Three values are missing from the shown data set. Given that the mode is forty-four, the mean is fifty-eight, the range is forty-four, and the data is listed in ascending order — so that’s from smallest up to largest — Find the missing values. And we’ve been given thirty-six, forty-four then another number then another number then seventy-four then another number, the largest number.
Now this is a bit of a mystery puzzle, but we’ve been given some interesting clues. The mode is forty-four, so the most frequently occurring number in that list is forty-four. So there must be at least one other forty-four in the list. The mean is fifty-eight. So there are six numbers entitled. So if I add them all up and divide by six, I should get an answer of fifty-eight. And the range is forty-four; that’s the difference between the largest number and the smallest number. So we told the range is forty-four. So the largest number minus the smallest number must equal forty-four.
Well we don’t know what the largest number is because we haven’t been told that. But we do know that the smallest number is thirty-six. So if I add thirty-six to both sides of that equation, I should be able to work out what the largest number is. Well forty-four plus thirty-six is eighty. And on the right-hand side, I’ve got the largest number minus thirty-six plus thirty-six, which is just the largest number. So the largest number is eighty. I can feel that in now. And I’ve used this piece of information. Now as I said before, the mode is forty-four; so there must be at least one other forty-four in the list. So I know this number here must be forty-four. It’s possible that this number here is also forty-four, but I don’t know that yet.
In fact for now let’s just call it 𝑥 because we don’t quite know what the number is going to be. So we’ve got to use the fact that the mean is fifty-eight to try to work out the value of 𝑥. So the formula we’re gonna be using for the mean is that we’re gonna sum the numbers and divide by the number of numbers. So the mean is gonna be thirty-six plus forty-four plus forty-four plus 𝑥 plus seventy-four plus eighty. And we’re gonna divide that by six because there were six numbers in the list. And the question told us that the mean is fifty-eight. So I can write that down here; I know what my answer is going to be. If I add together thirty-six, forty-four, forty-four, seventy-four, and eighty, I get two hundred and seventy-eight. So fifty-eight is equal to two hundred and seventy-eight plus 𝑥 all over six. Now if I multiply both sides of that equation by six, Well six times fifty-eight is three hundred and forty-eight. And over on the right-hand side, I’ve multiplied by six then I’m dividing by six. These sixes here and here will cancel out. So I’m just left with two hundred and seventy-eight plus 𝑥.
Now to work out the value of 𝑥, I’m gonna have to subtract two hundred and seventy-eight from the right-hand side. And if I subtract two hundred and seventy-eight from the right-hand side, I’ve got to subtract it from the left-hand side as well; otherwise, they won’t balance. Now on the right-hand side, two hundred and seventy-eight take away two hundred and seventy-eight is nothing. So those two things cancel out. So I’m just left with positive 𝑥 so 𝑥. And on the left-hand side, three hundred and forty-eight take away two hundred and seventy-eight is seventy. So I’ve got 𝑥 is equal to seventy. So I can now replace that on my list. So my answer is that the missing numbers were forty-four, seventy, and eighty.
Now obviously we used the clues that we were given in the question about the mode being forty-four, the mean being fifty-eight, and that the range being forty-four as well. But we did rely on the fact that the data was listed in ascending order from smallest down here up to largest up here. That helped us to answer this question as well.