Quartiles and the box plot

The median splits the ordered data into two halves: half the data are less than it or equal to it, and half are greater than it or equal to it.

The hourly pay of 11 young workers in a fast-food chain, in shekels, ordered from least to greatest:
$\displaystyle 32,\ 34,\ 35,\ 35,\ 36,\ 38,\ 40,\ 42,\ 45,\ 50,\ 70$

The median is the sixth value: 38 ₪.

The quartiles split each half in two again, so three numbers – $\displaystyle Q_1$, the median and $\displaystyle Q_3$ – split the data into four parts, each holding about a quarter of the data.

• The lower quartile $\displaystyle Q_1$ – the median of the lower half.
• The upper quartile $\displaystyle Q_3$ – the median of the upper half.
• When the number of data is odd, the median itself goes into neither half.

For the 11 workers:

• The lower half: $\displaystyle 32,\ 34,\ 35,\ 35,\ 36$ – so $\displaystyle {Q_1}={35}$.
• The upper half: $\displaystyle 40,\ 42,\ 45,\ 50,\ 70$ – so $\displaystyle {Q_3}={45}$.

When the number of data is even, the median is the mean of the two middle values, and every value is in one of the halves.

The hourly pay of 8 workers in a supermarket, in shekels:
$\displaystyle 30,\ 33,\ 34,\ 36,\ 40,\ 41,\ 45,\ 52$

• The median: $\displaystyle {\frac{36+40}{2}}={38}$.
• The lower half $\displaystyle 30,\ 33,\ 34,\ 36$: $\displaystyle {Q_1}={\frac{33+34}{2}}={33.5}$.
• The upper half $\displaystyle 40,\ 41,\ 45,\ 52$: $\displaystyle {Q_3}={\frac{41+45}{2}}={43}$.

Five numbers sum up the data: the least, $\displaystyle Q_1$, the median, $\displaystyle Q_3$ and the greatest. For the 11 workers: $\displaystyle 32,\ 35,\ 38,\ 45,\ 70$.

The interquartile range is the distance between the two quartiles – the middle half of the data lies within it.

For the 11 workers it is $\displaystyle {45-35}={10}$ ₪. The range is $\displaystyle {70-32}={38}$ ₪, because the outlying pay, 70 ₪, makes it larger. An outlier hardly changes the interquartile range.

A box plot draws the five numbers on one axis:

• The box – from $\displaystyle Q_1$ to $\displaystyle Q_3$: the middle half of the workers.
• The line inside the box – the median.
• The whiskers – from the box to the least value and to the greatest.

Each of the four parts – whisker, half box, half box, whisker – holds about a quarter of the workers. A long part means the data in it are spread out: the long right whisker shows that the high pay is far more spread out than the low.

Two box plots on the same axis compare two groups at a glance: where the centre is, and how spread out the data are.

The hourly pay of 11 workers in each of two branches:

• The medians are almost equal: 41 ₪ in branch A, 40 ₪ in branch B.
• The interquartile range: $\displaystyle {46-36}={10}$ ₪ in branch A, $\displaystyle {50-33}={17}$ ₪ in branch B.

Branch B has bigger gaps: a quarter of its workers earn 33 ₪ or less, and a quarter – 50 ₪ or more.

The hourly pay of 9 couriers in a delivery company, in shekels:
$\displaystyle 28,\ 30,\ 31,\ 33,\ 35,\ 38,\ 40,\ 44,\ 60$

Find the median.

Work out the lower quartile $\displaystyle Q_1$.

Work out the upper quartile $\displaystyle Q_3$.

Work out the interquartile range.

How many gigabytes of mobile data 10 pupils used in one month:
$\displaystyle 5,\ 7,\ 8,\ 8,\ 10,\ 12,\ 13,\ 15,\ 18,\ 22$

Work out the median.

Find $\displaystyle Q_1$.

Find $\displaystyle Q_3$.

A mobile company’s plan includes 15 gigabytes a month. For how many of the pupils is it enough? Explain using the quartiles.

200 customers called a bank’s phone service. The box plot shows how many minutes each of them waited for an answer:

What percent of the customers waited more than 14 minutes?

About how many customers waited between 5 and 14 minutes?

The bank’s website says: “The usual waiting time is 8 minutes.” Which number in the plot is it relying on? What else should it say?

The monthly pay in two small high-tech companies, in thousands of shekels:

In which company is the median higher? In which is the interquartile range larger?

Dana has been offered a job in both companies, and her pay will be set like an ordinary worker’s there. In which company is she more likely to earn 15 thousand ₪ or more? Explain.