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.

(In reply to re: solution by Dennis)

My idea, in more detail is:

If you have a configuration of an odd number of grades, with a 7 in the middle (when in order), such as:

some number n of grades<=7,  7, the same number n of grades>=7

that the presence of the 7 in the middle could lower the mean if the other numbers had a mean above 7, so one could effectively remove it if the remainder of the numbers had a mean below 8 (showing which was the goal of the rest of the post).

And if there were an even number of entries with two 7's in the middle, the same applies to the two 7's being removable if the mean of the rest of the numbers was <8.  In other words, the middle 7 or 7's can only bring down any average that was near 8, which would only be desirable if the rest had a mean above 8, which can't happen.

Posted by Charlie on 2007-05-24 14:55:02