Statistics How To

Interquartile range in statistics: What it is and How to find it

Main Index> Basic Statistics>How to find an interquartile range in statistics

 

A range is a measure of where the first and last data items are in a set; The interquartile range is a measure of where the “middle fifty” is in a data set. It’s where the bulk of the values lie, and that’s why it’s preferred over many other measures of spread (i.e. the average or median) when reporting things like school performance on SAT scores.

The interquartile range formula is the first quartile subtracted from the third quartile (What is a Quartile?): Q3-Q1.

Contents (click to skip to the page section):

  1. Solve the formula by hand.
  2. What if I have even numbers?
  3. Find an interquartile range for an odd set of numbers: Second Method
  4. Find the Interquartile Range with Technology (Excel, TI-83 etc.)
  5. What is an Interquartile range?
  6. What is an interquartile range used for?
  7. History of the Interquartile Range.

Solve the formula by hand.

Watch the video or read the steps below:


Steps:

  • Step 1: Put the numbers in order.
    1,2,5,6,7,9,12,15,18,19,27
  • Step 2: Find the median (How to find a median).
    1,2,5,6,7,9,12,15,18,19,27
  • Step 3: Place parentheses around the numbers above and below the median.
    Not necessary statistically–but it makes Q1 and Q3 easier to spot.
    (1,2,5,6,7),9,(12,15,18,19,27)
  • Step 4: Find Q1 and Q3
    Q1 can be thought of as a median in the lower half of the data. Q3 can be thought of as a median for the upper half of data.
    (1,2,5,6,7),  9, ( 12,15,18,19,27). Q1=5 and Q3=18.
  • Step 5: Subtract Q1 from Q3 to find the interquartile range.
    18-5=13.

Like the explanation? Check out the Practically Cheating Statistics Handbook, which has hundreds more step-by-step explanations, just like this one!

What if I Have an Even Set of Numbers?

Sample question: Find the IQR for the following data set: 3, 5, 7, 8, 9, 11, 15, 16, 20, 21.

  • Step 1: Put the numbers in order.
    3, 5, 7, 8, 9, 11, 15, 16, 20, 21.
  • Step 2: Make a mark in the center of the data:
    3, 5, 7, 8, 9, | 11, 15, 16, 20, 21.
  • Step 3: Place parentheses around the numbers above and below the mark you made in Step 2 –it makes Q1 and Q3 easier to spot.
    (3, 5, 7, 8, 9), | (11, 15, 16, 20, 21).
  • Step 4: Find Q1 and Q3
    Q1 is the median (the middle) of the lower half of the data, and Q3 is the median (the middle) of the upper half of the data.
    (3, 5, 7, 8, 9), | (11, 15, 16, 20, 21). Q1 = 7 and Q3 = 16.
  • Step 5: Subtract Q1 from Q3.
    16 – 7 = 9.
    This is your IQR.

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Find an interquartile range for an odd set of numbers: Alternate Method

As you may already know, nothing is “set in stone” in statistics: when some statisticians find an interquartile range for a set of odd numbers, they include the median in both both quartiles. For example, in the following set of numbers: 1,2,5,6,7,9,12,15,18,19,27 some statisticians would break it into two halves, including the median (9) in both halves:
(1,2,5,6,7,9),(9,12,15,18,19,27)
This leads to two halves with an even set of numbers, so you can follow the steps above to find the IQR.
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Technology Options

If you’d rather use technology to find the interquartile range, you have plenty of options, including (click for the link):

What is an Interquartile Range?


Imagine all the data in a set as points on a number line. For example, if you have 3, 7 and 28 in your set of data, imagine them as points on a number line that is centered on 0 but stretches both infinitely below zero and infinitely above zero. Once plotted on that number line, the smallest data point and the biggest data point in the set of data create the boundaries of an interval of space on the number line that contains all data points in the set. The interquartile range (IQR) is the length of the middle 50% of that interval of space.

what is an interquartile range

The interquartile range is the middle 50% of a data set. Box and whiskers image by Jhguch at en.wikipedia




If you want to know that the IQR is in formal terms, the IQR is calculated as: The difference between the third or upper quartile and the first or lower quartile. Quartile is a term used to describe how to divide the set of data into four equal portions (think quarter).

IQR Example

If you have a set containing the data points 1, 3, 5, 7, 8, 10, 11 and 13, the first quartile is 4, the second quartile is 7.5 and the third quartile is 10.5. Draw these points on a number line and you’ll see that those three numbers divide the number line in quarters from 1 to 13. As such, the IQR of that data set is 6.5, calculated as 10.5 minus 4. The first and third quartiles are also sometimes called the 25th and 75th percentiles because those are the equivalent figures when the data set is divided into percents rather than quarters.
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What is an Interquartile Range Used For?


The IQR is used to measure how spread out the data points in a set are from the mean of the data set. The higher the IQR, the more spread out the data points; in contrast, the smaller the IQR, the more bunched up the data points are around the mean. The IQR range is one of many measurements used to measure how spread out the data points in a data set are. It is best used with other measurements such as the median and total range to build a complete picture of a data set’s tendency to cluster around its mean.

Where Does the term Interquartile Range Come From?

Who invented the term “Interquartile Range?” In order to find that out, we have to go back to the 19th century.
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History

British physician Sir Donald MacAlister used the terms lower quartile and higher quartile in the 1879 publication, the Law of the Geometric Mean. Proc. R. Soc. XXIX, p. 374: ” “As these two measures, with the mean, divide the curve of facility into four equal parts, I propose to call them the ‘higher quartile’ and the ‘lower quartile’ respectively.”
origin of the word quartile Although he was a physician by trade, he was gifted with mathematics and achieved the highest score in the final mathematics exams at Cambridge University in 1877. He was also reported to have spoken nineteen languages including English, Czech and Swedish.

Macalister’s paper, the Law of the Geometric Mean was actually in response to a question but forward by Francis Galton. However, it wasn’t until 1882 that Galton (“Report of the Anthropometric Committee”) used the upper quartile and lower quartile values and the term “interquartile range” which was defined as twice the probable error. Galton wasn’t just a statistician — he was also an anthropologist, geographer, proto-genetecist and psychometrician who produced more than 340 books. He also coined the statistical terms “correlation” and “regression toward the mean.”

48 thoughts on “Interquartile range in statistics: What it is and How to find it

  1. Cathy Flanagan

    This article really helped me to understand interquartile range. I really like the hint of using parentheses to find Q1 and Q3! I will definently try it on my next problem.

  2. Philip Smith

    You make it all seem so easy. I really like the way you break it all down into easy-to-follow steps. More importantly, after reading the article and studying the steps, I feel like it’s something I will be able to remember.

  3. Evelyn Snyder

    Subtract Q1 from Q3 to find the interquartile range.
    The online textbook in MathZone helps, but your explanation on what the IQR represents and how it is found, makes it easier to grasp and solve the problem. The IQR is the middle point between Q1 and Q3. Now when I am asked to find the IQR for a report, I will have some knowledge as to “what?” I am being asked. In the past we have charted our ups and downs with a histrogram; but with this class, I will be able to use more pertinent charts to show a more realistic annual report, thanks!

  4. Lisa Barcomb

    Yeah you do make it seem easy but I worked these problems out and I did get them right and then when I went to take my test well that was a different story. I don’t know what happened with my thought process, but it went out the window. I also have a problem with those histograms.

  5. Vanessa DuBarry

    I loved how everything is in order and done step by step! this was really helpful thanks for these blogs!

  6. Shannon M

    Many thanks to the author of this page has helped me greatly, unfortunately I found it hard to believe that there is no decent standard deviation pages on this site yet the rest is so thorough….

    Regards

    Shannon

  7. Ela

    If you say in step 4 that: Q1=5 and Q3=18.
    then in Step 5. (Subtract Q1 from Q3 to find the interquartile range) You should get 5-18 = -13

  8. desperate student

    frst of all
    thankyou
    ive got exams tomaro n irealy had trouble with IQR
    n now i think ive becum n expert wen it comes to iqr..lol thanku again

  9. Molly

    Makes sense sort of, this way doesnt account for even numbers. but lets say for this set of data…. 1,1,2,2,3,3,4,4,5,5 the median is 3, Q1-2 and Q3- 4, so the IQR by your definition would be 2, but according to my statistics textbook the IQR is 2.5…. ????? So what do you do?

  10. Stephanie

    the IQR is subtracting Q1 from Q3. If you have a list of whole numbers, I don’t know why your text would come up with a fraction. Perhaps you could post your question in the forum (along with exactly what your book says).

  11. Dylan

    Very intuitive, but what if you have even amount of numbers.
    For example: 12, 15, 15, 22, 45, 45, 45, 60, 80, 200.

    The median is 45 because it’s between 45 and 45. Do I now add 12, 15, 15, 22 and 45 and divide by 5 to find Q1?

  12. Stephanie

    Dylan,

    Tricky question! the simple answer is to follow whatever your text/instructor says. In a basic stats class (at least, in mine) — you’ll never be “tricked” by being given an even number of figures to throw you off. In most textbooks you will be guided by an odd number, or a pair of numbers in the center that are similar (as in your question). If they are different, the median is *sometimes* the number you get by dividing the left and right numbers from the center. However, this doesn’t always work. for example, if you are talking about people — a median of 45.5 people doesn’t always make sense (because you can’t get half of a person).

    Q1 could be that middle figure — 15 — or it could be what you said (divide by 5). Again, it depends on the text — and whether the answer makes sense or not (depending upon the figures in the question).

    Think of it like a kindergarten question: if Bob has 10 apples and gives Maria 1 apple, how many apples does Bob have? The Kindergarten answer is — of course — 9. However, the answer depends upon whether Bob takes a bite, drops an apple or makes an apple pie.

    Stephanie

  13. Pramit

    I’m not sure if I’m correct, but here is my calculation:

    The given series is: 1,1,2,2,3,3,4,4,5,5
    Q1: 0.25(count+1)= 0.25*11 =2.75
    Q3: 0.75(count+1)= 0.75*11 =8.25

    now following the position, we get Q1p= 1.75 & Q3p= 4.25
    So IQR: Q3p- Q1p = 4.25-2.75 = 2.5

    Let me know if I did it wrong.

  14. Tony

    This is great to learn, how do I find the interquartile range?
    If I had the same numbers 0,1,3,5,0,2,8,2,1,3 and I had a list of names but the numbers would say something like the number of arrest. How do I figure this one out.

  15. Godwin yohanna

    Thanks i learn how to find interquartile range. what is the secret behind the use of percentile

  16. Jules Lee

    Thank you! I found your blog as the best alternative to my book. My book suck at explaining!

  17. charnay

    Thanks so much. This was a clear,understandable, answer on how to solve inter-quartile range.

  18. Captain Falcon

    Thanks that actually really helped, and you break it down so easy. love your work and keep it up

  19. Andale

    Brian…large data, you can use MS Excel. As for classes….my first thought would be that’s impossible (unless you have the raw data).

  20. Andale

    Kris,

    Time constraints prevent me from answering stats questions in the comments…but post on our forums and our mod will be happy to help :)

    Stephanie