Question

Construct a dataset of 7 elements whose median is 87?

Answer 1

An example: `12,13,14,87,246,357,790983234`.

Explanation:

For any collection of numbers (which we call a set), the median is the element of the set which has the same amount of elements both less than it and greater than it.

For example, in the set `{1,5,13,9,2,2,10},` the element `{5}` has three elements smaller `(1,2,2)` and three elements larger `(9,10,13).` Thus, for this set of seven elements, the median is 5.

To find the median of a set, a common first step is to arrange the elements in order. That way all you need to do is count in from both ends, one element at a time, until you find the middle.

Example: Find the median of `{1,6,23,2,7,89,3,2,4,70,18}.`

Answer: The median is 6.

How? First, arrange the elements in order:

`{1,2,2,3,4,6,7,18,23,70,89}`

Then "cross off" the lowest and highest element:

`{color(pink)1,2,2,3,4,6,7,18,23,70,color(pink)89}`

Keep going with what's left:

`{color(pink)1,color(pink)2,2,3,4,6,7,18,23,color(pink)70,color(pink)89}`
`{color(pink)1,color(pink)2,color(pink)2,3,4,6,7,18,color(pink)23,color(pink)70,color(pink)89}`
`{color(pink)1,color(pink)2,color(pink)2,color(pink)3,4,6,7,color(pink)18,color(pink)23,color(pink)70,color(pink)89}`
`{color(pink)1,color(pink)2,color(pink)2,color(pink)3,color(pink)4,6,color(pink)7,color(pink)18,color(pink)23,color(pink)70,color(pink)89}`

Since there's only one element left now, that element is the median.

If a set has an odd number of elements, the median will be the middle element of the set. If the set has an even number of elements, the median is the average of the two middle elements.

Example: Find the median of `{3,5,4,6}.`

Answer: The median is 4.5.

How? The smallest element is 3, and the largest is 6. Removing these leaves the elements 4 and 5. We can't remove these last two; that would leave us with no elements. So we take the average of 4 and 5, to get `(4+5)/2=9/2=4.5.`

To create a set with a given median `m`, all you do is add elements two at a time—one smaller than `m` and one larger than `m`. Repeat until you have as many elements as you need.