Deciles and quartiles

Statistics and probability, grade 10 (4 units) Lesson 9 of 20Learning

Deciles and quartiles

The lower decile and the upper decile

The median splits the ordered data into two halves. The deciles split them into ten parts, each holding 10% of the data.

The lower decile is the value that separates the smallest 10% of the data from the rest.
The upper decile is the value that separates the greatest 10% of the data from the rest.

In the news “the top decile” is also the group itself: the 10% of households with the highest income. The top decile deserves attention of its own: it shows the edge of the data, which the mean and the median do not show.

A decile’s place in an ordered list

A decile is found like the median: by its place in an ordered list of $\displaystyle n$ data.

\displaystyle 0.1\cdot n

This is the lower decile’s place; the upper decile’s place is $\displaystyle 0.9\cdot n$ .

When the place $\displaystyle k$ is whole – the decile is the mean of the $\displaystyle k$ -th datum and the $\displaystyle (k+1)$ -th.
When the place is not whole – the decile is the datum at the next whole place.

20 students were asked how many minutes a day they read, in order:
0, 5, 5, 10, 10, 10, 15, 15, 20, 20, 20, 25, 25, 30, 30, 35, 40, 45, 60, 90.

The lower decile: the place is $\displaystyle {0.1\cdot20}={2}$ , so $\displaystyle {\frac{5+5}{2}}={5}$ minutes.
The upper decile: the place is $\displaystyle {0.9\cdot20}={18}$ , so $\displaystyle {\frac{45+60}{2}}={52.5}$ minutes.

With 15 data the lower decile’s place is $\displaystyle {0.1\cdot15}={1.5}$ , and the lower decile is the second datum.

Deciles from a frequency table

In a frequency table, count up to the decile’s place with the cumulative frequency, as for the median.

Number of fouls	0	1	2	3	4
Frequency	4	16	9	6	5
Cumulative frequency	4	20	29	35	40

The lower decile: the place is $\displaystyle {0.1\cdot40}={4}$ . The fourth datum is 0, and the fifth 1, so the lower decile is $\displaystyle {\frac{0+1}{2}}={0.5}$ .
The upper decile: the place is $\displaystyle {0.9\cdot40}={36}$ . Data 36–40 are all 4, so the upper decile is 4.

The 10% of jumpers with the most fouls fouled four times each.

The quartiles

The quartiles split the ordered data into four parts, each holding about a quarter of the data.

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

In the fouls of the 40 jumpers, with median 1.5:

The lower half – data 1–20; its median is the mean of the 10th and 11th data, both 1. So $\displaystyle {Q_1}={1}$ .
The upper half – data 21–40; its median is the mean of the 30th and 31st data, both 3. So $\displaystyle {Q_3}={3}$ .

The box plot

Five numbers describe the data in short: the least datum, $\displaystyle Q_1$ , the median, $\displaystyle Q_3$ and the greatest datum. A box plot draws them on one axis.

The reading minutes of the 20 students:

The box stretches from $\displaystyle {Q_1}={10}$ to $\displaystyle {Q_3}={32.5}$ , and the line inside it is the median, 20.
Inside the box lies the middle half of the data; in each “whisker” – a quarter of the data.
The right whisker is long: a quarter of the students read between 32.5 and 90 minutes, and the data there are spread over a wide range.

Exercise 1

The hours of sleep of 20 students at night, in order:
5, 5.5, 6, 6, 6.5, 6.5, 7, 7, 7, 7, 7.5, 7.5, 8, 8, 8, 8.5, 8.5, 9, 9, 10.

Find the lower decile.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power