Variance of $$$1$$$, $$$3$$$, $$$4$$$, $$$6$$$, $$$1$$$, $$$7$$$
Your Input
Find the sample variance of $$$1$$$, $$$3$$$, $$$4$$$, $$$6$$$, $$$1$$$, $$$7$$$.
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 $$$\mu = \frac{11}{3}$$$ (for calculating it, see mean calculator).
Since we have $$$n$$$ points, $$$n = 6$$$.
The sum of $$$\left(x_{i} - \mu\right)^{2}$$$ is $$$\left(1 - \frac{11}{3}\right)^{2} + \left(3 - \frac{11}{3}\right)^{2} + \left(4 - \frac{11}{3}\right)^{2} + \left(6 - \frac{11}{3}\right)^{2} + \left(1 - \frac{11}{3}\right)^{2} + \left(7 - \frac{11}{3}\right)^{2} = \frac{94}{3}.$$$
Thus, $$$s^{2} = \frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1} = \frac{\frac{94}{3}}{5} = \frac{94}{15}$$$.
Answer
The sample variance is $$$s^{2} = \frac{94}{15}\approx 6.266666666666667$$$A.