Question

Find the standard deviation of the given data rounded to the nearest hundredth. 12, 53, 141, 219,500

Answer 1

STEP 1: Compute the mean

The mean of the dataset is the sum of the elements divided by their number:

$\dfrac{12+53+141+219+500}{5}=185$

STEP 2: Compute the residuals

You have to compute the difference between every element and the mean:

$12-185,\ 53-185,\ 141-185,\ 219-185,\ 500-185 = -173,\ -132,\ -44,\ 34,\ 315$

Step 3: Square the residuals:

$29929,\ 17424,\ 1936,\ 1156,\ 99225$

Step 4: Sum the squared the residuals:

$29929 + 17424 + 1936 + 1156 + 99225 = 149670$

Step 5: Divide by the number of elements in the dataset:

$\dfrac{149670}{5} = 29934$

Step 6: Consider the square root of the result:

$\sqrt{29934} \approx 173.01$

The formula we used is

$\sigma = \sqrt{\dfrac{\left(\sum_{i=1}^n (x_i-\mu)\right)^2}{n}}$

where $\sigma$ is the standard deviation, $x_i$ are the elements in the dataset, $n$ is the number of elements in the dataset, and $\mu$ is the mean of the dataset.