Measures of Spread and Outliers

What a measure of spread tells us

An average describes the centre of a distribution, but it does not reveal how tightly the values are grouped. Two basketball players could have the same median score even though one usually scores close to that median and the other alternates between very low and very high scores. A measure of spread describes this variation.

The two measures considered here are the range and the interquartile range. Both are measured in the same units as the original data.

Finding and interpreting the range

First put the values in order, then calculate

$\text{range}=\text{greatest value}-\text{least value}.$

For the scores $17,18,18,18,19,20,20,22,23,23,23,26,27,28,28$, the least value is 17 and the greatest is 28. Therefore,

$\text{range}=28-17=11\text{ points}.$

The scores extend across 11 points. A smaller range generally means that the values are more consistent.

The range is quick to find, but it uses only two values. If the largest value changed from 28 to 50, the range would jump from 11 to 33 even though all the other scores stayed unchanged. This makes the range very sensitive to extreme values and outliers.

Quartiles divide ordered data

The median divides ordered data into two halves. Quartiles extend this idea by dividing the ordered positions into four approximately equal parts:

• $Q_1$, the lower quartile, is about one quarter of the way through the data;
• the median, sometimes called $Q_2$, is halfway through;
• $Q_3$, the upper quartile, is about three quarters of the way through.

Always order raw data before locating these values. One convention for $n$ ordered values is:

1. calculate $\frac n4$ for $Q_1$ and $\frac{3n}{4}$ for ;

Different quartile conventions can produce different answers for some data-set sizes. Do not switch between position rules and informal “median of each half” methods: use one stated convention consistently, and follow any method or positions given in the question.

For the 15 ordered basketball scores, $\frac{15}{4}=3.75$, so $Q_1$ is the 4th value, 18. Also, $$, so is the 12th value, 26.

The interquartile range

The interquartile range is

$\text{IQR}=Q_3-Q_1.$

It measures the width of the middle 50% of the data. For the basketball scores,

$\text{IQR}=26-18=8\text{ points}.$

This means that the middle half of the scores is spread across 8 points. Because the IQR leaves out the lowest and highest quarters, an isolated extreme value usually affects it much less than it affects the range.

Quartiles in tables and diagrams

In a discrete frequency table, use cumulative frequencies as running totals. They show where the relevant quarter positions fall without requiring every observation to be written out. Once the positions of $Q_1$ and $Q_3$ have been located, read the corresponding data values and subtract.

Grouped data do not reveal the individual observations, so exact quartiles may not be available. A cumulative frequency graph can be used to read estimates at one quarter and three quarters of the total frequency. The IQR calculated from these readings is therefore also an estimate.

On a box plot, the left and right edges of the box represent $Q_1$ and $Q_3$. Consequently, the length of the box represents the IQR. The distance from the minimum to the maximum represents the range when the whiskers are drawn to those values.

Recognising and considering outliers

An outlier is unusually far from the main body of the data. It may be a genuine extreme result, or it may indicate a measuring, recording or data-entry error. It should be investigated rather than automatically deleted.

Sometimes an outlier is clear from a graph. When a numerical rule is required, a common rule classifies values as outliers if they lie

$\text{below }Q_1-1.5\times\text{IQR}$

or

$\text{above }Q_3+1.5\times\text{IQR}.$

For example, if $Q_1=20$ and $Q_3=32$, then the IQR is 12. The lower and upper boundaries are

$20-1.5(12)=2$

and

$32+1.5(12)=50.$

A value of 55 is above 50, so it is an outlier under this rule. Use a stated rule when the question supplies one; the word “unusual” on its own does not determine a unique cut-off.

Choosing and comparing measures of spread

Use the range when the full distance from the least to the greatest value matters. Use the IQR when the spread of the central data is more useful or when extreme values could distort the range.

A smaller range or IQR indicates less variation and therefore greater consistency. In the basketball data, Ben's IQR is 8 points and his range is 11 points. Sam's IQR is 12 points and his range is 20 points. Ben is more consistent because both measures of his spread are smaller.

A complete comparison should name the measure, quote the values and interpret them in context. Spread alone does not say which player generally scores more; that requires a suitable measure of central tendency as well.