Range of one standard deviation from the mean The table below show the number of women based on the number of their children according to the result of a survey conducted on $50000$ married women in the country.
The table
Determine the percentage of women whose number of children are within the range of one standard deviation from the mean.
My attempt,
$\bar{x}=\frac{\sum x}{n}$
$=\frac{111655}{50000}=2.23$
$\sigma=\sqrt{\frac{\sum x^2}{n}-(\frac{\sum x}{n})^2}$
$=\sqrt{\frac{318295}{50000}-(\frac{111655}{50000})^2}$
$=1.17$
Then, $\bar{x}\pm \sigma$
$2.23\pm 1.17$
How to proceed then? 
 A: I did not check your computations for the mean and standard
deviation. I'm assuming they are OK.
Because it does not make sense to talk about fractional
children,  we choose only the integers in the
interval $[1.06,3.40]$ suggested by @Barry. That would be 2 and 3.
The corresponding numbers in the table give: $$11633 + 19418 = 31051$$
mothers.
That amounts to $$(11633+19418)/50000 = 0.62102$$
or about 62% of the mothers in the table.
Perhaps this is an attempt to illustrate the Empirical Rule,
one part of which states that, for many populations, approximately 68% of a population
lies within one standard deviation of the mean. 
But this rule
doesn't work very well here because only integer values make sense.
If the standard deviation were just a little larger, you'd
include the mothers with only one child, and the proportion
would increase to about 79%. Also the rule usually works better
for symmetrical distributions, and yours has a 'tail' in the
direction of mothers with more children.
Maybe you can read about this rule in your text and see what you think of
this example.
