I have a sample of $121$ integers between $1$ and $1000$ both inclusive, with repetitions allowed. I am also given that the sample has a unique mode. What will be the greatest integer less than the largest possible value of the difference between the mode and the arithmetic mean of the sample?
I tried keeping the mode to be maximum, i.e. assuming $1000$ to appear $2$ times and the rest to be minimum by assuming them to be $1,2,3,\ldots$ all appearing only once. Any error in this?
Best Answer
Your idea of having $1000$ appear twice and the values $1,2,\ldots, 119$ appearing once each is a good idea
It should give you a difference between the most frequent value and the mean of about $924.4628$
But it is not the best option. Consider having $1000$ appear three times and the values $1,2,\ldots, 59$ appearing twice each. That should give you a difference between the most frequent value and the mean of about $945.9504$; this is a higher gap between the most frequent value and the mean as reducing the average of the low values from $60$ to $30$ more than offsets having an extra occurrence of $1000$
And even that is not the best option, but you can continue looking at this sort of solution. You will not have to look very far, as quite soon increasing the number of times $1000$ appears will tend to drive up the mean and not be sufficiently offset, so reducing the gap between the most frequent value and the mean
Rather than spoil the question for you, here is a graph of the maximum distance plotted against the number of times $1000$ appears - you are only interested in the top left values when $1000$ only appears a few times