Question

I know the number of students, mean and standard deviation - how do I calculate how many students will receive a score within 1 standard deviation? two standard deviations?

Answer 1

We know :

$\small \mu$ = Mean

$\small \sigma$ = Standard deviation .

n = Number of students

Now, If the data is normally distributed then :

$\small P\left ( \left | \frac{X-\mu}{\sigma} \right |\leq 1 \right ) = P(-1\leq Z\leq 1) = 2P(Z\leq 1)-1$

We can compute this using Excel function " NORMSDIST()"

$\small 2P(Z\leq 1)-1 = 2*NORMSDIST(1)-1 = 2*0.841345-1 = 0.6827$

Number of students who will receive score within 1 standard deviation = n * 0.6827

If the data is normally distributed then :

$\small P\left ( \left | \frac{X-\mu}{\sigma} \right |\leq 2 \right ) = P(-2\leq Z\leq 2) = 2P(Z\leq 2)-1$

We can compute this using Excel function " NORMSDIST()"

$\small 2P(Z\leq 2)-1 = 2*NORMSDIST(2)-1 = 2*0.97225-1 = 0.9545$

Number of students who will receive score within 2 standard deviation = n * 0.9545