A group of 36 students timed how long they can hold their breath while swimming underwater. The histogram shows information about the times achieved. Work out an estimate of the interquartile range. Show all your working.
The difference between a histogram and a frequency diagram or bar chart is it on the 𝑦-axis we have the frequency density instead of the frequency. This means that we can calculate the frequency or number of students in this case in each bar by working out its area. The area of any rectangle is calculated by multiplying its length by its width or its base by its height.
This means that the area of the first bar can be calculated by multiplying 25 by 0.2. This is equal to five. Therefore, there are five students in the first bar. The area of the second bar can be calculated by multiplying 10 by one. This is equal to 10. Therefore, there are 10 students in the second bar. The area of the third bar is equal to 10 multiplied by 1.5. This is equal to 15. Therefore, there are 15 students represented by the third bar. Finally, the area of the fourth bar is 15 multiplied by 0.4. This is equal to six. Therefore, there are six students in the fourth bar.
As these four numbers represent the number of students, they must all be integers. They must also have a sum of 36 as there were 36 students altogether. Five plus 10 plus 15 plus six is equal to 36. We could display this information in a grouped frequency table as shown. There were five students who held their breath for between zero and 25 seconds, 10 students between 25 and 35 seconds, 15 between 35 and 45 seconds, and six students between 45 and 60 seconds.
We were asked to estimate the interquartile range. This is equal to the upper quartile minus the lower quartile. The upper quartile is equal to the 75th percentile and the lower quartile is equal to the 25th percentile. This is because 25 percent equals one-quarter and 75 percent is equal to three-quarters. One-quarter of 36 is equal to nine. This means that when the times are listed from smallest to largest, the ninth person will be the lower quartile. Three-quarters of 36 is equal to 27. This means that the upper quartile will be the 27th person.
There were five students who held their breath for less than 25 seconds. This means there were 15 students that held their breath for less than 35 seconds as five plus 10 is equal to 15. The lower quartile or ninth person must therefore be in the group between 25 and 35 seconds. We can go one stage further and say that they would be the fourth person in this group as five plus four is equal to nine.
To calculate the lower quartile, we need to be four tenths of the way into this group of 10 people. As the lower bound of this group was 25, the lower quartile can be calculated by adding four tenths of 10 to 25. Four tenths of 10 is equal to four. Therefore, the lower quartile is 25 plus four. This is equal to 29. We could also have worked this out from the graph. As there are 10 squares between 25 and 35, we need to be four squares along from 25. This gives us a lower quartile value of 29.
The upper quartile was the 27th person. As there were 30 students who took less than 45 seconds, we know that the upper quartile is in the group 35 to 45. Once again, we can calculate the position within this group. The upper quartile will be the 12th person in this group as five plus 10 plus 12 is equal to 27. We need to be twelve fifteenths into this group. We need to calculate twelve fifteenths of 10 as the class width was 10 between 35 and 45 seconds.
The lower limit of this group was 35. Therefore, we need to add twelve fifteenths of 10 to 35. Twelve fifteenths can be simplified to four-fifths. And four fifths of 10 is equal to eight. Adding eight to 35 gives us 43. This means that our upper quartile is 43 seconds. Once again, we could have used the histogram to work this out. As four-fifths of 10 was equal to eight, we need to be eight squares along from 35. This confirms that the upper quartile is 43.
The interquartile range is, therefore, 43 minus 29. This is equal to 14. Therefore, the interquartile range of times is 14 seconds.