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!

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

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

Split the list in half:

Diagram showing the calculation of Q1, Q2, and Q3 for the data set 2, 4, 4, 5, 6, 7, 8 using medians of halves.

Gets us the median of 5

Now split into halves again:

Diagram showing the calculation of Q1, Q2, and Q3 for the data set 2, 4, 4, 5, 6, 7, 8 using medians of halves.

  • 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:

Diagram showing the calculation of Q1, Q2, and Q3 for the data set 1, 3, 3, 4, 5, 6, 6, 7, 8, 8 using medians of halves.

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.

Diagram showing the calculation of Q1, Q2, and Q3 for the data set 1, 3, 3, 4, 5, 6, 6, 7, 8, 8 using medians of halves.

And the result is:

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

Interquartile Range

The "Interquartile Range" is from Q1 to Q3:

Number line illustrating the interquartile range as the distance between Q1 and Q3, with Q2, minimum, and maximum also marked.

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

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

Example:

Diagram showing the calculation of Q1, Q2, and Q3 for the data set 2, 4, 4, 5, 6, 7, 8 using medians of halves.

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:

Generic box and whisker plot showing the minimum, Q1, median (Q2), 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 and whisker plot for the data set 3, 4, 4, 4, 7, 10, 11, 12, 14, 16, 17, 18, showing 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
Median Data Index