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Standard Deviation

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

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.

Population standard deviation ($\sigma$)

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

$\mu :{\text{population mean}};\ \ \ N:{\text{Number of scores in the population}}$

Sample standard deviation formula ($s$)

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

• Assuming the frequency distributions approximately normal, about 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 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).