STEP 1: Compute the mean
The mean of the dataset is the sum of the elements divided by their number:
[tex] \dfrac{12+53+141+219+500}{5}=185 [/tex]
STEP 2: Compute the residuals
You have to compute the difference between every element and the mean:
[tex] 12-185,\ 53-185,\ 141-185,\ 219-185,\ 500-185 = -173,\ -132,\ -44,\ 34,\ 315 [/tex]
Step 3: Square the residuals:
[tex] 29929,\ 17424,\ 1936,\ 1156,\ 99225 [/tex]
Step 4: Sum the squared the residuals:
[tex] 29929 + 17424 + 1936 + 1156 + 99225 = 149670 [/tex]
Step 5: Divide by the number of elements in the dataset:
[tex] \dfrac{149670}{5} = 29934 [/tex]
Step 6: Consider the square root of the result:
[tex] \sqrt{29934} \approx 173.01 [/tex]
The formula we used is
[tex] \sigma = \sqrt{\dfrac{\left(\sum_{i=1}^n (x_i-\mu)\right)^2}{n}} [/tex]
where [tex] \sigma [/tex] is the standard deviation, [tex] x_i [/tex] are the elements in the dataset, [tex] n [/tex] is the number of elements in the dataset, and [tex] \mu [/tex] is the mean of the dataset.
