Start by considering a data set written in increasing order. Are you able to modify some of the data values so that the third quartile is equal to the maximum value?
No, see solution.
Practice makes perfect
We are asked if the third quartile of a data set is always less than the maximum value. To answer this question, let's first consider some example data set. For simplicity, we will start with the data set written in increasing order.
The third quartile Q_3 is the median of the upper half of the data. We will also highlight the maximum data value.
In this example, the third quartile is indeed less than the maximum value. We can now modify the last two data values to be equal to Q_3, while keeping the increasing order of the data set.
Since the order is preserved, the value of Q_3 is exactly the same as before. However, the maximum value of the modified data set is now equal to 15. Therefore, the third quartile is equal to the maximum value, which means that it is not always less than the maximum value.
Q_3and MaxComparison
Data Set: 15 < 21
Modified Data Set: 15 = 15
Note that, if we further decreased the last two terms, the order of the data values would change and 15 would no longer be the third quartile. Therefore, we can summarize that the third quartile will always be less than or equal to the maximum value.
