Question 53·Medium·One-Variable Data Distributions; Measures of Center and Spread
The table shows the exam scores (in points) for a class of 12 students.
| Score | Number of students |
|---|---|
| 60 | 2 |
| 70 | 4 |
| 80 | 3 |
| 90 | 3 |
What is the interquartile range (IQR) of the exam scores?
(Express the answer as an integer)
For IQR questions from a frequency table, first expand the table into an ordered list that repeats each value according to its frequency so you can clearly see all data points. Count how many data values you have, locate the middle to split the data into a lower and an upper half, then find Q1 and Q3 as the medians of those halves. Finally, subtract from to get the IQR, being careful not to confuse quartiles with just the minimum, maximum, or distinct score levels.
Hints
Expand the frequency table
Try writing out all 12 scores explicitly (for example, 60, 60, 70, 70, …) before doing any quartile calculations.
Recall what IQR measures
The interquartile range is . Think about how to find the first quartile (Q1) and third quartile (Q3) once the data are in order.
Finding quartiles in an even-sized data set
With 12 data points, the median lies between the 6th and 7th values. Use the first 6 values to find Q1 and the last 6 values to find Q3 by taking the medians of those halves.
Desmos Guide
Enter the data as a list
In Desmos, type a list with all the scores, for example:
L = [60,60,70,70,70,70,80,80,80,90,90,90]
Use Desmos statistics functions to find the IQR
On a new line, type IQR(L) (or quartile(L,3) - quartile(L,1)). The value that Desmos outputs is the interquartile range of the exam scores.
Step-by-step Explanation
Write out all the scores in order
Use the table to list every student's score, repeating each score according to its frequency, then sort them.
- Two students scored 60
- Four students scored 70
- Three students scored 80
- Three students scored 90
So the ordered list of 12 scores is:
60, 60, 70, 70, 70, 70, 80, 80, 80, 90, 90, 90
Split the data set into lower and upper halves
There are 12 scores, so the middle lies between the 6th and 7th values.
- The 6th value is 70
- The 7th value is 80
For quartiles, we split the data into two halves:
- Lower half (first 6 scores): 60, 60, 70, 70, 70, 70
- Upper half (last 6 scores): 80, 80, 80, 90, 90, 90
Find the first quartile (Q1)
Q1 is the median (middle value) of the lower half.
Lower half: 60, 60, 70, 70, 70, 70
- There are 6 numbers, so Q1 is the average of the 3rd and 4th values.
- The 3rd value is 70
- The 4th value is 70
So .
Find Q3 and compute the interquartile range
Q3 is the median of the upper half.
Upper half: 80, 80, 80, 90, 90, 90
- There are 6 numbers, so Q3 is the average of the 3rd and 4th values.
- The 3rd value is 80
- The 4th value is 90
So .
The interquartile range is :
So the interquartile range of the exam scores is .