# Variance and Standard Deviation

StatisticsMeanExpected ValueWeighted Average → Variance and Standard Deviation

Consider the two data sets:

Data Set 1: 4, 3, 5, 7, 6
Data Set 2: 2, 10, 1, 9, 3

The mean (or average) of the two data sets is same, that is 5. But in data set 2, the values are more spread out as compare to data set 1. So, Mean alone is not sufficient to represent the data. We also need to calculate the standard deviation of the given data.

Variance
Variance is the average of the squared differences from the mean.

If $\mu$ is the mean of the values $x_i$, then variance, denoted by $\sigma^2$ is given by

$\sigma^2=\frac{\sum_{i=1}^n(x_i-\mu)^2}{n}$

where $n$ is total number of values.

Standard Deviation
Standard deviation is a measure of how spreads out points are from the mean.  The standard deviation is the square root of variance.

$\textrm{Standard Deviation}=\sqrt{\sigma^2}=\sigma$

So, variance is denoted by $\sigma^2$ and standard deviation is denoted by $\sigma$.

Example. Calculate variance and standard deviation of the above two data sets.

The mean of data set 1 is 5, and there are total 5 numbers. So, $\mu=5$ and $n=5$. So, by using the variance formula

$\sigma^2=\frac{\sum_{i=1}^n(x_i-\mu)^2}{n}$

we have

$\sigma^2=\frac{(4-5)^2+(3-5)^2+(5-5)^2+(7-5)^2+(6-5)^2}{5} \\ = 2.5$

And, the standard deviation is given by

$\sigma=\sqrt{2.5}=1.58$

Similarly, the variance and standard deviation of data set 2 are given by (NOTE: The mean of data set 2 is also 5, and there are also 5 values.)

$\sigma^2=\frac{(2-5)^2+(10-5)^2+(1-5)^2+(9-5)^2+(3-5)^2}{5} \\ = 17.5$

$\sigma=\sqrt{17.5}=4.18$

In investment and Finance, Risk is calculated by calculating the standard deviation.

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