# How to find percentiles of a Normal distribution?

A lecturer wishes to "grade on the curve". The students' marks seem to be normally distributed with mean 70 and standard deviation 8. If the lecturer wants to give 20% A's, what should be the threshold between an A grade and a B grade?

• Some hints are provided by other homework questions related to the normal distribution, such as stats.stackexchange.com/q/5504/919 . – whuber Mar 8 '11 at 20:28
• A further tip: what you actually call the threshold will translate to a quantile on the ${\cal N}(0;1)$ PDF. – chl Mar 8 '11 at 21:31
• @chl when you say $N(0; 1)$ did you mean $N(70; 8)$? – Gavin Simpson Mar 8 '11 at 22:01
• @Gavin Ah, indeed :( – chl Mar 8 '11 at 22:17
• @chl @Gavin May I suggest you're both correct? Pedagogically, the merit of standardizing and z-scores (that is, relating statistics to N(0,1)) is that you learn one reference distribution and using it habituates you to thinking in units of standard deviation. Familiarity with that process makes light work of this question... – whuber Mar 8 '11 at 23:11

A normal distribution has about 20% of its distribution more than 0.842 standard deviations above the mean; using the cumulative distribution of standard normal $\Phi$, $$\Phi(0.842) \approx 0.8$$ so the threshold should be about $70 + 8\times 0.842 \approx 76.7$.