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.

What is the highest possible mean when the median is 7 and the mode is 8? (Well, we want it to be 8, at least.)

For a mean of 8, the presence of an actual 7 would lower the mean, so let's leave out the 7 and just assume there are an even number of grades with the same number below 7 as above. Also, let's say there are n 8's.  To balance around the median we need n numbers less than 7. There can be only at most n-1 6's, so there must be at least one 5. At this point the mean is clearly below 7, and all the more so below 8.

But if we try to increase the mean by adding 10's, the highest grades we can add below 7 to keep the median in balance is 5's. Adding 10's and 5's in equal numbers can't bring the mean up to 8.

 

Posted by Charlie on 2007-05-24 10:35:23