Median and quartiles

Median

We know that the median is the middle number in a set of ordered data. It divides the set of ordered data into two halves.

We also know that for a set of \(n\) ordered numbers, the median is the \(\frac{(n+1)^{th}}{2}\) number.

Example

If there are \(13\) numbers,the median is \(\frac{13+1}{2}=\,7^{th}\,number\)

Quartiles

Similarly, the lower quartile divides the bottom half of the ordered data into two halves

The upper quartile divides the upper half of the ordered data into two halves.

The lower quartile is the \(\frac{(n+1)^{th}}{4}\) value and the upper quartile is the \(\frac{3(n+1)^{th}}{4}\) value.

Diagram showing the lower quartile, median and upper quartiles

Question

Find the median, lower quartile and upper quartile for the following data:

\[11,4,6,8,3,10,8,10,4,12,31\]

Ordering the data, we get \(3,4,4,6,8,8,10,10,11,12,31\)

There are \(11\) numbers.

The median is the \(\frac{11+1}{2}=\,6^{th}\,value.\)

The lower quartile (\(Q_1\)) is the \(\frac{11+1}{4}=\,3^{rd}\,value\)

The upper quartile (\(Q_3\)) is the \(\frac{3(11+1)}{4}=\,9^{th}\,value\)

3,4,\(4\),6,8,\(8\),10,10,\(11\),12,31

Therefore the median is \(8\), the lower quartile is \(4\) and the upper quartile is \(11\).

The quartiles are often denoted by \(Q_1\) and \(Q3\) (the median is technically \(Q_2\)).

So we could write for the above example:

Median\( = 8,\; Q_1 = 4,\; Q_3 = 11\)