Question

given the data set {4, 6, 8}, calculate the standard deviation. use the following formula to calculate standard deviation: $sigma=sqrt{\frac{sum_{i = 1}^{n}(x_{i}-\text{mean})^{2}}{n}}$ where $x_{i}$ is each data point, and $n$ is the number of data points.

Explanation:

Step1: Calculate the mean

The data - set is $\{4,6,8\}$. The mean $\bar{x}=\frac{4 + 6+8}{3}=\frac{18}{3}=6$.

Step2: Calculate the squared differences

For $x_1 = 4$: $(x_1-\bar{x})^2=(4 - 6)^2=(-2)^2 = 4$.
For $x_2 = 6$: $(x_2-\bar{x})^2=(6 - 6)^2=0^2 = 0$.
For $x_3 = 8$: $(x_3-\bar{x})^2=(8 - 6)^2=2^2 = 4$.

Step3: Calculate the sum of squared differences

$\sum_{i = 1}^{3}(x_i-\bar{x})^2=4 + 0+4=8$.

Step4: Calculate the standard deviation

Using the formula $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\text{mean})^2}{n}}$, with $n = 3$, we have $\sigma=\sqrt{\frac{8}{3}}\approx1.63$.

Answer:

$1.63$