How do I find the probability of getting an average when selecting from a normal distribution? If I have that scores follow a normal distribution, I know that the probability of selecting any person at random and them getting a score between 1 and 2 sd's above the mean is 13.6%. If I randomly select 4 people what is the probability that their scores will average to a value between 1 and 2 sd's above the mean? I dont care where any of their individual scores are only that they average to a value between 1 and 2 sds above the mean.
 A: Let's call $\bar X =\frac 14(X_1 + \cdots + X_4)$ with $X_i\sim N(\mu,\sigma^2)$.
Under the given conditions we have
$$\bar X \sim N\left(\mu,\frac{\sigma^2}{4}\right)$$
Hence,
$$P(\sigma \leq \bar X - \mu \leq 2\sigma ) = P\left(2\leq \frac{\bar X - \mu}{\frac{\sigma}{2}}\leq 4\right)=\Phi(4)-\Phi(2)$$
where $\Phi$ is the cumulative distribution function of the standard normal distribution.
A: Simulation: $n = 4,  \mu = 68,  \sigma = 4.$
Perhaps normally distributed heights in inches of four students.
set.seed(217)
n = 4;  mu = 68;  sg=4
a = replicate(10^6, mean(rnorm(n,mu,sg))) # 10^6 sample averages
mean((a > mu+sg) & (a < mu+2*sg))
[1] 0.022626
2*sd((a > mu+sg) & (a <= mu+2*sg))/1000
[1] 0.0002974161  # aprx 95% margin of simulation error

With a million iterations, one can expect about three place
accuracy, so this result is in essential agreement with @trancelocation's result (+1) $\Phi(4) = \Phi(2) = 0.023.$
pnorm(4)-pnorm(2)
[1] 0.02271846

Because the average of four $A = \bar X_4 \sim
\mathsf{Norm}(68, 2)$ we can get the same numerical
result directly (without standardizing) in R:
diff(pnorm(c(72,76), mu, sg/2))
[1] 0.02271846

The histogram below shows simulated values of $A$ along
with its normal distribution (solid orange curve). Vertical dashed lines
mark the interval $(\mu + \sigma, \mu + 2\sigma) = (72, 76).$ The dotted density curve of $\mathsf{Norm}(68, 4)$  is for each individual observation.

R code for figure:
hdr = "n=4: Dist'n of Sample Means"
hist(a, prob=T, br=30, col="skyblue2", xlim=c(55,80), main=hdr)
 abline(v = c(mu+sg, mu+2*sg), lwd=2, col="darkgreen", lty="dotted")
 curve(dnorm(x, mu, sg/sqrt(n)), add=T, lwd=2, col="orange")
 curve(dnorm(x, mu, sg), add=T, lwd=2, lty="dotted")
 abline(v = c(mu+sg, mu+2*sg), lwd=2, col="darkgreen", lty="dashed")

