Quartiles

Quartiles are the values that divide a list of numbers into quarters.

To do this we halve the list at the median , then halve each of those halves, to end up with quarters!

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

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

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

Split the list in half:

Sorted numbers 2, 4, 4, 5, 6, 7, 8 with the median 5 highlighted in the center

Gets us the median of 5

Now split into halves again:

Sorted numbers with Q1 marked at 4, Q2 at 5, and Q3 at 7

Quartile 1 (Q1) = 4
Quartile 2 (Q2), the median, = 5
Quartile 3 (Q3) = 7

Sometimes a "split" is between two numbers ... so we average the two numbers.

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

The numbers are already in order

split the list in the middle:

Sorted even data set split in the middle with arrows pointing to 5 and 6

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

Q2 = 5+62 = 5.5

Now split the data again, but include numbers 5 and 6 in the lower and upper halves when finding Q1 and Q3.

Sorted even data set showing Q1 at 3, Q2 at 5.5, and Q3 at 7

And the result is:

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

Note: Quartiles can be calculated in different ways (including or excluding the median). If your app gives a slightly different answer, don't worry.

Interquartile Range

The Interquartile Range (IQR) is from Q1 to Q3:

It shows how spread out the middle half of the data is.

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

Example:

Data set showing Q1 as 4 and Q3 as 7

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:

Box and whisker plot labeling the minimum, Q1, median, Q3, and maximum values

Putting It All Together

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

The median is between 10 and 11

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

Q2 = 10 + 112 = 10.5

Now quartiles:

3, 4, 4 | 4, 7, 10 (10.5) 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:

Box plot for data set with minimum 3, Q1 4, median 10.5, Q3 15, and maximum 18

And the Interquartile Range is:

Q3 − Q1 = 15 − 4 = 11

697, 1470, 3066, 698, 1471, 3067, 3800, 3801, 3802, 3803