Question 1094341: invent a data set with 8 values that has a median of 21, a mean of 25, and a mode of 23.
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
If there are 8 values with a median of 21, then the average of the 4th and 5th numbers must be 21.
One way to do that would be to have the 4th and 5th numbers both equal to 21. But with a mode of 23, that would mean the 6th, 7th, and 8th numbers would all have to be 23. But then it would be impossible to have a mean of 25.
So the next thing to try is having the 4th and 5th numbers 20 and 22. Then we would only need the 6th and 7th numbers equal to 23 to make 23 the mode. Now the 1st, 2nd, 3rd, and 8th numbers can be almost anything we want; they only have to be all different (to keep 23 the mode), and they have to make the mean 25.
So we can make the first three numbers 17, 18, and 19, giving us
17, 18, 19, 20, 22, 23, 23
as the first 7 of the 8 numbers. Then a little arithmetic tells us that to make the mean 25 the last number has to be 58.
With the last number in our solution being so much larger than the 7th number, it should be clear that we could have chosen smaller numbers for the 1st, 2nd, and 3rd.
And there are also solutions where the middle two numbers are 19 and 23 instead of 20 and 22, as in our solution.
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