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
$s=\sqrt{\frac{1}{n-1}(\sum\limits_{i=1}^n{X^2_i-n\bar{X}^2})}$
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 $n$:
$\bar{X}=\frac{1}{n}(\sum\limits_{i=1}^n{X_i})$
$\bar{X}=\frac{1}{100}\big[(0*27)+(1*22)+(2*30)+(3*12)+(4*7)+(5*2)\big] = 1.56$
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.