Question

over the past three years, freddies car maintenance sessions cost $250, $270, and $260. what is the standard deviation? use the following formula for 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 points are $x_1 = 250$, $x_2=270$, $x_3 = 260$. The number of data - points $n = 3$. The mean $\bar{x}=\frac{250 + 270+260}{3}=\frac{780}{3}=260$.

Step2: Calculate the squared differences

$(x_1-\bar{x})^2=(250 - 260)^2=(- 10)^2 = 100$; $(x_2-\bar{x})^2=(270 - 260)^2=10^2 = 100$; $(x_3-\bar{x})^2=(260 - 260)^2=0^2 = 0$.

Step3: Calculate the sum of squared differences

$\sum_{i = 1}^{3}(x_i-\bar{x})^2=100 + 100+0=200$.

Step4: Calculate the standard deviation

Using the formula $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_i - \bar{x})^2}{n}}$, we substitute $n = 3$ and $\sum_{i = 1}^{3}(x_i-\bar{x})^2=200$. So $\sigma=\sqrt{\frac{200}{3}}\approx8.16$.

Answer:

$8.16$