Standard deviation

The standard deviation indicates the spread of the values around the mean value (arithmetic mean). The standard deviation is calculated using the square root of the variance

The standard deviation can be determined as the sample standard deviation for a partial quantity or for the total quantity. Different formulas apply to the total quantity or the sample.

To perform the calculation, enter a series of numbers. Then click the 'Calculate' button.

Input format

The data can be entered as a series of numbers, separated by semicolons or spaces. You can enter the data as a list (one value per line). Or from a column from Excel spreadsheet by copy & paste

Standard deviation formula

Calculation of the standard deviation of a sample

\(\displaystyle s=\sqrt{ \frac{1}{n-1} \sum^n_{i=1} (x_i-\overline{x})^2} \)

\(s\)	Standard deviation
\(n\)	Number of data points
\(x_i\)	Single data point
\(\overline{x}\)	Mean of the sample

Calculation of the standard deviation of a total quantity

\(\displaystyle σ=\sqrt{ \frac{1}{n} \sum^n_{i=1} (x_i-µ)^2} \)

\(σ\)	Standard deviation
\(n\)	Number of data points
\(x_i\)	Single data point
\(µ\)	Mean of all data points

Example to the standard deviation of a sample

data set \( \displaystyle x= 3, 5, 7, 8 \)

mean \( \displaystyle \overline{x}= \frac{3+ 5+ 7+ 8}{4} =5.75\)

\( \displaystyle s=\sqrt{\frac{1}{4-1}\cdot((3-5.75)^2+(5-5.75)^2+(7-5.75)^2+(8-5.75)^2)}\)

\( \displaystyle s=\sqrt{\frac{1}{3}\cdot(7.5625+0.5625+1.5625+5.0625)}\)

\( \displaystyle s=\sqrt{\frac{1}{3}\cdot 14.75} =\sqrt{ 4.9167}=\color{blue}{2.2174}\)