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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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.

Answers to Questions

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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