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Lesson 6: Interquartile range (IQR)

# Interquartile range review

Calculate interquartile range from small data sets with even or odd numbers of data points.

## 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 with an odd number of data points

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.
The median is $4$.
Note that the median is not part of either the lower or upper half of the data.
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.
${\text{Q}}_{1}=\frac{3+3}{2}=3$
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.
${\text{Q}}_{3}=\frac{4+6}{2}=5$
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.

### Example with an even number of data points

The foot lengths of the members of a dance troupe are in centimeters.
Find the IQR of these foot lengths:
$25.5$, $25.5$, $26.5$, $28.5$, $29$, $30.5$, $31.5$, $31.5$, $32$, $32.5$
Step 1: The data is already in order.
Step 2: Find the median. There are $10$ lengths, so the median is the average of the middle two lengths.
$\text{median}=\frac{29+30.5}{2}=29.75$
The median is $29.75$.
Step 3: Find ${\text{Q}}_{1}$, which is the median of the data to the left of the median.
There is an odd number of lengths to the left of the median, so we need the middle length.
The first quartile is $26.5$.
Step 4: Find ${\text{Q}}_{3}$, which is the median of the data to the right of the median.
There is an odd number of lengths to the right of the median, so we need the middle length.
The third quartile is $31.5$.
Step 5: Calculate the IQR.
$\begin{array}{rl}\text{IQR}& ={\text{Q}}_{3}-{\text{Q}}_{1}\\ \\ & =31.5-26.5\\ \\ & =5\end{array}$
The IQR is $5\phantom{\rule{0.167em}{0ex}}\text{cm}$.

### 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?

• 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.
• guys i forgot to take my fish on a walk today
• um. what does that have to to with math?
• 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.
• the only thing teachers teach is how to be a teacher 💀
• yes i agree
• khan ingtrquartile range is very fun to do. Thanks
• you spelled it wrong
• why is it called "quartile" when there're only 2 parts? I'm not a native so it's a bit confusing to me
• there are 4 quartiles in total, namely Q1,Q2,Q3,Q4. Below Q1 lies 25% of the data, below Q2 lies 50% of the data(Q2 is also the median of the data), below Q3 lies 75% of the data and below Q4 lies 100% of the data. So, IQR = q3 - q1, 75% - 25%, i.e 50%
• For those in the comments who waste time asking when we will apply this knowledge in the real world, read this post.

Several professions and fields may require knowledge of finding the interquartile range (IQR), including:

Statistics and Data Analysis: Statisticians, data analysts, and data scientists often use the interquartile range as a measure of variability or spread in a dataset.
Finance and Economics: Professionals in finance and economics use IQR for analyzing financial data, such as stock prices, to understand the dispersion of returns or volatility.
Healthcare and Medicine: Researchers and epidemiologists might use IQR to analyze health-related data, such as patient outcomes or clinical trial results.
Education: Teachers and educators may teach students about descriptive statistics, including measures of central tendency and measures of dispersion like the interquartile range.
Quality Control and Six Sigma: Professionals involved in quality control and process improvement may use IQR to identify variability in manufacturing processes and to monitor process stability.
Environmental Science: Environmental scientists might use IQR to analyze data related to pollution levels, biodiversity, or climate variables.
Market Research: Market analysts and researchers may use IQR to analyze consumer data and understand variations in preferences or behaviors.
Social Sciences: Researchers in fields like psychology, sociology, and political science may use IQR to analyze survey data or other forms of social research.
Engineering: Engineers might use IQR to analyze data related to structural integrity, material properties, or performance of systems.
Business Analytics: Professionals in various business sectors use IQR to analyze operational data, customer feedback, and market trends.
These are just a few examples, but the application of interquartile range extends to various fields where understanding data variability is crucial for making informed decisions.