Question

find the standard deviation for the group of data items. 17, 18, 19, 20, 21 the standard deviation is (round to two decimal places as needed.)

Explanation:

Step1: Calculate the mean

The data set is \(17, 18, 19, 20, 21\). The mean \(\bar{x}\) is calculated as the sum of the data divided by the number of data points \(n\).
The sum of the data is \(17 + 18 + 19 + 20 + 21=\frac{(17 + 21)\times5}{2}=95\) (using the formula for the sum of an arithmetic series \(S_n=\frac{n(a_1 + a_n)}{2}\), where \(n = 5\), \(a_1=17\), \(a_n = 21\)). The number of data points \(n = 5\). So the mean \(\bar{x}=\frac{95}{5}=19\).

Step2: Calculate the squared differences from the mean

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

• For \(x_1 = 17\): \((17 - 19)^2=(- 2)^2 = 4\)
• For \(x_2=18\): \((18 - 19)^2=(-1)^2=1\)
• For \(x_3 = 19\): \((19 - 19)^2=0^2 = 0\)
• For \(x_4=20\): \((20 - 19)^2=1^2 = 1\)
• For \(x_5=21\): \((21 - 19)^2=2^2=4\)

Step3: Calculate the variance

The variance \(s^2\) (for a sample, but here since it's the entire population, we use \(\sigma^2\)) is the average of the squared differences. The sum of the squared differences is \(4 + 1+0 + 1+4 = 10\). The variance \(\sigma^2=\frac{10}{5}=2\).

Step4: Calculate the standard deviation

The standard deviation \(\sigma\) is the square root of the variance. So \(\sigma=\sqrt{2}\approx1.41\) (rounded to two decimal places).

Answer:

\(1.41\)