Standard Deviation Calculator

Instantly calculate the mean, variance, and standard deviation for both population and sample datasets. This is the perfect tool for checking your statistics homework and understanding data dispersion.

The Mathematical Formulas

Standard deviation measures 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, while a high standard deviation indicates that the values are spread out over a wider range.

Population Standard Deviation ($\sigma$)

Use this formula when your dataset includes the entire population you are studying:

$\sigma = \sqrt{\frac{\sum_{i=1}^N (x_i - \mu)^2}{N}}$

Where:

• $N$ = Total number of data points in the population.
• $\mu$ = Population mean.
• $x_i$ = Each individual data point.

Sample Standard Deviation ($s$)

Use this formula when your dataset is a sample drawn from a larger population. We divide by $n-1$ (Bessel's correction) to correct for bias:

$s = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}}$

Where:

• $n$ = Number of data points in the sample.
• $\bar{x}$ = Sample mean.

Frequently Asked Questions

Should I use Population or Sample?

In 99% of statistics homework problems and real-world scenarios, you are working with a sample of data, not the entire population. If the problem doesn't explicitly state it's a population, default to the Sample Standard Deviation.