Quartiles - Math Is Fun

Quartiles

Quartiles are the values that divide a list of numbers into quarters. There are different methods, here we use the simple method of splitting the ordered list into four equal parts:

• Put the list of numbers in order
• Then split the list into four equal parts
• The quartiles are at the "split points"

Like this:

Example: 5, 7, 4, 4, 6, 2, 8

Put them in order: 2, 4, 4, 5, 6, 7, 8

Split the list into quarters:

And the result is:

• Quartile 1 (Q1) = 4
• Quartile 2 (Q2), which is also the Median, = 5
• Quartile 3 (Q3) = 7

Sometimes a "split" is between two numbers ... the Quartile is the average of the two numbers.

Example: 1, 3, 3, 4, 5, 6, 6, 7, 8, 8

The numbers are already in order

split the list into quarters:

In this case Quartile 2 is half way between 5 and 6:

Q2 = (5+6)/2 = 5.5

And the result is:

• Quartile 1 (Q1) = 3
• Quartile 2 (Q2) = 5.5
• Quartile 3 (Q3) = 7

Another Method to Find Quartiles

We can also split at the median, then split the lower and upper halves of the list:

1. Put the numbers in order
2. Find the median (this is Quartile 2, Q2)
3. Look at the numbers below the median. Their median is Quartile 1 (Q1)
4. Look at the numbers above the median. Their median is Quartile 3 (Q3)

This gives the same results as our "split into four equal parts" idea.

Interquartile Range

The "Interquartile Range" is from Q1 to Q3:

To calculate it: subtract Quartile 1 from Quartile 3, like this:

Example:

The Interquartile Range is:

Q3 − Q1 = 7 − 4 = 3

Box and Whisker Plot

We can show all the important values in a "Box and Whisker Plot", like this:

A final example covering everything:

Example: Box and Whisker Plot and Interquartile Range for

4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11

Put them in order:

3, 4, 4, 4, 7, 10, 11, 12, 14, 16, 17, 18

split it into quarters:

3, 4, 4 | 4, 7, 10 | 11, 12, 14 | 16, 17, 18

In this case all the quartiles are between numbers:

• Quartile 1 (Q1) = (4+4)/2 = 4
• Quartile 2 (Q2) = (10+11)/2 = 10.5
• Quartile 3 (Q3) = (14+16)/2 = 15

Also:

• The Lowest Value is 3,
• The Highest Value is 18

So now we have enough data for the Box and Whisker Plot:

And the Interquartile Range is:

Q3 − Q1 = 15 − 4 = 11

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