Question

find the standard deviation for the group of data items. 18, 19, 20, 21, 22 the standard deviation is (round to two decimal places as needed)

Explanation:

Step1: Calculate the mean ($\mu$)

The data set is \(18, 19, 20, 21, 22\). The mean is the sum of the data divided by the number of data points (\(n = 5\)).

$$ \mu=\frac{18 + 19 + 20 + 21 + 22}{5}=\frac{100}{5}=20 $$

Step2: Calculate the squared differences from the mean

For each data point \(x\), calculate \((x - \mu)^2\):

• For \(x = 18\): \((18 - 20)^2=(-2)^2 = 4\)
• For \(x = 19\): \((19 - 20)^2=(-1)^2 = 1\)
• For \(x = 20\): \((20 - 20)^2=0^2 = 0\)
• For \(x = 21\): \((21 - 20)^2=1^2 = 1\)
• For \(x = 22\): \((22 - 20)^2=2^2 = 4\)

Step3: Calculate the variance ($\sigma^2$)

Variance is the average of the squared differences from the mean.

$$ \sigma^2=\frac{4 + 1 + 0 + 1 + 4}{5}=\frac{10}{5}=2 $$

Step4: Calculate the standard deviation ($\sigma$)

Standard deviation is the square root of the variance.

$$ \sigma=\sqrt{2}\approx1.41 $$

Answer:

\(1.41\)