Modal Median

Probability Level 1

Consider the set of numbers

{ 1 , 2 , 2 , 3 , 3 , 3 , … , 10 , 10 , 10 , 10 , 10 , 10 , 10 , 10 , 10 , 10 } , \{1, 2, 2, 3, 3, 3, \ldots, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10\},

where the number n n appears n n times for n = 1 n=1 to 10. What is the (absolute value) of the difference between the mode and the median of this set?

Details and assumptions

The mode of a set is the value that appears most often. The median of a set is the middle value when the data is arranged in numerical order.


The answer is 3.

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Arron Kau by Arron Kau

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1 solution

Arron Kau Staff
May 13, 2014

The mode of this set is 10. To calculate the median, there are 10 × 11 2 = 55 \frac {10 \times 11}{2} = 55 numbers, hence we seek the 28 28 th number. Starting from 10, there are 10 10's, 9 9's, 8 8's, hence the 28th number will be 7.

Thus, the absolute value of the difference between these numbers is 10 − 7 = 3 10-7 = 3 .

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