals for $\sigma$.
Example
The coefficient of variation $CV = s/\bar{x}$ expresses standard deviation as a proportion of the mean, enabling comparison of variability across datasets with different scales or units (e.g., comparing volatility of stock prices at different price levels).
Key Insight
Chebyshev's inequality gives a distribution-free lower bound: $P(|X-\mu| \ge k\sigma) \le 1/k^2$ for any distribution with finite variance. At least $75\%$ of data falls within $2\sigma$ and at least $89\%$ within $3\sigma$, regardless of distribution shape.