Calculating standard deviation without a data set.

Question

I know how to calculate SD when given data points by using: $ \displaystyle \mathrm{SD} = \sqrt{\sum(x^2 - \text{mean}^2) / n} $.

I have been given just the sum of $x$ and sum of $x^2$. How do I calculate SD from this?!

An example question I am stuck on: Sum of $x = 1303$

Sum of $x^2 = 123557.$

There are 14 years for which the data is given - I would assume this is n...

Answer 1

In a not confusing manner (hopefully) the way i would start is to work out the variance using: Sxx = (Sum of)x^2 - n(Mean)^2

Then from there to find the standard deviation i would use:

srqroot(Sxx/n-1)

hopefully that has helped!

Answer 2

Hint: You need a formula, where you can enter the sum of squares and the square of the sum. Let us first define these as follows: $$ SSQ =\sum\limits_{k=1}^{n} x_k^2 \quad\text{and}\quad SQS = \sum\limits_{k=1}^{n} x_k \;. $$

A famous formula of the (population)¹ variance is $$ \mathrm{Var}(X) = \dfrac{1}{n} \sum\limits_{k=1}^{n} x_k^2 - \left( \dfrac{1}{n} \sum\limits_{k=1}^{n} x_k \right)^2 = \dfrac{SSQ}{n} - \left( \dfrac{SQS}{n} \right)^2 \;. $$ The (population) standard deviation, therefore, is $$ \boxed { \mathrm{SD}(X) = \sqrt{\dfrac{SSQ}{n} - \left( \dfrac{SQS}{n} \right)^2} } \;. $$ You see that you also need $n$, the number of samples.

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^{1 The population statistics can be corrected, when sample statistics are required.}