The Median of a set of Numbers

The median is the number exactly in the middle of a sorted set of numbers. A set of numbers is sorted when it is either in ascending or descending order. In other words, the numbers are sorted from least to greatest, or from greatest to least.

Finding the median depends on whether there are an odd or even amount of numbers in the set. We will cover both situations in this lesson, starting with an odd set of numbers and then progressing to an even set of numbers.

Finding the Median in an odd set of Numbers

When the quantity of numbers in the set is odd, the median is the number in the middle. Here are the steps to follow for finding the median in an odd set of numbers:

1. First, make sure the numbers are sorted in order. The numbers may be sorted in order from least to greatest, or in order from greatest to least.
2. Determine how many numbers there are.
3. Divide the quantity of numbers by two, then round up to the nearest whole number.
4. The median will be that rounded up amount of numbers into the set, counted from one end of the set.

That may have sounded like a mouthful, so let's go through a quick example to see the steps in action:

Consider the number set 1, 4, 7, 11, 14.

1. The numbers are already sorted from least to greatest.
2. There are 5 numbers which is an odd quantity.
3. Dividing the quantity by two and rounding up to the nearest whole number, we get 5/2 = 2.5 → 3.
4. We can count in from the left end or the right end of the set, let's do it from the right end in this example.
   The first number is 14, the second number is 11, and the third number is 7. Therefore, the median of this set of numbers is 7. It is the number in the middle of the sorted set.

Finding the Median in an Even set of Numbers

When the quantity of numbers in the set is even, the median is between the two middlemost numbers. Here are the steps to follow for an even set of numbers:

1. First, make sure the numbers are sorted in order. The numbers may be sorted in order from least to greatest, or in order from greatest to least.
2. Determine how many numbers there are.
3. Divide the quantity of numbers by two.
4. Count that amount of numbers in from one end of the set. The median is the average of that number and the next one over.

For example, let's find the median of the number set 50, 45, 29, 10, 3, -5.

1. The numbers are already sorted from greatest to least.
2. There are 6 numbers which is an even quantity of numbers.
3. Dividing the quantity of numbers by two, we get 6/2 = 3.
4. Let's count from the left end of the set (as stated earlier, we can count from either end of the set).
   The first number is 50, second is 45, and third is 29. The next number over from the third number is 10. Finally, we must find the average of 29 and 10.
   ^{(29 + 10)}⁄₂ = ³⁹⁄₂ = 19.5.
   The median of the set of numbers is 19.5. Although 19.5 is not one of the numbers in the set, it is the number in the middle of the sorted set, and therefore is the median.