# 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
Commented Mar 8, 2011 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
Commented Mar 8, 2011 at 21:31
• @chl when you say $N(0; 1)$ did you mean $N(70; 8)$? Commented Mar 8, 2011 at 22:01
• @Gavin Ah, indeed :(
– chl
Commented Mar 8, 2011 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
Commented Mar 8, 2011 at 23:11

Ten days later this is probably worth an answer:

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 $70 + 8\times 0.842 \approx 76.7$.

I do wonder slightly why the lecturer would do this instead of just giving an A to the top 20% of students.

• Henry Normally we try to provide useful advice and guidance on homework questions without actually doing the homework for people. The lack of an answer for ten days was restraint on the part of this community, not ignorance.
– whuber
Commented Mar 18, 2011 at 22:14
• @whuber: I realise that, which I waited ten days. But I do think that questions that have an answer should eventually get one. Happy to discuss on meta if you prefer. Commented Mar 19, 2011 at 9:20
• This discussion already occurred on meta at meta.stats.stackexchange.com/questions/12/… . That would be a good place to share your thoughts. Also take a look at meta.stats.stackexchange.com/questions/725/… .
– whuber
Commented Mar 19, 2011 at 13:29