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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 -

Tag(s) : Statistics

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# 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 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 mean is $$m$$, the formula is $$\sigma(X)= \sqrt{ \frac{1}{n-1} \sum_{i=1}^{n}(x_{i}-m)^2 }$$

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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