Question

find the standard deviation for the group of data items. 19, 19, 20, 20, 21, 21 the standard deviation is (simplify your answer. 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, $n = 6$, and $\sum_{i=1}^{6}x_i=19 + 19+20 + 20+21 + 21=120$.
So, $\bar{x}=\frac{120}{6}=20$.

Step2: Calculate the squared - deviation from the mean

For each data point $x_i$, calculate $(x_i-\bar{x})^2$.

• When $x_i = 19$: $(19 - 20)^2=(-1)^2 = 1$
• When $x_i = 19$: $(19 - 20)^2=(-1)^2 = 1$
• When $x_i = 20$: $(20 - 20)^2=0^2 = 0$
• When $x_i = 20$: $(20 - 20)^2=0^2 = 0$
• When $x_i = 21$: $(21 - 20)^2=1^2 = 1$
• When $x_i = 21$: $(21 - 20)^2=1^2 = 1$

The sum of squared - deviations $\sum_{i = 1}^{n}(x_i-\bar{x})^2=1 + 1+0 + 0+1 + 1=4$.

Step3: Calculate the variance

The variance $s^2$ (for a sample) is given by $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}$.
Since $n = 6$, $s^2=\frac{4}{6 - 1}=\frac{4}{5}=0.8$.

Step4: Calculate the standard deviation

The standard deviation $s=\sqrt{s^2}$.
So, $s=\sqrt{0.8}\approx0.89$.

Answer:

$0.89$