Question Video: Finding a Variable Value in a Data Set Using the Median Mathematics • 6th Grade

Video Transcript

Given that the median of the values 𝑎 minus four, 𝑎 plus six, 𝑎 minus three, 𝑎 plus seven, and 𝑎 plus nine is 30, determine the value of 𝑎.

In this problem, we have five different values: 𝑎 minus four, 𝑎 plus six, 𝑎 minus three, 𝑎 plus seven, and 𝑎 plus nine. Now because 𝑎 will not change, then the values will change only because of the value that is added or subtracted from it. We are given information relating to the median, and we can recall that the median of a data set is the middle value when the data is ordered, either in ascending or descending order.

So, let’s put the values we were given in ascending order. We’ll need to consider the values that are added or subtracted with values that have a larger number subtracted from it being the smallest overall value, which means that the smallest value must be 𝑎 minus four. The next smallest value would be 𝑎 minus three, then 𝑎 plus six, and finally 𝑎 plus seven and 𝑎 plus nine. These values would still be in ascending order even if the value of 𝑎 is a negative number.

So now we need to find the median, the middle value. There are five different values, so when we have an odd number of values, as we have here, the middle value is the actual middle number. If there were an even number of values, we would have to find the midpoint of the two values in the middle. So, the middle value is 𝑎 plus six.

Now, we were given that the median is 30. So, putting both these pieces of information together, we know that 𝑎 plus six must equal 30. And subtracting six from both sides of the equation, we have 𝑎 equals 24. And so, we can give this as our answer.

As a check, we can substitute the value of 𝑎 equals 24 into each of the original values, which means the first value is 20. And the remaining values are 30, 21, 31, and 33. In order, we have 20, 21, 30, 31, and 33. The median of this ordered set is 30, which confirms our answer that 𝑎 equals 24.