Question

find the standard deviation for the group of data items.
16, 17, 18, 19, 20
the standard deviation is
(round to two decimal places as needed)

Explanation:

Step1: Calculate the mean

The mean $\bar{x}$ of a data set $x_1,x_2,\cdots,x_n$ is given by $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$.
Here, $n = 5$, $x_1=16,x_2 = 17,x_3=18,x_4=19,x_5=20$.
$\bar{x}=\frac{16 + 17+18+19+20}{5}=\frac{90}{5}=18$.

Step2: Calculate the squared differences

The formula for the squared - difference $(x_i-\bar{x})^2$:
For $x_1 = 16$: $(16 - 18)^2=(-2)^2 = 4$.
For $x_2=17$: $(17 - 18)^2=(-1)^2 = 1$.
For $x_3 = 18$: $(18 - 18)^2=0^2 = 0$.
For $x_4=19$: $(19 - 18)^2=1^2 = 1$.
For $x_5=20$: $(20 - 18)^2=2^2 = 4$.

Step3: Calculate the variance

The variance $s^2$ for a sample (when we assume this is a sample, and the formula for sample variance is $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}$).
$\sum_{i=1}^{5}(x_i-\bar{x})^2=4 + 1+0+1+4=10$.
$s^2=\frac{10}{5 - 1}=\frac{10}{4}=2.5$.

Step4: Calculate the standard deviation

The standard deviation $s=\sqrt{s^2}$.
Since $s^2 = 2.5$, then $s=\sqrt{2.5}\approx1.58$.

Answer:

$1.58$