The lowest test score is 10, and the highest is 100. The 25th percentile is 30, the median is 48, and the 75th percentile is 70; no student scored exactly 30, 48, or 70. If 120 students scored less than 70, then how many scored above 48?
1 Answer
80 students gained more than 48 marks.
Explanation:
A box plot divides a data set into 4 regions, marked by 5 lines. The lowest line marks the lowest score, and the highest line marks the highest score. Each region contains 25% of the data points; the middle line marks the median of the scores.
For example, in this box plot, we have lines at 10, 30, 48, 70, and 100. We are also told that no student scored exactly 30, 48, or 70, so we can infer the following:
It is common to "box" the middle two regions, because it is the interquartile range (IQR)—it contains the middle 50% of the data.
We are told that 120 students scored less than 70 on the test. We also know each region contains 25%, and there are 3 regions below 70, so that means 3 x 25% (or 75%) of the students scored below 70.
In other words: 120 students = 75%.
So if 3 regions contain a total of 120 students, then each region contains
We want to find out how many students scored greater than 48. How many regions is this? From the box plot, we can see this is 2 regions, and so, since each region contains 40 students, that means 2 regions contain 2 x 40 (or 80) students.
Note:
Sometimes a box plot will have dots below (or above) the lowest (or highest) region. These points are called outliers; they are extreme values that seem very far form the rest of the data. If we extended the region to include these extreme values, we'd need the horizontal line to extend further out, which makes it look like there is more data along that extension of the line, when there is really just empty space.
To avoid this, we use a formula to determine if any of the extreme values qualify as outliers, but for the sake of this 25% rule, the outliers count as part of the region they are closest to.
