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Standard Deviation

https://statistics.laerd.com/statistical-guides/measures-of-spread-standard-deviation.php

The standard deviation is a measure of the spread of scores within a set of data. Usually, we are interested in the standard deviation of a population. However, as we are often presented with data from a sample only, we can estimate the population standard deviation from a sample standard deviation. These two standard deviations - sample and population standard deviations - are calculated differently. In statistics, we are usually presented with having to calculate sample standard deviations, and so this is what this article will focus on, although the formula for a population standard deviation will also be shown.

Population standard deviation ($\sigma$)

$\sigma ={\sqrt {\frac {\sum _{i=1}^{N}(x_{i}-\mu )^{2}}{N}}}$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu: \text{population mean};\ \ \ N: \text{Number of scores in the population}}

Sample standard deviation formula (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s} )

Sometimes our data is only a sample of the whole population. In this case, we can still estimate the Standard deviation; but when we use a sample as an estimate of the whole population, the Standard deviation formula changes to this:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s = \sqrt{\frac{\sum_{i=1}^{n}(x_{i} - \bar{x})^2}{n -1}}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \bar{x}: \text{Sample mean};\ \ \ n: \text{Number of scores in the sample}}

• The Standard Deviation is the square root of the variance.

• This measure is the most widely used to express deviation from the mean in a variable.

• The higher the value the more widely distributed are the variable data values around the mean.

• Assuming the frequency distributions approximately normal, about 68% of all observations are within Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle +/-\ 1 } standard deviation.

• Approximately Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 95%} of all observations fall within two standard deviations of the mean (if data is normally distributed).