Solved: A collection of data consists of five... [IB Mathematics Analysis and Approaches (AA)]

Question

HLPaper 1

A collection of data consists of five values $x_{1},\,x_{2},\,x_{3},\,x_{4},\,x_{5}$ which have been arranged in ascending order.

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

[2]

Verified

Solution

${LQ = \frac{{x_1 + x_2}}{2},\ UQ = \frac{{x_4 + x_5}}{2},\ IQR = \frac{{x_4 + x_5 - x_1 - x_2}}{2}}$ [2 marks]

Hence, demonstrate that a data set with only 5 values 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 values. AG

[5 marks]

Provide an example of a collection of data with 7 values 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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