If a set of data has 5 numbers has a range of 9 a mode of 9 a median of 11 and a mean of 12 what are the 5 numbers?

Answer 1

This is a logical puzzle. You can read the statement this way:

1. There are five numbers.
2. The highest number is 8 greater than the lowest number
3. The number that repeats the most is 9
4. The number in the middle of the set is 11
5. The average of the numbers is 12

We know that there are five numbers [statement #1], so let's call them a, b, c, d and e, in ascending order.

Given: (a + b + c + d + e) / 5 = 12 [statement #5]

∴ a + b + c + d + e = 60

Given: c = 11 [statement #4]

∴ a + b + 11 + d + e = 60

If the middle number is 11, and the most repetitious number is 9 [statement #3], then a and b must both equal 9.

∴ 9 + 9 + 11 + d + e = 60

Given: e = a + 8 [statement #2], and the knowledge that a = 9, we know then that e = 9 + 8, or 17:

∴ 9 + 9 + 11 + d + 17 = 60

and with that, we can calculate the value of our final number, d:

∴ d = 60 - 9 - 9 - 11 - 17
∴ d = 60 - 46
∴ d = 14

So the set of data is:

{9, 9, 11, 14, 17}

You can quickly check these results by using the original statements on them:

"... has a range of 9 ..." - correct. 9 through 17 inclusive is indeed a range of 9.
"... a mode of 9 ..." - correct, that is the most frequently occurring number.
"... a median of 11 ..." - correct, that is our original number
"... a mean of 12 ..." (9 + 9 + 11 + 14 + 17) / 5 = 12, - correct, that is our mean.