12.1 Summarising data

$\{1,3,5,8,9,12,24,25,28,30,41,50\}$

There are 12 values in the data set.

The first quartile occurs between data position 3 and 4 and is the average of data values 5 and 8. The second quartile occurs between positions 6 and 7 and is the average of data values 12 and 24. The third quartile occurs between positions 9 and 10 and is the average of data values 28 and 30.

The first quartile = 6,5. ( $Q_1$ )

The second quartile = 18. ( $Q_2$ )

The third quartile = 29. ( $Q_3$ )

The range = 91 - 20 = 71. This tells us that the marks are quite widely spread. (Remember, however, that 'wide' and 'large' are relative terms. If you are considering one hundred people, a range of 71 would be 'large', but if you are considering one million people, a range of 71 would likely be 'small', depending, of course, on what you were analyzing).

i.e. $M=\frac{46+53}{2}=\frac{99}{2}=49,5$

i.e. $Q_1=\frac{40+43}{2}=\frac{83}{2}=41,5$

i.e. $Q_3=\frac{63+70}{2}=\frac{133}{2}=66,5$

The quartiles are 41,5, 49,5 and 66,5. These quartiles tell us that 25 $\%$ of the marks are less than 41,5; 50 $\%$ of the marks are less than 49,5 and 75 $\%$ of the marks are less than 66,5. They also tell us that 50 $\%$ of the marks lie between 41,5 and 66,5.

The Interquartile Range = 66,5 - 41,5 = 25. This tells us that the width of the middle 50 $\%$ of the data values is 25.

The Semi-interquartile Range = $\frac{25}{2}$ = 12,5