Question

the average weekly unemployment benefits (in dollars) for a random selection of states are listed below. 272 239 289 214 276 252 190 209 251 272 310 321 327 356 416 send data to excel part: 0 / 3 part 1 of 3 find the standard deviation. round your answer to at least one decimal place. the standard deviation is

Explanation:

Step1: Calculate the mean

Let the data set be \(x_1,x_2,\cdots,x_n\). Here \(n = 16\), and \(\sum_{i = 1}^{16}x_i=272 + 239+289+214+276+252+190+209+251+272+310+321+327+356+416=4304\). The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{4304}{16}=269\).

Step2: Calculate the squared - differences

For each data point \(x_i\), calculate \((x_i-\bar{x})^2\). For example, for \(x_1 = 272\), \((272 - 269)^2=9\); for \(x_2 = 239\), \((239 - 269)^2=900\), and so on. Then sum up all these squared - differences \(\sum_{i = 1}^{n}(x_i - \bar{x})^2\).
\((272-269)^2+(239 - 269)^2+(289-269)^2+(214-269)^2+(276-269)^2+(252-269)^2+(190-269)^2+(209-269)^2+(251-269)^2+(272-269)^2+(310-269)^2+(321-269)^2+(327-269)^2+(356-269)^2+(416-269)^2\)
\(=9 + 900+400 + 3025+49+289+6241+3600+324+9+1681+2704+3364+7569+21609\)
\(=49783\).

Step3: Calculate the variance

The variance \(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}\) (for a sample). Here \(n=16\), so \(s^2=\frac{49783}{16 - 1}=\frac{49783}{15}\approx3318.867\).

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^2}=\sqrt{3318.867}\approx57.6\).

Answer:

57.6