Standard Deviation

Standard deviation is a measure of spread, which helps give us a meaningful estimate of the variability in a data set. A small standard deviation means most scores are close to the mean. Conversely, a large standard deviation means the scores are very spread out.

The standard deviation is found by calculating the square root of the variance.

Variance is the average of the squared differences from the mean. Here is its formula.

$\sigma^2=\frac{1}{n}\Sigma\left(x_i-\mu\right)^2$σ2=1n​Σ(xi​−μ)2

The following are the steps required. It is clear that for data sets of only moderate size, the amount of calculation needed is quite large and this makes calculations time-consuming and error-prone. For this reason, modern statistics depends heavily on automation by computer software and by hand-held calculators. 

To Calculate a Population Standard Deviation

This is the formula by which a calculator calculates the standard deviation of a data set from a full population. That is, it is the formula used for census data rather than sample data.

$\sigma=\sqrt{\frac{1}{n}\Sigma\left(x_i-\mu\right)^2}$σ=√1n​Σ(xi​−μ)2

In this formula, the numbers $x_i$xi​ are the values in the data set. There is one value for each subscript $i$i.
There are $n$n numbers $x_i$xi​ in the data set. So, $i$i goes from $1$1 to $n$n in the summation.
The symbol $\mu$μ (Greek letter 'mu') is the population mean.
The Greek letter $\sigma$σ (sigma) is used for the population standard deviation.
The symbol $\Sigma$Σ (upper case sigma) is the summation symbol. 

steps

1. Calculate the mean.  $\mu=\frac{1}{n}\Sigma_{i=1}^n\ x_i$μ=1n​Σni=1​ xi​ 
2. Find the difference from the mean for each score. $x_i-\mu$xi​−μ
3. Square each of the differences.    $\left(x_i-\mu\right)^2$(xi​−μ)2
4. Sum the squared differences.  $\Sigma\left(x_i-\mu\right)^2$Σ(xi​−μ)2
5. Divide the sum by the number of scores. $\frac{1}{n}\Sigma\left(x_i-\mu\right)^2$1n​Σ(xi​−μ)2
6. Take the square root.  $\sigma=\sqrt{\frac{1}{n}\Sigma\left(x_i-\mu\right)^2}$σ=√1n​Σ(xi​−μ)2

Worked Examples

Question 1

Question 2