Word Problem about Statistics

Question

The median of 5 numbers is 15. The mode is 12. The mean is 15. What are the 5 numbers?

I've done some algebra and now I know that the last 2 digit numbers must add up to 36. How can I determine those two digits without having to use trial and error which would consume a lot of time?

Answer 1

So, let's just assume that $a,b,c,d,e$ are increasing in order. Then since the median of the five numbers is 15, we must have that $c=15$. You know that the mean is 15, so $(a+b+c+d+e)/5 = 15$, or equivalently $a+b+d+e = 60$. Since the mode is 12, and we said that the list of variables is increasing in order, then $a=b=12$.

Now, we have that $d+e=36$. Since they both have to be bigger than $15$ (otherwise, 15 would not be the median and the mode would not be 12), what two numbers will do the job?

After that, you can easily double check that the list of numbers you get meets the criteria.

Answer 2

Assume without loss of generality $a\leq b\leq c\leq d\leq e$.

for the mean : $a+b+c+d+e = 75$.

assuming $1$ mode : $a =12 ,b = 12$.

for the median and because there's just $1$ mode : $c=15, d> 15,e> 15 $.

so $d+e=36$ , so $(d = 16, e = 20)$ or $(d = 17, e = 19)$.