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Variance of $$$1$$$, $$$2$$$, $$$3$$$, $$$4$$$, $$$5$$$
Your Input
Find the sample variance of $$$1$$$, $$$2$$$, $$$3$$$, $$$4$$$, $$$5$$$.
Solution
The sample variance of data is given by the formula $$$s^{2} = \frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1}$$$, where $$$n$$$ is the number of values, $$$x_i, i=\overline{1..n}$$$ are the values themselves, and $$$\mu$$$ is the mean of the values.
Actually, it is the square of standard deviation.
The mean of the data is $$$\mu = 3$$$ (for calculating it, see mean calculator).
Since we have $$$n$$$ points, $$$n = 5$$$.
The sum of $$$\left(x_{i} - \mu\right)^{2}$$$ is $$$\left(1 - 3\right)^{2} + \left(2 - 3\right)^{2} + \left(3 - 3\right)^{2} + \left(4 - 3\right)^{2} + \left(5 - 3\right)^{2} = 10$$$.
Thus, $$$s^{2} = \frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1} = \frac{10}{4} = \frac{5}{2}$$$.
Answer
The sample variance is $$$s^{2} = \frac{5}{2} = 2.5$$$A.