Jeanette received an integer grade between 1 and 10 inclusive, for each of her lab reports. She said that the arithmetic mean, median, and mode of all her lab grades were 8, 7, and 8 respectively.

Is this possible? If so, find a grade distribution consistent with the data; if not, prove it.

Since the mode is 8 and thus the most frequent score, and since these are also "to the right" of median score 7 if the scores are lined up sequentially, there must be an equal quantity of scores "to the left" of median 7. These could be more 7s, but never in a greater quantity that the 8s.

7s and 8s together will never comprise a mean of 8. If we were to assume that there were some 10s at the far right of our line of scores that would help us, but these would have to be counterbalanced by more 7s to the left of median 7, but not to exceed our quantity of 8s.

Let n=# of scores of 8. In order to maximize our opportunity to achieve a mean of 8, assume then that we have (n-1) scores of 7. In order for one of these to be the median, not only do the other (n-2) lie to the left of median 7, but we will have to add 2 more "non-7" scores to the far left of our line. Might as well make then 6s. Clearly this line of scores won't have a mean of 8.

What if there were some score higher that 8 though? Well, ok. Let n=# of 8s, (n-1)=# of 7s, and maybe k=# of 10s, where k<n. Still in this case, we are going to have to add k additional scores to the far left of our line in order to preserve our median 7. We can make them 6's as long as we don't surpass the quantity of 8s. All these extra 6s and 10s have a combined mean of 8, but we needed it to be greater than 8 to counteract the low mean we alread had. Pushing these scores out in longer and longer quantities while trying to preserve 8 as the mode will not provide us with a mean that will reach 8, no matter how large we choose to make our original n.

Posted by Mike C on 2007-05-24 09:59:30