If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Lesson 6: 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?

• 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. :)
• 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.
• How is this useful? Like if you work in a factory?
(1 vote)
• That depends, are you just working on the floor and following orders from the boss and supervisors, or are you the boss or accountant, or several other positions? What kind of factory are you talking about? Do you want to learn the logic and patterns that might help you in any job? Too many questions and possibilities for someone so young.
• do you use zeros
• Yes you do
(1 vote)
• for the interqaurtiles do i have to divide the 2 meians that is on the side"s of the median
• No. For the IQR once you find the interquartiles you subtract the greater value from the lesser value (like 12-3) not divide. Hope that helps!
(1 vote)
• this is so dum
(1 vote)
• I am sorry you feel that way :(
• What do you do if you have an even number of values? For example:
1, 2, 3, 4, 5, 6

When you split the values (after finding the median, 3.5), do you calculate the Q1 to Q3 as:
(123)(456) or (12)34(56), the brackets representing the ranges of values that will be "quartiled".
(1 vote)
• I would say(123)(456) is right.
• Is it possible for the IQR to be less than the given data points? i.g. the data points being 1st: 8.3, and 3rd: 9.8, IQR = 1.5. 1.5 < 8.3, 9.8
(1 vote)
• What if there are a lot of numbers?
(1 vote)
• The number of numbers does not matter as long as you have a minimum of 4, so you first have to put the numbers in numeric order.
Then, you need to know 4 patterns which start at 4 and repeat every 4 numbers.
4 numbers mean there are one in each quartile
5 numbers mean the middle number is between 2nd and 3rd quartile
6 numbers mean you will have numbers between 1st and 2nd quartile and between 3rd and 4th quartile
7 numbers mean you will have numbers between each of the 1st and 2nd, 2nd and 3rd, and 3rd and 4th quartiles
Each of these numbers do not count as part of any of the quartiles, but might be used for the interquartile range
8 numbers repeats 4 (but 2 numbers in each quartile). To find which of these to use, divide by 4 which gives the number of elements in each quartile. The remainder will determine if there are any intermediate elements between quartiles (0 remainder is like 4, 1 remainder like 5, 2 remainders like 6, and 3 remainders like 7. The IQR could use these numbers if necessary.
Does this help some?
(1 vote)
• How do you suppose to write your answer in the box for checking.
(1 vote)
• If you talking about the article you should just have to type in the answer in the small box (Just click it) and press check
(1 vote)