# Standard Deviation Calculator

Calculate the standard deviation and variance for the given numbers using the selected method.

**Calculation:**

## What is Standard Deviation?

Standard Deviation is a statistical measure that tells you how spread out the values in a data set are from the mean (average). It quantifies the amount of variation or dispersion in a set of data.

A low standard deviation suggests that most data points are close to the mean, while a high standard deviation indicates that data points are more spread out.

Standard Deviation is widely used in various fields, including finance, science, and quality control. It helps in risk assessment, decision-making, and understanding the distribution of data. In finance, for example, it's used to measure investment volatility, while in manufacturing, it's used to monitor product quality.

This calculator allows you to calculate Standard Deviation using either the Population or Sample method. Additionally, it computes the arithmetic mean, variance, deviation, and other parameters. It also displays the entire calculation process, starting from the mathematical formula.

## How to calculate Standard Deviation?

Population Standard Deviation is used when you have data for an entire population. It calculates the precise measure of how spread out the data is, without any estimation involved.

**Standard Deviation (population):**$$\sigma = \sqrt{\dfrac{\sum_{i=1}^{n}(x_i - \mu)^{2}}{n}}$$

- n = Count Deviation
- μ = Arithmetic Mean

**Example:**Calculate the Population Standard Deviation of: 5, 12, 15, 29.

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

$$\sigma = \sqrt{\dfrac{(5 - 15.25)^{2} + ... + (29 - 15.25)^{2}}{4}}$$

$$\sigma = \sqrt{\dfrac{304.75}{4}}$$

$$\sigma = \sqrt{76.1875}$$

$$\sigma = 8.728545125048045$$

Sample Standard Deviation is used when you have data for only a part of the population, known as a sample. It's used to estimate the variability in the entire population based on the sample data.

**Standard Deviation (sample):**$$\sigma = \sqrt{\dfrac{\sum_{i=1}^{n}(x_i - \overline{x})^{2}}{n - 1}}$$

- n = Count Deviation
- x̄ = Arithmetic Mean

**Example:**Calculate the Sample Standard Deviation of: 8, 15, 30, 42.

$$\sigma = \sqrt{\dfrac{\sum_{i=1}^{n}(x_i - \overline{x})^{2}}{n - 1}}$$

$$\sigma = \sqrt{\dfrac{(8 - 23.75)^{2} + ... + (42 - 23.75)^{2}}{4 - 1}}$$

$$\sigma = \sqrt{\dfrac{696.75}{3}}$$

$$\sigma = \sqrt{232.25}$$

$$\sigma = 15.239750654128171$$