Proportion of standard deviation

Question

suppose the distribution of a variable is normally distributed. what proportion of the data will be within 0.75 standard deviations of the mean?

Answer 1

z = (x - mu)/sigma

The normal distribution has what area between mu + 0.75 sigma and mu - 0.75 sigma?

This is the area excluding the two tails.

Look up the normal distribution and find the area in one tail,that is greater than mu + 0.75 sigma,

1 - N(0, 0.75) = 0.2266 where the table is for a normal distribution of mean = 0 and sigma=1.

Double that, because you want both tails

0.4532

Then subtract this from 1.0000,

to get the interior of the total distribution, a bit over 50%.