A set of data comprises of five numbers x 1 , x 2 , x 3 , x 4 , x 5 which have been placed in ascending order.

Question

SLPaper 1

A set of data comprises of five numbers ${x_{1\,}}{\text{,}}\,\,{x_2}{\text{,}}\,\,{x_3}{\text{,}}\,\,{x_4}{\text{,}}\,\,{x_5}$ which have been placed in ascending order.

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.

[2]

Verified

Solution

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

[2 marks]

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

[5]

Verified

Solution

$UQ + 1.5IQR = 1.25{x_4} + 1.25{x_5} - 0.75{x_1} - 0.75{x_2} \geqslant {x_5}$ M1A1

Since $1.25{x_4} + 0.25{x_5} \geqslant 0.75{x_1} + 0.75{x_2}$ due to the ascending order. R1

Similarly $LQ - 1.5IQR = 1.25{x_1} + 1.25{x_2} - 0.75{x_4} - 0.75{x_5} \leqslant {x_1}$ M1A1

Since $0.25{x_1} + 1.25{x_2} \leqslant 0.75{x_3} + 0.75{x_4}$ due to the ascending order.

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

[5 marks]

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.

[2]

Verified

Solution

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

[2 marks]

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A set of data comprises of five numbers x 1 , x 2 , x 3 , x 4 , x 5 which have been placed in ascending order.

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