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### 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 $50\mathrm{%}$ of a dataset.
In other words, it is the distance between the first quartile $\left({\text{Q}}_{1}\right)$ and the third quartile $\left({\text{Q}}_{3}\right)$.
$\text{IQR}={\text{Q}}_{3}-{\text{Q}}_{1}$
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 $\left({\text{Q}}_{1}\right)$. 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 $\left({\text{Q}}_{3}\right)$. 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 ${\text{Q}}_{3}-{\text{Q}}_{1}$.

### 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 ${\text{Q}}_{1}$, 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.
$1$, $3$, $3$, $3$
${\text{Q}}_{1}=\frac{3+3}{2}=3$
The first quartile is $3$.
Step 4: Find ${\text{Q}}_{3}$, 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.
$4$, $4$, $6$, $6$
${\text{Q}}_{3}=\frac{4+6}{2}=5$
The third quartile is $5$.
Step 5: Calculate the IQR.
$\begin{array}{rl}\text{IQR}& ={\text{Q}}_{3}-{\text{Q}}_{1}\\ \\ & =5-3\\ \\ & =2\end{array}$
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?

• wait what? when will we ever use this irl?
• never, so idk why we need to learn this :')
• 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. :)
• why do i need to know this?
• Interquartile range is useful when analyzing data. For example, let 50, 100, 200, 300, 400 be 5 people's money before they work, and 100, 100, 200, 350, 700 be those 5 people after they work. Now if you want to claim that at least 50% of those people has their money increased, you can use interquartile range to evaluate the correctness of this claim.
• the only thing teachers teach is how to be a teacher 💀
• yes i agree
• Hey people. You may not be interested but I just took down a bunch of vapers in my school!
• You are my hero <3
• 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.
• why is it called "quartile" when there're only 2 parts? I'm not a native so it's a bit confusing to me