
help!
If you have a random sample of 1000 social work clients, and the average value of some sample statistic (e.g., age, weight, income, etc.) is ‘50’, with a standard deviation of 10, about how many people from your sample will be included in the range of values between 40 and 60 on this measure of the sample statistic? Why? About how many will be included between the range of 30 – 70? Why?

Standard Deviation
It depends ultimately on the underlying distribution, but for most we can assume an approximately normal distribution. For a normal distribution, the confidence intervals are approximately 68.27% for $\displaystyle \mu \pm 1\sigma$ and 95.45 for $\displaystyle \mu \pm 2\sigma$. As the mean $\displaystyle \mu=50$ and standard deviation $\displaystyle \sigma=10$, this gives the range 40 to 60 as $\displaystyle \mu \pm \sigma$, so we should expect about $\displaystyle 0.6827 \cdot 1000 = 683$ in this range. Similarly, 30 to 70 is $\displaystyle \mu \pm 2\sigma$, so we expect about $\displaystyle 0.9545 \cdot 1000 = 955$ within that range. These formulae are ultimately derived from numerical integration of the probability density function of the standard normal distribution (normal distribution with mean $\displaystyle \mu=0$ and variance $\displaystyle \sigma^2 =1$), which is
$\displaystyle \dfrac{1}{\sqrt{2\pi}}e^{\tfrac{x^2}{2}}$.
Thus the fraction of the population within n standard deviations of the mean is approximated by
$\displaystyle \int_{n}^{n}\dfrac{1}{\sqrt{2\pi}}e^{\tfrac{x^2}{2}}\, dx$
Kevin C.