Boxplots The boxplot is an informative way of displaying the distribution of a numerical variable.. It uses the five-figure summary: minimum, lower quartile, median, upper quartile and maximum.

Boxplots The median The median is the midpoint of a distribution; that is, 50% of values are lower than or equal to the median and 50% are higher than or equal to the median.

Boxplots Unordered data: The median is located by listing all the data values in numerical order The median... and then finding the point that divides the distribution into two equal parts. For example:

Boxplots Just as the median is the point that divides a distribution in half, quartiles are the points that divide a distribution into quarters. We use the symbols Q 1, Q 2, and Q 3 to represent the quartiles. Of course, Q 2 is the median.

Boxplots Q 1 Q 3 a box is used to represent the middle 50% of scores whisker minimummaximum whisker m The median is shown by a vertical line drawn within the box. Lines (called whiskers) are extended out from the lower and upper ends of the box to the smallest and largest data values of data set respectively.

Boxplots For example, these are the marks (out of 20) obtained by a group of 15 students on an assignment: The median is 10 The maximum is 18The minimum is 2 Q 3 is 15Q 1 is 5

Boxplots...and the boxplot looks like this. minimumQ1Q1 mQ3Q3 maximum

Boxplots From the boxplot we can see: shape centre spread

Boxplots So we can say that the distribution of scores on the assignment for this group of student is symmetric, with about 50% of the students scoring 10 or more out of 20. The marks on the assignment were quite variable, with the middle 50% of scores ranging from 5 to 15.

Boxplots We can also construct a more complicated boxplot which also show the outliers (if any). But first of all, we need a definition of an outlier.

Boxplots Suppose we have a data set for which the IQR range is represented by the following box: Then any data value which falls outside of these lines is called an outlier. IQR 1.5  IQR Imaginary lines are drawn 1.5 interquartile ranges below and above the box.

Boxplots IQR 1.5  IQR We join the whiskers from the box to the largest data value which is not an outlier... …and use an asterisk or similar to display the outlier. *

Boxplots The times (in minutes) taken for a sample of 18 people to assemble a new pre-packed TV cabinet are: We will demonstrate a boxplot with outliers using the following data.

Boxplots maxQ3Q3 Q1Q1 min By putting the data in order we can soon find the five-figure summary m=32 IQR = Q 3  Q 1 = 36  28 =  IQR = 1.5  8 = 12

Boxplots The imaginary lines are drawn at: 28  12 = 16 and = 48 This means we have one outlier (14),and the boxplot looks like this: