Quartiles, range and box plots

Order the data. The median splits the ordered list; the lower and upper quartiles \(Q_1,Q_3\) are the medians of the lower and upper halves. In this method exclude the overall median from both halves when the count is odd.

Range \(=\text{maximum}\hspace{0.5em}-\text{minimum}\); interquartile range \(\mathrm{IQR}=Q_3-Q_1\). The range uses the extremes, while the IQR describes the spread of the middle half and is less affected by a single extreme.

Example

For \(2,4,4,6,8,10,10,12\), the median is \(7\), \(Q_1=4\), \(Q_3=10\). Thus range \(=10\) and IQR \(=6\). The five-number summary is \(2,4,7,10,12\).

The box spans \(Q_1\) to \(Q_3\), its internal line marks the median, and the whiskers reach the minimum and maximum. Use a uniform horizontal scale. It summarises centre and spread but does not show all values, their frequencies or the sample size.

Quartiles divide ordered positions into four parts. Ties can mean more than \(25\%\) of observations are at or below \(Q_1\); do not assume each visible interval contains exactly a quarter of the observations in every small data set.