Quartiles and box plots Worked Solutions — GCSE Maths

Write down the rule for the quartile positions

For a list of raw data the lower quartile sits at position $\frac{n+1}{4}$, the median at position $\frac{n+1}{2}$ and the upper quartile at position $\frac{3(n+1)}{4}$. These are POSITIONS in the ordered list, not values.

Work out the position of the lower quartile

With $n = 11$, $\frac{n+1}{4} = 3$. Position $3$ is a whole number, so the quartile is simply the $3$th value in the ordered list.

Read off the lower quartile

Counting to position $3$ in the ordered list gives a lower quartile of $13$ seconds. A quarter of the data lies below it.

Write down the rule for the quartile positions

For a list of raw data the lower quartile sits at position $\frac{n+1}{4}$, the median at position $\frac{n+1}{2}$ and the upper quartile at position $\frac{3(n+1)}{4}$. These are POSITIONS in the ordered list, not values.

Work out the position of the median

With $n = 11$, $\frac{n+1}{2} = 6$. Position $6$ is a whole number, so the quartile is simply the $6$th value in the ordered list.

Read off the median

The median is $25$ centimetres: half of the plants are below this value and half are above it.

Write down the rule for the quartile positions

For a list of raw data the lower quartile sits at position $\frac{n+1}{4}$, the median at position $\frac{n+1}{2}$ and the upper quartile at position $\frac{3(n+1)}{4}$. These are POSITIONS in the ordered list, not values.

Work out the position of the upper quartile

With $n = 15$, $\frac{3(n+1)}{4} = 12$. Position $12$ is a whole number, so the quartile is simply the $12$th value in the ordered list.

Read off the upper quartile

The upper quartile is $73$ grams. Three quarters of the data lies below it, so only a quarter lies above it.

Work out the positions of the two quartiles

With $n = 7$ the lower quartile sits at position $2$ and the upper quartile at position $6$ in the ordered list.

Read off the lower and upper quartiles

Counting to position $2$ gives $Q_1 = 20$ and counting to position $6$ gives $Q_3 = 33$.

Work out the interquartile range

The interquartile range is the upper quartile minus the lower quartile: $33 - 20 = 13$ marks. It is the width of the middle half of the data.

Put the values in order of size, smallest first

Quartiles are found by counting through the data in order, so the very first thing to do is sort it. Everything after this depends on the list being in order.

Write down the smallest and the largest value

Once the list is in order the smallest value is the first, $21$, and the largest is the last, $50$.

Work out the range

The range is largest minus smallest: $50 - 21 = 29$ years. It uses only the two most extreme values.