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Re: When median = mean, is the set always evenly spaced? [#permalink]

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17 Nov 2013, 23:14

No. For example, in the list: {2, 5, 9, 10, 19}, the median is 9 and the arithmetic mean is also 9. However, the numbers in the list are not uniformly spaced.

Here the deviations of the two numbers to the left of the median of 9 are -7 (equal to 2-9) and -4 (equal to 5-9), whereas it is +1 and +10 for the two numbers on the right. Here the deviations of -7, -4, +1, +11, all add up to 0, and thus lead to a median equal to the mean. This means that as long as the deviations of all numbers from the median add up to zero, then mean=median, however the spacing need not be the same between consecutive terms.

Is the reverse also true? I.e. when the mean and median are the same, is the set always evenly spaced?

Also note that it is very easy to have mean = median in any set i.e. the constraints are few.

Say, 10, 16, 28, 30

Here mean = 21 Just add 21 to the list: 10, 16, 21, 28, 30 Now Mean = Median

As long as there are equal number of elements below and above the mean, you can easily create a set where mean = median. An evenly spaced set is a very specific set.
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