how do I figure the standard deviation on a frequency distribution?

Question

What is the formula to figure this out? I need easy steps, because I am missing something somewhere and my answers keeps coming out wrong.

Class Frequency
0 Up to52
5 Up to107
10 Up to1512
15 Up to206
20 Up to253

Answer 1

Is the third line supposed to be
10 Up to 151
or
10 Up to 152?

Since you don't know the original values for the entire population, you assume that each member of a class has a value equal to the midpoint of that class.

For example, you have 5 values that are greater than 52 and less than or equal to 107, so the midpoint is (107+53)/2 = 80. I assume that the only valid values are integers? If not, then the midpoint is (107+52)/2.

Then, you compute the mean and standard deviation in the usual way. Since you have many repeated values, you can simplify things somewhat.

For instance, for the mean, you have 50 values altogether, and the mean is

m = (5*80/2 + 10*(108+151)/2 + 15*(152+206)/2 + 20*(207+253)/2) / 50

and then the variance is

1/49(5*(80-m)^2 + 10*( (108+151)/2 - m )^2 + ... )