# Standard Deviation Calculator

## Standard Deviation Calculator

First, select "population" or "sample", input two or more numbers, then it will calculate the "total", "average", "variance", and "standard deviation"

## What is Standard Deviation

Standard deviation is a measure of dispersion within a data set commonly used in statistics, represented by σ, indicating the deviation of each data point from the mean. A lower standard deviation means data points are closer to the mean, and a higher standard deviation indicates a wider distribution of data points from the mean.

Standard deviation is often used in fields such as finance and scientific research to quantify the variability or risk level.

## Difference between Population Standard Deviation and Sample Standard Deviation

`Standard deviation usually refers to the population standard deviation unless specified`

Population standard deviation refers to the standard deviation calculated for the entire data set (population), whereas sample standard deviation is based on a subset (sample) randomly selected from the population. The main difference lies in the adjustment of degrees of freedom: population standard deviation calculates the average deviation of all data directly, while sample standard deviation decreases the number of data points by one (N-1) when calculating variance. This adjustment corrects potential biases when estimating population parameters using only a subset of data, making the sample standard deviation closer to the true population standard deviation.

## Standard Deviation Formulas

### Population Standard Deviation

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

- $N$: Population size
- $x_i$: Each value
- $\mu$: Population mean

### Sample Standard Deviation

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

- $n$: Sample size
- $x_i$: Each value
- $\bar{x}$: Sample mean

## What is Variance

Variance is a statistical measure that represents the average of the squared differences from the mean. It measures how far each number in the set is from the mean.

`Standard deviation is the square root of variance`