Answer to Question #280300 in Statistics and Probability for loverlie

Question

I have five numbers. Their mean is 4. Their median is 3. Their mode is 3. Can you find all the different sets of five positive whole numbers that satisfy these conditions?

Answer 1

Solution:

n = 5

Let the numbers in increasing order be a,b,c,d,e.

Given that median = 3

So, median = $\dfrac{n+1}2=\dfrac{5+1}2=\dfrac 62=3rd$ term

$\Rightarrow$ c = 3

Now, mean $=\dfrac{a+b+c+d+e}5$

$\Rightarrow 4=\dfrac{a+b+3+d+e}5 \\\Rightarrow a+b+d+e=17\ ...(i)$

Next, mode = 3

It means at least two numbers are 3 out of 5, one is c=3, another one could be b or d only, as they are in ascending order.

Case I: When b=3

Using (i)

$\\ a+3+d+e=17 \\ \Rightarrow a+d+e=14$

Now, a can take values 1 or 2.

If a=1, d+e=13 and $d\ne e; d,e>3; d<e$

So, pairs of (d,e) = (4,9), (5,8), (6,7).

Now, a can take values 1 or 2.

If a=2, d+e=12 and $d\ne e; d,e>3; d<e$

So, pairs of (d,e) = (4,8), (5,7).

Case II: When d=3

Using (i)

$\\ a+b+3+e=17 \\ \Rightarrow a+b+e=14$

Now, a can take values 1 or 2.

If a=1, b+e=13

Since c = 3, b can take only one value, i.e., 2

So, b=2.

Then, 2+e=13, so, e=11

So, only pair of (b,e) = (2,11).

Thus, different sets are (1,3,3,4,9), (1,3,3,5,8), (1,3,3,6,7), (2,3,3,4,8), (2,3,3,5,7), (1, 2, 3, 3, 11).