Data sets can be compared using averages and measures of spread. A box plot displays information about the range, the median and the quartiles.

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.

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

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.

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