Question

find the standard deviation for the group of data items. 19, 15, 19, 15, 19, 15, 19, 15 the standard deviation is (round to two decimal places as needed.)

Explanation:

Step1: Calculate the mean

The formula for the mean $\bar{x}$ of a data set $x_1,x_2,\cdots,x_n$ is $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$.
Here, $n = 8$, and $\sum_{i=1}^{8}x_i=(19 + 15+19 + 15+19 + 15+19 + 15)=19\times4 + 15\times4=(19 + 15)\times4=34\times4 = 136$.
So, $\bar{x}=\frac{136}{8}=17$.

Step2: Calculate the squared - deviation from the mean

The formula for the squared - deviation $(x_i-\bar{x})^2$.
For $x = 19$: $(19 - 17)^2=2^2 = 4$.
For $x = 15$: $(15 - 17)^2=(-2)^2 = 4$.
Since there are 4 values of 19 and 4 values of 15, $\sum_{i = 1}^{n}(x_i-\bar{x})^2=4\times4+4\times4=16 + 16=32$.

Step3: Calculate the variance

The formula for the variance $s^2$ (for a sample) is $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}$. Here, $n = 8$, so $s^2=\frac{32}{8 - 1}=\frac{32}{7}\approx4.5714$.

Step4: Calculate the standard deviation

The formula for the standard deviation $s$ is $s=\sqrt{s^2}$.
So, $s=\sqrt{\frac{32}{7}}\approx2.14$.

Answer:

$2.14$