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## Statistics and probability

### Course: Statistics and probability>Unit 3

Lesson 3: Interquartile range (IQR)

# Interquartile range review

## 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, start text, Q, end text, start subscript, 1, end subscript, right parenthesis and the third quartile left parenthesis, start text, Q, end text, start subscript, 3, end subscript, right parenthesis.
start text, I, Q, R, end text, equals, start text, Q, end text, start subscript, 3, end subscript, minus, start text, Q, end text, 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, start text, Q, end text, 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, start text, Q, end text, 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 start text, Q, end text, start subscript, 3, end subscript, minus, start text, Q, end text, 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 start text, Q, end text, 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
start text, Q, end text, 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 start text, Q, end text, 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
start text, Q, end text, 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.
\begin{aligned} \text{IQR} &= \text{Q}_3-\text{Q}_1 \\ \\ &= 5-3 \\ \\ &= 2 \end{aligned}
The IQR is 2 points.

### 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.
Find the interquartile range (IQR) of the data set.
classes

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

## Want to join the conversation?

• When will I even need to use this in the real world?...
• To calculate the spread of the middle 50 percent of the ACT scores at a college that you are considering.
• what if there are two numbers in the middle?
• it's just the first number with a '.5' so its like half. Like if the two numbers was 13,14 its 13.5
Or what ever number in in the middle of them like 13,15 then its 14. :)
• wait what? when will we ever use this irl?
• never, so idk why we need to learn this :')
• how can you find the median
• The median is simply the middle number in the data set (if you have your data set ordered from least to greatest). This is easy if you have an odd number of data.

If your number set has an even amount of data, then there's no central number. You would then take the average (or mean) of the two middle numbers to obtain the median for the data set.

Someone else gave an example of 1,2,2,3,5. Since there are an odd number of data, the median would simply be the (third) middle number of '2'.

Had the data set looked like this (with an even number of data)-
1,2,2,3,5,9

...then you would take the middle two number, and find the average (mean) of them. In this case, 2 & 3, the median would be 2+3, divided by 2, which would be 2.5.
• Hey people. You may not be interested but I just took down a bunch of vapers in my school!
• You are my hero <3
(1 vote)
• this is stupid
• when and how would i use this in the real world
• What if there are a lot of numbers?