Quartiles, Box Plots and Comparing Distributions

Begin with the kind of data

A distribution describes how the values of a variable are spread: which values occur, where the centre lies and how much variation there is. A univariate data set concerns one variable, such as the journey time for each pupil in a class.

It is useful to identify how the data arose. Data you collect yourself is primary data; data previously collected by somebody else is secondary data. This distinction concerns the source of the data, not the diagram used to display it.

You must also distinguish discrete from continuous data. Discrete values occur as separate possibilities: the number of matches won can be $0,1,2,\ldots$, but not $2.4$. Continuous data is measured and can, in principle, take any value in a range. Height and time are continuous even when a measuring instrument records them only to the nearest centimetre or second.

When there are many observations, they may be put into classes. For example, heights might be grouped as

$150<h\leq160,\qquad160<h\leq170.$

Grouping makes the overall pattern easier to see, but loses the exact individual values within each class.

Choose a representation that suits the data

A useful graph should reveal the feature you need to analyse. Separate categories can be shown with a bar chart, while ungrouped discrete numerical data can be shown with a vertical line chart. Histograms are suited to grouped continuous data, and cumulative frequency graphs help estimate medians and quartiles. A box plot gives a compact comparison of centre and spread.

Always read the scale, labels and units before interpreting a graph. Consider both the typical value and the variation rather than judging a distribution from one isolated feature.

Quartiles divide ordered data

Quartiles and box plots are Higher-tier content.

First place the observations in ascending order. The median, also called $Q_2$, is the middle of the whole data set. The lower quartile $Q_1$ is the middle of the lower half, and the is the middle of the upper half. They mark positions approximately , and of the way through the ordered data.

A reliable procedure is:

1. order every value;
2. find the overall median;
3. if there is an odd number of values, leave that median out when forming the two halves;
4. find the median of the lower half to obtain $Q_1$;
5. find the median of the upper half to obtain $Q_3$.

If a half contains two middle values, take their mean. Quartile conventions can differ for some data-set sizes, so follow any method or quartile positions specified in the question.

For the ordered data

$2,2,3,4,8,8,9,11,12,12,13,$

the overall median is $8$. Leaving that value out gives the lower half $2,2,3,4,8$, whose median is $3$, and the upper half $9,11,12,12,13$, whose median is . Thus , and .

Measure the spread of the middle half

The range uses the two extremes:

$\text{range}=\text{maximum}-\text{minimum}.$

The interquartile range uses the two quartiles:

$IQR=Q_3-Q_1.$

For the data above, the range is $13-2=11$, while the interquartile range is $12-3=9$. The IQR describes the spread of the middle $50\%$ of the observations. Because it ignores the lowest and highest quarters, it is less affected than the range by an extreme value or outlier.

The equation can also recover a missing quartile. If $Q_3=52$ and $IQR=35$, then

$Q_1=Q_3-IQR=52-35=17.$

Construct a box plot from five values

A box plot is determined by the five-number summary:

$\text{minimum},\quad Q_1,\quad \text{median},\quad Q_3,\quad \text{maximum}.$

Draw and label a suitable linear scale extending at least from the minimum to the maximum. Mark all five values accurately. Draw a box from $Q_1$ to $Q_3$, place a line inside it at the median, and join the box to the minimum and maximum with lines often called whiskers.

The left and right edges of the box locate $Q_1$ and $Q_3$, so the width of the box represents the IQR. The complete distance from the minimum to the maximum represents the range. A box plot does not show every observation, the frequency of each individual value, the mean or the mode.

Interpret and compare distributions

A sound comparison normally considers both centre and spread.

• Compare the medians to decide which group tends to have higher values.
• Compare the IQRs to decide which middle half is more variable or more consistent.
• If useful, compare the ranges or the positions of the minimum and maximum, remembering that these are more sensitive to extreme values.

Suppose group A has a higher median but a larger IQR than group B. In context, group A generally has higher values, but its middle half is more spread out; group B is more consistent. Neither statement alone is a complete comparison.

Comparative wording must follow the variable. A smaller IQR means more consistent results, but whether a higher median is desirable depends on what was measured: a higher test mark may be favourable, whereas a higher journey time may not be. Box plots being compared should use the same scale so that their positions and widths can be judged fairly.