# How to calculate the mean with percentages

Scores for a exam are normally distributed with a standard deviation of 7. to be eligible for employment, you must score in the top 5%. The lowest score you can get and be eligible for employment is an 85%. What is the mean score for the exam?

I have no clue how to start this and how to finish it. To me the question throws me off...if you have to score in the top 5%, but you can get an 85%...I am confused, doesn't that just mean that you need to get an 95% or above, or am I reading this wrong?

• There may be some context missing. I cannot find enough information for an answer. – mlc May 3 '17 at 19:32

You have been told the minimum requirements to "pass" the exam twice, each time in a different way.

Suppose all the questions on the exam are worth $100$ points altogether. Then to "pass" the exam (to be eligible for employment) you must get at least $85$ points, which is $85\%$ of the highest possible score.

Now suppose $1000$ people take the exam. Line up all the applicants according to their exam scores, highest score first. The first $50$ applicants in this line all have scores that are as high as or higher than than the scores of any of the other $950$ applicants. Those $50$ applicants are the "top $5\%$" of all applicants who took the exam.

What the question (as worded) does not say is whether these two criteria describe exactly the same set of eligible applicants, or whether these are just two criteria that were previously determined to be "passing" criteria. (The employer could have said, "We won't take anyone who scores less than $85,$ but in any case we won't consider anyone who wasn't in the top $50$ even if their score is greater than $85.$")

In the case that these are two separate criteria, the question can't be answered. So let's assume the two criteria describe the same set of applicants. Then $950$ out of $1000$ applicants score less than $85\%$ on the exam.

In order to answer the question, we also have to make some assumption about how to translate between scores stated as simple numbers (two scores that differ by the $7$-point standard deviation) and scores stated as percentages. The simplest assumption is that the "$85\%$" score is a score of $85$ points.

Now you have a normal distribution with mean $\mu$ and standard deviation $7,$ with a $5\%$ probability for the random variable to be greater than $85.$ Find $\mu.$

I give this question low marks because of the huge unsupported assumptions it requires us to make in order to find any answer. But those are the assumptions that I would guess the question-writer intended us to make.

• Thank you, that sort of clears it up. What would be the formula that I would put this into to find the mean? – Kathryn Hutton May 3 '17 at 19:51
• @KathrynHutton You need to calculate the CDF to get a value of 95%. You can use a mean of zero for this part. You can just iterate through different values of the random variable ($x$). Since the stddev is 7, you could try 7, then 14. If you overshoot reduce the value of $x$. Once you've done this to whatever accuracy you want, you will know how many points above the mean the top 5% corresponds to. Since the top 5% corresponds to 85, then you'd subtract the number of points you determined from 85. You can find online CDF calculators. You can save some work by finding an inverse CDF calculator! – Χpẘ May 3 '17 at 21:14