Is it possible to find the mean and standard deviation when given the normal z-scores for the bottom 25% and top 25%

Question

I want to apply to a university where the 25th /75th percentiles for the SAT Math are 490 / 620 respectively, but I am curious how would I find the mean and the standard deviation assuming that the data is normally distributed? I know that we are talking about the middle 50th-percentile and $z_{0.25} = 620$ and $z_{0.75} = 490$

Answer 1

Using z-score tables for standard normal variable, $$ z_{0.25} = \mu - \sigma \cdot 0.675 \qquad z_{0.75} = \mu + \sigma \cdot 0.675 $$ You now have two equations for two unknowns.

In[213]:= Solve[{Quantile[NormalDistribution[mu, si], 0.25] == 490, 
  Quantile[NormalDistribution[mu, si], 0.75] == 620}, {mu, si}]

Out[213]= {{mu -> 555., si -> 96.3691}}

Answer 2

If you want an EXCEL Solution, here it is:

The two equations are:

$\frac{490-\mu}{\sigma}$ = Normsinv(.25) = -0.67449

$\frac{620-\mu}{\sigma}$ = Normsinv(.75) = 0.67449

Solve for $\mu$ and $\sigma$