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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 ($\sigma$)

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

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

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:

$s={\sqrt {\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}{n-1}}}$

${\bar {x}}:{\text{Sample mean}};\ \ \ n:{\text{Number of scores in the sample}}$

• 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 $+/-\ 1$ standard deviation.

• Approximately $95\%$ of all observations fall within two standard deviations of the mean (if data is normally distributed).