Five Number Summary
Five Number Summary
A common way of summarising the overall data set is with the five number summary and the box-and-whisker plot. These two represent exactly the same information, numerically in the case of the five number summary and graphically in the case of the box-and-whisker plot.
The five number summary consists of the minimum value, the maximum value and the three quartiles. Another way of saying this is that the five number summary consists of the following percentiles: \(0^{\text{th}}\), \(25^{\text{th}}\), \(50^{\text{th}}\), \(75^{\text{th}}\), \(100^{\text{th}}\).
The box-and-whisker plot shows these five percentiles as in the figure below. The box shows the interquartile range (the distance between \(Q_1\) and \(Q_3\)). A line inside the box shows the median. The lines extending outside the box (the whiskers) show where the minimum and maximum values lie. This graph can also be drawn horizontally.
This video explains how to draw a box-and-whisker plot for a data set.
Example
Question
Draw a box and whisker diagram for the following data set:
\[\{\text{1.25}; \text{1.5}; \text{2.5}; \text{2.5}; \text{3.1}; \text{3.2}; \text{4.1}; \text{4.25}; \text{4.75}; \text{4.8}; \text{4.95}; \text{5.1}\}\]Determine the minimum and maximum
Since the data set is already sorted, we can read off the minimum as the first value (\(\text{1.25}\)) and the maximum as the last value (\(\text{5.1}\)).
Determine the quartiles
There are \(\text{12}\) values in the data set. Using the percentile formula, we can determine that the median lies between the sixth and seventh values, making it:
\[\cfrac{\text{3.2} + \text{4.1}}{2} = \text{3.65}\]The first quartile lies between the third and fourth values, making it:
\[\cfrac{\text{2.5} + \text{2.5}}{2} = \text{2.5}\]The third quartile lies between the ninth and tenth values, making it:
\[\cfrac{\text{4.75} + \text{4.8}}{2} = \text{4.775}\]This provides the five number summary of the data set and allows us to draw the following box-and-whisker plot.
This lesson is part of:
Statistics and Probability