Maximum Difference between mean and mode of the given set

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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?




statistics means




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  • What do you think?
    – iamwhoiam
    Jul 29 at 10:25







  • 1




    I tried keeping the mode to be maximum, ie assuming 1000 to appear 2 times and the rest to be minimum by assuming them to be 1,2,3,.... all appearing only once. Any error in this?
    – saisanjeev
    Aug 4 at 7:55














up vote
0
down vote

favorite












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?




statistics means




share|cite|improve this question





















  • What do you think?
    – iamwhoiam
    Jul 29 at 10:25







  • 1




    I tried keeping the mode to be maximum, ie assuming 1000 to appear 2 times and the rest to be minimum by assuming them to be 1,2,3,.... all appearing only once. Any error in this?
    – saisanjeev
    Aug 4 at 7:55












up vote
0
down vote

favorite









up vote
0
down vote

favorite











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?




statistics means




share|cite|improve this question













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?





statistics means





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 4 at 22:30









Henry

92.8k469147




92.8k469147









asked Jul 29 at 9:53









saisanjeev

362210




362210











  • What do you think?
    – iamwhoiam
    Jul 29 at 10:25







  • 1




    I tried keeping the mode to be maximum, ie assuming 1000 to appear 2 times and the rest to be minimum by assuming them to be 1,2,3,.... all appearing only once. Any error in this?
    – saisanjeev
    Aug 4 at 7:55
















  • What do you think?
    – iamwhoiam
    Jul 29 at 10:25







  • 1




    I tried keeping the mode to be maximum, ie assuming 1000 to appear 2 times and the rest to be minimum by assuming them to be 1,2,3,.... all appearing only once. Any error in this?
    – saisanjeev
    Aug 4 at 7:55















What do you think?
– iamwhoiam
Jul 29 at 10:25





What do you think?
– iamwhoiam
Jul 29 at 10:25





1




1




I tried keeping the mode to be maximum, ie assuming 1000 to appear 2 times and the rest to be minimum by assuming them to be 1,2,3,.... all appearing only once. Any error in this?
– saisanjeev
Aug 4 at 7:55




I tried keeping the mode to be maximum, ie assuming 1000 to appear 2 times and the rest to be minimum by assuming them to be 1,2,3,.... all appearing only once. Any error in this?
– saisanjeev
Aug 4 at 7:55










1 Answer
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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



enter image description here






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  • yeah I figured out the answer. Thanks for your help, but can you help me give a rigorous approach to this question
    – saisanjeev
    5 hours ago










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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
0
down vote













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



enter image description here






share|cite|improve this answer





















  • yeah I figured out the answer. Thanks for your help, but can you help me give a rigorous approach to this question
    – saisanjeev
    5 hours ago














up vote
0
down vote













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



enter image description here






share|cite|improve this answer





















  • yeah I figured out the answer. Thanks for your help, but can you help me give a rigorous approach to this question
    – saisanjeev
    5 hours ago












up vote
0
down vote










up vote
0
down vote









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



enter image description here






share|cite|improve this 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



enter image description here







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Aug 4 at 22:29









Henry

92.8k469147




92.8k469147











  • yeah I figured out the answer. Thanks for your help, but can you help me give a rigorous approach to this question
    – saisanjeev
    5 hours ago
















  • yeah I figured out the answer. Thanks for your help, but can you help me give a rigorous approach to this question
    – saisanjeev
    5 hours ago















yeah I figured out the answer. Thanks for your help, but can you help me give a rigorous approach to this question
– saisanjeev
5 hours ago




yeah I figured out the answer. Thanks for your help, but can you help me give a rigorous approach to this question
– saisanjeev
5 hours ago












 

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