Below is a grid of five numbers. 10, four, three, seven, 14. One of the numbers is removed from the grid. The median of the remaining numbers is seven. Which number was removed from the grid? You must show your working.
Let’s begin by reminding ourselves what is meant by the median of a set of data. The median is the middle value when the data have been put into an ordered list. They must be in order from smallest to largest. It’s not just a middle value as the data are originally written. Let’s write these five numbers then in order from smallest to largest.
The smallest number is three, then four, then seven, 10, and finally 14. We can see that the middle value in our ordered list of all five numbers is seven. So actually the median of all five numbers is seven. But the question tells us that one of the numbers is removed from the grid and then then the median is still equal to seven. Let’s think about how we can decide which number was removed.
If we remove one number from a list of five, then there are four numbers remaining in that list. When we have four values or indeed any even number of values, there won’t be a single middle value; it will be the average of the two values in the middle. In fact, we can use the formula 𝑛 plus one over two where 𝑛 represents the number of values to work out the position of the median in a dataset.
In our dataset with one number removed, the number of values is four. So the position of the median will be four plus one over two. That’s five over two which is equal to 2.5. Now what does this actually mean? Well, it means that in order to find the median in a set of data with four values, we take the average — that’s the mean average — of the second and third values.
If we label the ordered values as 𝑥 one, 𝑥 two, 𝑥 three, and 𝑥 four, then the median will be 𝑥 two plus 𝑥 three over two. That’s the average of the second and third values. In this case, we know that the median will be equal to seven, which means we have an equation: 𝑥 two plus 𝑥 three over two is equal to seven. By multiplying both sides of this equation by two, we can make it simpler: 𝑥 two plus 𝑥 three is equal to 14.
So now, we know that in our new dataset with only four values, the sum of the second and third values must be equal to 14. We can use this to work out which value was removed. If we remove the three from our list, then the second and third values are seven and 10 and the sum of these two values is 17 which tells us that we didn’t remove the number three from the list.
If, however, we remove the number four from the list, then our ordered list is three, seven, 10, 14. So the second and third values are still seven and 10 which sum to 17. This tells us that it wasn’t the number four that we removed from the list. If, however, we remove the number seven from the list then the second and third values will be four and 10 and the sum of four and 10 is 14. So the average of four and 10 will be 14 divided by two which is equal to seven.
Removing the number seven from the list, therefore, gives us the correct median of seven for the remaining four numbers. If we removed either the 10 or the 14 from the list, then the sum of the second and third values would not be 14.
So this tells us that the number which was removed from the grid in order to keep the median a seven was the number seven.