Recalling definitions, such as the Lower Quartile is the $\frac{n+1}{4}th$ piece of data with the data placed in order, find an expression for the Interquartile Range.

Hence, show that a data set with only 5 numbers in it cannot have any outliers.

Give an example of a set of data with 7 numbers in it that does have an outlier, justify this fact by stating the Interquartile Range.

$LQ=\frac{x_1+x_2}{2}\text{,}UQ=\frac{x_4+x_5}{2}\text{,}IQR=\frac{x_4+x_5-x_1-x_2}{2}$       M1A1

[2 marks]

$UQ+1.5IQR=1.25x_4+1.25x_5-0.75x_1-0.75x_2⩾x_5$      M1A1

Since $1.25x_4+0.25x_5⩾0.75x_1+0.75x_2$ due to the ascending order.      R1

Similarly $LQ-1.5IQR=1.25x_1+1.25x_2-0.75x_4-0.75x_5⩽x_1$      M1A1

Since $0.25x_1+1.25x_2⩽0.75x_3+0.75x_4$ due to the ascending order.

So there are no outliers for a data set of 5 numbers.      AG

[5 marks]

For example 1, 2, 3, 4, 5, 6, 100 where $IQR=4$     A1A1

[2 marks]