Standard Deviation
Standard Deviation
Since the variance is a squared quantity, it cannot be directly compared to the data values or the mean value of a data set. It is therefore more useful to have a quantity which is the square root of the variance. This quantity is known as the standard deviation.
Definition: Standard deviation
Let a population consist of \(n\) elements, \(\{x_1; x_2; \ldots; x_n\}\), with a mean of \(\overline{x}\). The standard deviation of the data is \[\sigma = \sqrt{\cfrac{\sum_{i=1}^n (x_i - \overline{x})^2}{n}}\]
In statistics, the standard deviation is a very common measure of dispersion. Standard deviation measures how spread out the values in a data set are around the mean. More precisely, it is a measure of the average distance between the values of the data in the set and the mean. If the data values are all similar, then the standard deviation will be low (closer to zero). If the data values are highly variable, then the standard variation is high (further from zero).
The standard deviation is always a positive number and is always measured in the same units as the original data. For example, if the data are distance measurements in kilogrammes, the standard deviation will also be measured in kilogrammes.
The mean and the standard deviation of a set of data are usually reported together. In a certain sense, the standard deviation is a natural measure of dispersion if the centre of the data is taken as the mean.
Optional Investigation: Tabulating results
It is often useful to set your data out in a table so that you can apply the formulae easily. Complete the table below to calculate the standard deviation of \(\{\text{57}; \text{53}; \text{58}; \text{65}; \text{48}; \text{50}; \text{66}; \text{51}\}\).
- Firstly, remember to calculate the mean, \(\overline{x}\).
- Complete the following table.
| index: \(i\) | datum: \(x_i\) | deviation: \(x_i - \overline{x}\) | deviation squared: \((x_i - \overline{x})^2\) |
| \(\text{1}\) | \(\text{57}\) | ||
| \(\text{2}\) | \(\text{53}\) | ||
| \(\text{3}\) | \(\text{58}\) | ||
| \(\text{4}\) | \(\text{65}\) | ||
| \(\text{5}\) | \(\text{48}\) | ||
| \(\text{6}\) | \(\text{50}\) | ||
| \(\text{7}\) | \(\text{66}\) | ||
| \(\text{8}\) | \(\text{51}\) | ||
| \(\sum x_i = \ldots\) | \(\sum (x_i - \overline{x}) = \ldots\) | \(\sum (x_i - \overline{x})^2 = \ldots\) |
- The sum of the deviations is always zero. Why is this? Find out.
- Calculate the variance using the completed table.
- Then calculate the standard deviation.
Example
Question
What is the variance and standard deviation of the possibilities associated with rolling a fair die?
Determine all the possible outcomes
When rolling a fair die, the sample space consists of \(\text{6}\) outcomes. The data set is therefore \(x=\{1;2;3;4;5;6\}\) and \(n = 6\).
Calculate the mean
The mean is: \begin{align*} \overline{x} &= \cfrac{1}{6}(1+2+3+4+5+6) \\ &= \text{3.5}\end{align*}
Calculate the variance
The variance is: \begin{align*} \sigma^2 &= \cfrac{\sum {(x-\overline{x})}^{2}}{n} \\ &= \cfrac{1}{6}(\text{6.25}+\text{2.25}+\text{0.25}+\text{0.25}+\text{2.25}+\text{6.25}) \\ &= \text{2.917}\end{align*}
Calculate the standard deviation
The standard deviation is: \begin{align*}\sigma &= \sqrt{\text{2.917}} \\ &= \text{1.708}\end{align*}
This lesson is part of:
Statistics and Probability