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Math Help - Missing Date

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    Missing Date

    The mean of thirteen-item data set is 320. Eleven of the items are 300, 320, 199, 175, 325, 520, 156, 225, 326, 421, and 504. The median is 325. Find the remaining two items in the set if it is known that these items have the greatest possible difference.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by aznmartinjai View Post
    The mean of thirteen-item data set is 320. Eleven of the items are 300, 320, 199, 175, 325, 520, 156, 225, 326, 421, and 504. The median is 325. Find the remaining two items in the set if it is known that these items have the greatest possible difference.
    let's arrange these numbers in increasing order, we get:

    156, 175, 199, 225, 300, 320, 325, 326, 421, 504, 520

    now, 325 is the median. since there should be 13 numbers, we need to have 6 numbers on either side of 325. we already have 6 on the left of 325, so with the addition of the two missing numbers we will have 6 on the right of 325. so a tentative list of our numbers is as follows:

    let the two numbers be x and y, we have:

    156, 175, 199, 225, 300, 320, 325, 326, 421, 504, 520, x, y

    we are given that the mean is 320, so:

    (156 + 175 + 199 + 225 + 300 + 320 + 325 + 326 + 421 + 504 + 520 + x + y)/13 = 320

    => (3471 + x + y)/13 = 320
    => x + y = 13(320) - 3471
    => x + y = 689

    now we are also told that the numbers x and y have the greatest difference, so, one has to be the smallest possible number either can be. what is the smallest number either can be? well, the median is 325, so to be on the right of 325 when arranged in ascending order, one of the numbers have to be at least 325, that is, greater than or equal to 325. so the least it can be is 325.

    now if one of the numbers is 325, the other has to be 689 - 325 = 364

    so the two missing numbers are 325 and 364
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