# Calculating standard deviation without a data set.

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...

• would you like to describe how do you compute SD given data points? which formula do you use? Jan 24 '18 at 18:44
• Hint: The variance is $s^2 = \frac{1}{n-1}(Q - T^2/n),$ where $Q = \sum_i X_i^2$ and $T = \sum_i X_i.$ This is proved by performing the square in $\frac{1}{n-2}\sum_i(X_i - \bar X)^2,$ using the distributive law, and using the definition of $\bar X.$ Jan 24 '18 at 19:12

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!

• Your formula for SD seems slightly wrong. Assuming the mean $m$ is known then the variance $V=\frac{\sum (x_k-m)^2}{n}=\frac{\sum x_k^2}{n}-m^2$ so $SD=\sqrt{\frac{\sum x_k^2}{n}-m^2}$. When the mean is being estimated by the average of the $x's$, then the division is by $n-1$ rather than $n$. Jul 13 '18 at 3:02

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.

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