MathIt BlogHow to make a box plot, read it and find the interquartile range
How do you find the interquartile range?
To find the interquartile range, sort the data, find the median, then find Q1 (the median of the lower half) and Q3 (the median of the upper half) and subtract: IQR = Q3 − Q1. For the test scores 52, 61, 64, 68, 70, 73, 75, 78, 80, 84 and 95, Q1 is 64 and Q3 is 80, so the interquartile range is 16.
Key takeaways
- A box plot is built from five numbers: the minimum, the first quartile Q1, the median, the third quartile Q3 and the maximum.
- The interquartile range is Q3 − Q1, the spread of the middle half of the data, so the scores 52, 61, 64, 68, 70, 73, 75, 78, 80, 84 and 95 have an IQR of 80 − 64 = 16.
- The interquartile range ignores the extreme values, so changing the lowest score from 52 to 31 moves the range from 43 to 64 while the IQR stays at 16.
- Two box plots are compared by their medians first and their box widths second, because the median shows the typical value and the box width shows how consistent the middle half is.
- The 1.5 × IQR rule marks a value as a possible outlier when it lies more than 1.5 × IQR below Q1 or above Q3.
A report card says the class median was 73 and shows a small graph with a box and two lines sticking out of it. That graph is a box plot, also called a box and whisker plot, and the number that sets the width of its box is the interquartile range. You can draw one with a pencil in about five minutes once you know which five numbers to find.
What a box plot shows: the five-number summary
A box plot shows where the middle of a data set sits and how spread out the values are, using five numbers. OpenStax Introductory Statistics 2e lists them: "A box plot is constructed from five values: the minimum value, the first quartile, the median, the third quartile, and the maximum value." Together they form the five-number summary.
The quartiles cut the sorted data into four parts of about equal size: Q1 is the middle of the lower half, the median M is the middle of the whole set, and Q3 is the middle of the upper half. If the median needs a refresher, the guide to mean, median and mode shows how to find it for odd and even counts.
Box plots are a grade 6 topic in the US. Common Core standard 6.SP.B.4 says students should "Display numerical data in plots on a number line, including dot plots, histograms, and box plots", and 6.SP.B.5c asks for measures of "variability (interquartile range and/or mean absolute deviation)".
Why the interquartile range beats the range when there is an outlier
The interquartile range measures the spread of the middle half of the data, so one extreme value cannot stretch it. OpenStax section 2.3 describes the IQR as "a number that indicates the spread of the middle half or the middle 50% of the data", and the formula is IQR = Q3 − Q1.
Take 11 test scores: 52, 61, 64, 68, 70, 73, 75, 78, 80, 84 and 95. The range is 95 − 52 = 43 and the IQR is 80 − 64 = 16. If one student had a bad day and scored 31 instead of 52, the range jumps to 95 − 31 = 64, while Q1, the median and Q3 stay at 64, 73 and 80, so the IQR stays at 16.
How to make a box plot step by step
A box plot takes six steps, shown here with the same 11 test scores.
- Sort the data. Write the values from smallest to largest: 52, 61, 64, 68, 70, 73, 75, 78, 80, 84, 95. There are 11.
- Find the median. With 11 values, the median is the 6th value, because 5 values sit on each side of it. The median is 73.
- Find Q1 and Q3. Q1 is the median of the lower half, 52, 61, 64, 68, 70, which is 64. Q3 is the median of the upper half, 75, 78, 80, 84, 95, which is 80. The interquartile range is 80 − 64 = 16.
- Note the minimum and maximum. The minimum is 52 and the maximum is 95, so the five-number summary is 52, 64, 73, 80, 95.
- Draw a scale. Draw a number line that covers every value, here 50 to 100, with equal steps of 5 or 10. Reading it is the same skill as reading a ruler.
- Draw the box and the whiskers. Draw a box from 64 to 80 with a vertical line at 73. Then draw one whisker from 64 left to 52 and one from 80 right to 95.
The second pass checks the quartiles by counting. Two scores sit below Q1 (52 and 61), two between Q1 and the median (68 and 70), two between the median and Q3 (75 and 78) and two above Q3 (84 and 95). Equal counts in the four gaps mean the quartiles are right.
Play free in your browserFind the AverageGrade 4–6The part people get wrong: the median in or out of the halves
With an odd count, people disagree on whether the median belongs to the lower and upper halves. Common Core leaves it out: its glossary defines the first quartile as "the median of the data values less than M", and a footnote admits that "many different methods for computing quartiles are in use." This post uses the school method, which excludes the median.
If you include the median, the lower half becomes 52, 61, 64, 68, 70, 73, so Q1 = (64 + 68) ÷ 2 = 66, and the upper half gives Q3 = (78 + 80) ÷ 2 = 79. The IQR shrinks to 13. A spreadsheet's QUARTILE.INC function and many calculators return those values, so check which method your teacher or textbook uses before you call an answer wrong.
Range vs interquartile range vs mean absolute deviation vs standard deviation
The range, the interquartile range, the mean absolute deviation and the standard deviation all measure spread, but each looks at a different part of the data. The range uses only the two extremes, the IQR uses the middle half, and the other two use every value through its distance from the mean.
| Measure | What it measures | Hurt by one outlier? | Where it is taught (Common Core) |
|---|---|---|---|
| Range | Maximum − minimum | Yes, strongly | Alongside mean, median and mode |
| Interquartile range | Q3 − Q1, the middle half | Barely | Grade 6 (6.SP.B.5c) |
| Mean absolute deviation | Average distance from the mean | Somewhat | Grade 6 (6.SP.B.5c) |
| Standard deviation | A typical distance from the mean, from squared distances | Yes | High school (S-ID.A.2) |
Which measure of spread to pick
The interquartile range is the measure to pair with the median, and the mean absolute deviation is the one to pair with the mean. Use the median and IQR when the data has an outlier, such as a test where one absent student scored 0, and the mean and MAD when the values have no extremes.
How to read and compare two box plots
A box plot is read from the middle out: the median line first, then the box, then the whiskers. Each of the four parts, two whiskers and two halves of the box, holds about a quarter of the values.
Box plots earn their keep when you put two on one scale. The NIST/SEMATECH e-Handbook of Statistical Methods calls them "an excellent tool for conveying location and variation information in data sets," and says the box "represents the middle 50% of the data--the 'body' of the data."
Class A has the five-number summary 52, 64, 73, 80, 95, and class B has 58, 70, 76, 82, 90. Class B's median is 3 points higher, its box is 12 points wide against 16 for class A, and its range is 32 against 43. Class B scored a little higher and much more evenly.
Free printable worksheetMixed Operations — 6th GradersWorksheet · PDFThe rule: compare medians first, then box widths
Two box plots are compared in a fixed order: the medians, then the box widths, then the whiskers. The medians tell you which group is higher in a typical case, and the box widths tell you which group is more consistent.
The size of the gap matters as much as its direction. Common Core 7.SP.B.3 asks grade 7 students to judge "the degree of visual overlap of two numerical data distributions" by "expressing it as a multiple of a measure of variability." A 3-point gap in medians is a quarter of class B's IQR of 12, and the two boxes overlap from 70 to 80, so the difference between the classes is small.
How to check progress with box plots
Five short checks, done without notes, show whether a student understands box plots.
- Five numbers: for 3, 5, 6, 8, 10, 11, 15, can you give 3, 5, 8, 11, 15 and an IQR of 11 − 5 = 6?
- Even count: for 2, 4, 5, 7, 9, 10, 12, 15, can you find the median (7 + 9) ÷ 2 = 8, Q1 = (4 + 5) ÷ 2 = 4.5 and Q3 = (10 + 12) ÷ 2 = 11?
- Reading: looking at a box plot, can you say what fraction of the values lies inside the box? The answer is about half.
- Comparing: given two box plots, can you name the group with the higher median and the more consistent group in one sentence each?
- Explaining: can you say why the IQR barely moves when the lowest value drops a long way?
A five-minute test with real data
A five-minute test uses numbers from your own week, such as 11 quiz scores or 9 daily step counts in thousands. Sort them, find the five-number summary, draw the plot and check the quarters by counting.
Score the method, then the arithmetic. If the quartiles are right but a subtraction is off, the gap is in number facts, and the math game for 6th graders or the math game for adults rebuilds that speed. Parts of the data also read as percentages: the box holds about 50% and each whisker about 25%, which the guide on how to calculate percentages explains.
Common box plot mistakes and edge cases
Four mistakes cause most wrong box plots, and each one has a quick fix.
- Mistake: skipping the sort. The middle of an unsorted list is a random value. Sort first.
- Mistake: reading a long whisker as more data. Every whisker holds about a quarter of the values; a long one means they are spread out.
- Mistake: an uneven scale. Steps of 50, 60, 80, 100 stretch some parts of the plot. Use equal steps, and start near the minimum instead of at 0.
- Mistake: mixing quartile methods. Including the median in one half and excluding it from the other gives a wrong IQR. Pick one method and use it for both halves.
Edge case: outliers and the 1.5 × IQR fence
An outlier is a value far from the rest, and the 1.5 × IQR rule gives a test for it. OpenStax states that "a value is suspected to be a potential outlier if it is less than (1.5)(IQR) below the first quartile or more than (1.5)(IQR) above the third quartile," and adds that "potential outliers always require further investigation."
Use the scores with 31 in place of 52: Q1 is 64, Q3 is 80 and the IQR is 16. Then 1.5 × 16 = 24, the lower fence is 64 − 24 = 40 and the upper fence is 80 + 24 = 104. The score 31 is below 40, so it is a possible outlier, and nothing is above 104.
On a box plot with fences, the outlier is drawn as a dot at 31. NIST says the whisker then goes "to the smallest point that is greater than" the lower fence, so the left whisker stops at 61.
Your box plot checklist and next steps
- Sort and count: list the values from smallest to largest and write the count.
- Median: find the middle value, or the mean of the two middle values for an even count.
- Quartiles: take the median of each half, with the median itself left out for an odd count.
- IQR and fences: subtract Q3 − Q1, then check for values beyond 1.5 × IQR.
- Draw: use an evenly spaced scale, a box from Q1 to Q3, a line at the median and whiskers to the extremes.
- Check: count the values in each quarter; the four counts should match or nearly match.
The free find the average game in the browser practices the mean, the other measure of center you will meet next to every box plot, with no sign-up.
Practise it in the MathIt app
About 90 math games with adaptive difficulty, daily practice and 2–5 player battles — free on iOS and Android.
MathIt is a free app for iOS and Android with about 90 game types and adaptive difficulty, so a few minutes of daily practice keeps the sorting, subtraction and halving behind quartiles quick.
Frequently asked questions
What is the interquartile range?
The interquartile range is the distance between the first quartile and the third quartile of a data set, written IQR = Q3 − Q1. It measures how spread out the middle half of the values is. Because it ignores the lowest and highest quarters, one very high or very low value barely changes it.
What is a box plot?
A box plot is a graph that shows a data set with five numbers on a number line: the minimum, Q1, the median, Q3 and the maximum. A box runs from Q1 to Q3 with a line at the median, and two whiskers reach out to the smallest and largest values. Common Core standard 6.SP.B.4 introduces box plots in grade 6.
What is a box and whisker plot?
A box and whisker plot is the same graph as a box plot; the two names mean one thing. The box shows the middle half of the data, from Q1 to Q3, and the whiskers are the lines that reach out to the minimum and the maximum.
How do you read a box plot?
A box plot is read from the middle out. The line inside the box is the median, the box holds the middle half of the values, and each whisker holds about a quarter of them. A long whisker or a long half of the box means the values in that quarter are spread out, not that there are more of them.
How do you find outliers with the interquartile range?
Outliers are found with the 1.5 × IQR rule. Multiply the interquartile range by 1.5, subtract the result from Q1 and add it to Q3; any value outside those two fences is a possible outlier. With Q1 = 64, Q3 = 80 and IQR = 16, the fences are 40 and 104.
