Quartiles and the five-number summary

What you will be able to do

• find $Q_1$, the median and $Q_3$ for an odd or an even number of values;
• write the five-number summary and the interquartile range;
• explain what a percentile means, and why two programs can give slightly different quartiles.

Fifteen phones

A consumer group runs a made-up test. It plays videos on 15 phones until each battery is empty. The battery lives, in hours and in order, are

$7.5, 8.2, 8.8, 9.1, 9.4$, $9.6, 9.9, 10.3, 10.5, 10.8$, $11.2, 11.6, 12.0, 12.9, 14.7$.

The median is the 8th value, $10.3$ hours: seven phones last less and seven last more. But one number cannot tell you how much the phones differ. The quartiles can.

The rule for quartiles

You met this rule in First Steps. Here it is again, with care for the details.

1. Sort the data and find the median.
2. Split the data into a lower half and an upper half. If the number of values is odd, leave the median out of both halves.
3. $Q_1$, the lower quartile, is the median of the lower half. $Q_3$, the upper quartile, is the median of the upper half.

For the phones, the lower half is the first seven values, and its middle is $Q_1 = 9.1$ hours. The upper half is the last seven values, and its middle is $Q_3 = 11.6$ hours.

With an even number of values, the halves are simply the first half and the second half. For the ten values $2, 4, 5, 7, 8$, $10, 11, 13, 14, 17$, the median is $\dfrac{8 + 10}{2} = 9$. The lower half $2, 4, 5, 7, 8$ gives $Q_1 = 5$, and the upper half $10, 11, 13, 14, 17$ gives $Q_3 = 13$.

The five-number summary

The five-number summary is the minimum, $Q_1$, the median, $Q_3$ and the maximum, in that order.

For the phones it is $7.5$, $9.1$, $10.3$, $11.6$ and $14.7$ hours. The interquartile range is $\text{IQR} = Q_3 - Q_1 = 11.6 - 9.1 = 2.5$ hours. It is the width of the middle half of the data. A box plot draws the five numbers: a box from $Q_1$ to $Q_3$ with a line at the median, and whiskers out to the smallest and largest values.

Quartiles and percentiles

A percentile tells you what percentage of the values lie below a point. The 90th percentile of a test score has about $90\%$ of the scores below it. The quartiles are close to the 25th, 50th and 75th percentiles. So about a quarter of the values lie below $Q_1$, a quarter between $Q_1$ and the median, a quarter between the median and $Q_3$, and a quarter above $Q_3$. With few values, "about" matters: 15 values cannot be cut into four equal parts.

One data set, several rules

There is more than one rule for quartiles. Spreadsheets, calculators and Python's statistics.quantiles each use their own, and they can give slightly different answers for the same data. For the 15 phones the rules happen to agree. For the 12 reaction times of the next lesson they do not. When you report quartiles, say which rule you used. In this chapter we always use the medians of the halves.

You should find that each quartile only moves while the amber value passes through a small stretch. $Q_1$ moves between 5 and 6 minutes, the median between 9 and 11, and $Q_3$ between 14 and 15. Outside those stretches the amber value can go anywhere, and the box does not move. Only the whiskers follow it.

Practise with new numbers

Try it in Python

The program writes a function for the five-number summary with the rule of this lesson, prints the summary of the 15 phones, and then compares it with statistics.quantiles.

Common mistakes

• Not sorting first. Quartiles need the data in order.
• Keeping the median in a half when the count is odd. Leave it out.
• Giving $Q_3$ as the IQR. The IQR is the width $Q_3 - Q_1$.
• Expecting every program to agree. Different rules give slightly different quartiles. Say which rule you used.

Remember these

• $Q_1$ and $Q_3$ are the medians of the lower and upper halves; leave the median out when the count is odd.
• Five-number summary: minimum, $Q_1$, median, $Q_3$, maximum.
• IQR = $Q_3 - Q_1$, the width of the middle half.
• About a quarter of the values lie in each of the four parts.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find $Q_1$ and $Q_3$ of $3, 4, 6, 7, 9$, $12, 15$.

2. Find the median, $Q_1$ and $Q_3$ of $10, 12, 13, 15, 18$, $20, 21, 25$.

3. A five-number summary is 4, 11, 16, 19 and 30 kg. What is the IQR, and what is the range?

4. In a class of 40, a mark is at the 75th percentile. About how many students scored less?

5. Two programs give $Q_1 = 21.5$ and $Q_1 = 21.75$ for the same data. Has one of them made a mistake?

References

• OpenStax, Introductory Statistics, Chapter 2 (Descriptive Statistics), sections on measures of the location of the data and on box plots.