Interquartile Range (IQR)

Interquartile range is the amount of spread in the middle 50, percent of a dataset.
In other words, it is the distance between the first quartile left parenthesis, Q, start subscript, 1, end subscript, right parenthesis and the third quartile left parenthesis, Q, start subscript, 3, end subscript, right parenthesis.
I, Q, R, equals, Q, start subscript, 3, end subscript, minus, Q, start subscript, 1, end subscript
Here's how to find the IQR:
Step 1: Put the data in order from least to greatest.
Step 2: Find the median. If the number of data points is odd, the median is the middle data point. If the number of data points is even, the median is the average of the middle two data points.
Step 3: Find the first quartile left parenthesis, Q, start subscript, 1, end subscript, right parenthesis. The first quartile is the median of the data points to the left of the median in the ordered list.
Step 4: Find the third quartile left parenthesis, Q, start subscript, 3, end subscript, right parenthesis. The third quartile is the median of the data points to the right of the median in the ordered list.
Step 5: Calculate IQR by subtracting Q, start subscript, 3, end subscript, minus, Q, start subscript, 1, end subscript.

Example

Essays in Ms. Fenchel's class are scored on a 6 point scale.
Find the IQR of these scores:
1, 3, 3, 3, 4, 4, 4, 6, 6
Step 1: The data is already in order.
Step 2: Find the median. There are 9 scores, so the median is the middle score.
1, 3, 3, 3, 4, 4, 4, 6, 6
The median is 4.
Step 3: Find Q, start subscript, 1, end subscript, which is the median of the data to the left of the median.
There is an even number of data points to the left of the median, so we need the average of the middle two data points.
1, 3, 3, 3
Q, start subscript, 1, end subscript, equals, start fraction, 3, plus, 3, divided by, 2, end fraction, equals, 3
The first quartile is 3.
Step 4: Find Q, start subscript, 3, end subscript, which is the median of the data to the right of the median.
There is an even number of data points to the right of the median, so we need the average of the middle two data points.
4, 4, 6, 6
Q, start subscript, 3, end subscript, equals, start fraction, 4, plus, 6, divided by, 2, end fraction, equals, 5
The third quartile is 5.
Step 5: Calculate the IQR.
IQR=Q3Q1=53=2\begin{aligned} \text{IQR} &= \text{Q}_3-\text{Q}_1 \\ \\ &= 5-3 \\ \\ &= 2 \end{aligned}
The IQR is 2 points.
Want to learn more about calculating IQR? Check out this video.

Practice problem

The following data points represent the number of classes that each teacher at Broxin High School teaches.
Sort the data from least to greatest.
  • 5
  • 9
  • 1
  • 3
  • 4
  • 1
  • 5
  • 7
  • 6
  • 4
  • 5
Find the interquartile range (IQR) of the data set.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
classes

Want to practice more problems like these? Check out this exercise on interquartile range (IQR).