Question

sample standard deviation
the following are distances (in miles) traveled to the workplace by 6 employees of a certain brokerage firm.
3, 14, 28, 18, 37, 14
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find the standard deviation of this sample of distances. round your answer to two decimal places.
(if necessary, consult a list of formulas.)

Explanation:

Step1: Calculate the mean

The mean $\bar{x}$ of a sample $x_1,x_2,\cdots,x_n$ is given by $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$.
Here, $n = 6$, $x_1=3,x_2 = 14,x_3=28,x_4=18,x_5=37,x_6=14$.
$\sum_{i=1}^{6}x_i=3 + 14+28+18+37+14=114$.
$\bar{x}=\frac{114}{6}=19$.

Step2: Calculate the squared - differences

The squared - difference for each data point is $(x_i-\bar{x})^2$.
For $x_1 = 3$: $(3 - 19)^2=(-16)^2 = 256$.
For $x_2=14$: $(14 - 19)^2=(-5)^2 = 25$.
For $x_3=28$: $(28 - 19)^2=9^2 = 81$.
For $x_4=18$: $(18 - 19)^2=(-1)^2 = 1$.
For $x_5=37$: $(37 - 19)^2=18^2 = 324$.
For $x_6=14$: $(14 - 19)^2=(-5)^2 = 25$.

Step3: Calculate the sample variance

The sample variance $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}$.
$\sum_{i = 1}^{6}(x_i-\bar{x})^2=256+25+81+1+324+25=712$.
$s^2=\frac{712}{6 - 1}=\frac{712}{5}=142.4$.

Step4: Calculate the sample standard deviation

The sample standard deviation $s=\sqrt{s^2}$.
$s=\sqrt{142.4}\approx11.93$.

Answer:

$11.93$