Search for a tool
Standard Deviation

Tool to calculate the standard deviation of a list of values. Standard deviation is a statistical value characterizing the dispersion of a sample or distribution.

Results

Standard Deviation -

Tag(s) : Statistics

Share dCode and you

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Team dCode likes feedback and relevant comments; to get an answer give an email (not published). It is thanks to you that dCode has the best Standard Deviation tool. Thank you.

# Standard Deviation

## Standard Deviation Calculator

Tool to calculate the standard deviation of a list of values. Standard deviation is a statistical value characterizing the dispersion of a sample or distribution.

### What is the standard deviation? (Definition)

The standard deviation measures the dispersion of a value series around its mean. This value, denoted $\sigma$ (Greek letter sigma lowercase) characterizes how the data $X$ (random variable) are scattered, their volability, by measuring the square root of the differences between each value (of the variable) and the mean $m$ (or expectation). $$\sigma(X) = \sqrt{ \mathbb{E} \left[(X - m)^{2}\right] }$$

### How to calculate the standard deviation from a list of numbers? (Formula)

From a list of numbers $x_i$ of a random variable $X$ whose distribution is unknown but with a mean $m$, the formula is $$\sigma(X)= \sqrt{ \frac{1}{n} \sum_{i=1}^{n}(x_{i}-m)^2 }$$ however, this estimator has a biais and the following formula is preferred $$\sigma(X)= \sqrt{ \frac{1}{n-1} \sum_{i=1}^{n}(x_{i}-m)^2 }$$

Example: The (unbiased) standard deviation of the series of 3 numbers 4,5,9 whose average is 6 is $\sqrt{ \frac{1}{3-1} \left( (4-6)^2 + (5-6)^2 + (9-6)^2 \right) } = \sqrt{ 14/2 = 7 } \approx 2.646$

### What is the relation between standard deviation and variance?

The value of the standard deviation is the square root of the variance. Knowing the value of the variance $V$, $\sigma$ can be calculated with the relation: $$\sigma(X) = \sqrt{ V(X) }$$

## Source code

dCode retains ownership of the source code of the script Standard Deviation online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Standard Deviation script for offline use on PC, iPhone or Android, ask for price quote on contact page !