I am going back through my old homework assignments to study for an upcoming Statistics test and one of the questions gives a table with number of children and number of women who have that number of children:

`Children | 0 | 1 | 2 | 3 | 4 | 5 ----------------+--------+---------+---------+---------+---------+-------- Number of Women | 27 | 22 | 30 | 12 | 7 | 2 `

The question is:

**Find the sample standard deviation of the number of children.**

As I understand it, the sample standard deviation is

$s=sqrt{frac{1}{n-1}(sumlimits_{i=1}^n{X^2_i-nbar{X}^2})}$

And in order to do this, I would need to sum up each of the individual values given within the set of $n$ values. However, since there are 100 values (collapsed as the table shows), How might I do this by hand, that is, without the aid of a program on a test?

I was able to (I think) calculate $bar{X}$ by summing the product of the number of children by the number of women and dividing by $n$:

$bar{X}=frac{1}{n}(sumlimits_{i=1}^n{X_i})$

$bar{X}=frac{1}{100}big[(0*27)+(1*22)+(2*30)+(3*12)+(4*7)+(5*2)big] = 1.56$

But I'm unsure how to do the same with the sample standard deviation calculation.

I tried searching and looking online for the answer to this, but I'm obviously a novice at this, so I might just be searching for the wrong things.

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#### Best Answer

Your formula for the mean is correct.

Presumably you can work out the mean from the variance.

Have you seen the formula for the population variance $text{Var}(X)=E(X^2)-E(X)^2$?

The usual sample variance is the same, with Bessel's correction –

$text{Var}(x)=frac{n}{n-1} [overline{x^2}-overline{x}^2]$.

(Where $overline{x^2}$ means 'take the mean of the squared observations')

From that you can use exactly the same method you did for the mean to work out the mean of the $x^2$ values and hence obtain the variance and then the standard deviation.

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