Quartiles, box plots and outliers

What you will be able to do

• find the quartiles and the interquartile range of a list of data;
• draw and read a box plot from the five-number summary;
• use the fence rule to find outliers, and say how they affect the mean and the standard deviation.

Splitting the data into quarters

The median splits ordered data into two halves. Now split each half again. The three cut points are the quartiles.

• The lower quartile $Q_1$ has a quarter of the data below it.
• The median $Q_2$ has half of the data below it.
• The upper quartile $Q_3$ has three quarters of the data below it.

How to find them

There are several rules for the quartiles, and they can differ a little for a small data set. This chapter uses the rule that most textbooks and many calculators use.

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

Take twelve readings in order: $8, 9, 11, 12, 13, 14,$ then $15, 16, 18, 19, 21, 48$. The median is $\dfrac{14 + 15}{2} = 14.5$. The lower half is $8, 9, 11, 12, 13, 14$, so $Q_1 = \dfrac{11 + 12}{2} = 11.5$. The upper half is $15, 16, 18, 19, 21, 48$, so $Q_3 = \dfrac{18 + 19}{2} = 18.5$.

A word of caution. Spreadsheet programs and the default setting of numpy interpolate between values and give $11.75$ and $18.25$ for the same data. Some A-Level boards use a position rule that rounds up. A question will either state the rule, or choose data where the rules agree. When in doubt, say which rule you used.

The interquartile range

The interquartile range is

It is the width of the middle half of the data. For the twelve readings it is $18.5 - 11.5 = 7$. Unlike the range and the standard deviation, it is not affected by the value $48$. Some Indian books use half of it, the quartile deviation $\dfrac{Q_3 - Q_1}{2}$, and the coefficient of quartile deviation $\dfrac{Q_3 - Q_1}{Q_3 + Q_1}$.

If every value is multiplied by $a$, the IQR is multiplied by $\lvert a \rvert$. If a number is added to every value, the IQR does not change.

The five-number summary and the box plot

The five-number summary is the smallest value, $Q_1$, the median, $Q_3$ and the largest value. A box plot (also called a box-and-whisker diagram) draws it:

• a box from $Q_1$ to $Q_3$, with a line at the median;
• whiskers that run from the box to the smallest and the largest values that are not outliers;
• a cross for each outlier.

Outliers

An outlier is a value that lies far from the rest. The most common rule uses fences:

A value below the lower fence or above the upper fence is an outlier. The twelve readings have fences at $11.5 - 10.5 = 1$ and $18.5 + 10.5 = 29$. So $48$ is an outlier, and the right whisker stops at $21$, the largest value inside the fence.

A-Level questions sometimes define an outlier as a value more than $2$ standard deviations from the mean, or give their own rule. IB and AP use the $1.5\,\text{IQR}$ rule above.

What outliers do

An outlier pulls the mean and increases the standard deviation. The median, the quartiles and the IQR hardly notice it. For the twelve readings the mean is $17$ with the outlier and $14.18$ without it. That is why box plots and the median are used for data with a few extreme values. An outlier is not always a mistake. It may be a real, rare event. Check it before you remove it.

Practise with new numbers

Try it in Python

The program finds the quartiles, the IQR, the fences and the outliers of the twelve readings. It also shows what the default quartile rule of numpy gives.

Common mistakes

• Keeping the median in both halves. For an odd number of values, leave the median out.
• Forgetting the $1.5$. The fences are $1.5$ interquartile ranges from the quartiles.
• Ending the whisker at the outlier. The whisker stops at the last value inside the fence.
• Treating every outlier as an error. Check the source first.
• Mixing the rules. Use one quartile rule throughout, and say which.

Remember these

• $Q_1$ and $Q_3$ are the medians of the lower and the upper half. $\text{IQR} = Q_3 - Q_1$.
• Five-number summary: minimum, $Q_1$, median, $Q_3$, maximum.
• Fences: $Q_1 - 1.5\,\text{IQR}$ and $Q_3 + 1.5\,\text{IQR}$. A value outside them is an outlier.
• Outliers move the mean and the standard deviation much more than the median and the IQR.

Check yourself

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

1. Find the median, $Q_1$, $Q_3$ and the IQR of $3, 5, 6, 8, 9, 11, 13, 14, 20$.

2. The quartiles of a data set are $Q_1 = 30$ and $Q_3 = 50$. What are the two fences?

3. A data set has a smallest value $4$, $Q_1 = 10$, a median $14$, $Q_3 = 20$ and a largest value $60$. Is $60$ an outlier?

4. Why does the IQR describe the spread better than the range when one value is far away from the rest?

5. Every value in a data set is multiplied by $3$ and then increased by $10$. The IQR was $8$. What is it now?

References

• OpenStax, Introductory Statistics, Chapter 2, sections on box plots, quartiles and outliers.
• NCERT, Mathematics for Class XI, Chapter 13 (Statistics). The quartiles, box plots and the fence rule are not in this chapter.