Question

A teacher gives 5 students a multiple choice test, in which each problem is worth 1 point. The median and mean scores turn out to be 9 and 10 points, respectively.

1. What is the minimum possible top score?
2. What is the maximum possible top score?
3. What is the minimum possible standard deviation?
4. What is the maximum possible standard deviation?

Answer 1

Let be the 5 scores such that . that is the scores are arranged in an ascending order.

We know that the median sore is 9. Since the number of observations is 5 and it is odd, the median value is (5+1)/2=3rd observation.

Hence

The mean is 10, that is

Each problem is worth 1 point. That means fractional scores are not possible.

We can say that and

a) To get the minimum possible top score, we need to get the lower values as close to 10 as possible. The closest we can get the values lower than the median is the median value 9. That is we can set and still get the median value =9

With that, we can get the sum of x4 and x5 as

Since x4 and x5 have to be integers, and both have to be 9 or more, the value 11 and 12 will give the lowest top value

Hence the scores for the minimum possible top score are

The mean then is

and the median is 9

ans: the minimum possible top score is 12

b) To get the maximum possible top score, we need to get the lower values as faraway from 10 as possible. But the lowest that a student can score is 0. Hence we will set

We can get the sum of x4 and x5 as

the lowest that we can push x4 is 9 and hence with x4=9 we can get x5=32 to get a sum of 41

Hence the values are

giving us a mean =10 and median=9

ans: the maximum possible top score 32

c) the minimum possible standard deviation is when the values are close together, which is the case of part a).

The standard deviation when the values are is

ans: the minimum possible standard deviation is 1.4142

d)  the maximum possible standard deviation is when the values are spread out, which is the case of part b).

The standard deviation when the values are is

ans: the maximum possible standard deviation is 13.0958