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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.

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Standard Deviation -

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# Standard Deviation

## Standard Deviation Calculator

### 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 volatility, 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: $$\sigma(X) = \sqrt{ V(X) }$$

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