Quartiles, Box Plots and Comparing Populations

Describing a population with a few numbers

A population is the whole group an investigation concerns: for example, all pupils at a school. A sample is a smaller group selected from that population. Statistics calculated from a sample can help describe the population, but the conclusions depend on how well the sample represents it.

A summary statistic condenses part of a data set into one number. An average describes a typical value, while a measure of spread describes how much the values vary. Both are needed: two groups can have the same average but very different spreads.

For a fair comparison, use like-for-like summaries. Compare means with means or medians with medians, rather than comparing one group’s mean with another’s median. The measurements must also be comparable: distances travelled over ten minutes should not be compared directly with distances travelled over twenty minutes.

Choosing useful summaries

The mean uses every value, which makes it useful when all measurements should contribute to the summary. However, an unusually large or small value can pull it towards that extreme. The median depends on the middle position in ordered data, so it is less affected by extreme values.

The mode identifies the most common value, but there may be no mode or more than one mode. It can also give a poor impression of the rest of the distribution because it does not take every value into account.

The range measures the full span from the minimum to the maximum. It is useful for describing the overall spread, but depends on just two values and is strongly affected by extremes. Quartiles provide a way to describe spread without relying on the endpoints.

Quartiles divide an ordered distribution

Imagine arranging measurements from smallest to largest. Quartiles mark the quarter-way points through this ordered distribution:

• The lower quartile, $Q_1$, has about 25% of the data below it.
• The median, $Q_2$, has about 50% below it.
• The upper quartile, $Q_3$, has about 75% below it.

The interval from the lower quartile to the upper quartile contains the middle 50% of the data. Its width is the interquartile range:

$$\mathrm{IQR}=Q_3-Q_1$$

A smaller IQR means that the middle half of the measurements are more closely grouped. Because the IQR does not use the minimum and maximum, it is less affected by extreme values than the range. Its limitation is that it does not describe how far the outer values extend.

Reading quartiles from a cumulative frequency graph

A cumulative frequency graph shows running totals. Its horizontal axis gives the measured quantity, and its vertical axis gives cumulative frequency. The final cumulative frequency reached by the graph is the total number of observations, $N$.

To estimate the quartiles, find these heights on the cumulative frequency axis:

$$Q_1:\ \frac{N}{4},\qquad \text{median}:\ \frac{N}{2},\qquad Q_3:\ \frac{3N}{4}$$

From each height, move horizontally to the curve, then vertically down to the horizontal axis. The horizontal-axis reading is the estimated quartile value, in the measurement’s units.

For a constructed illustration, the graph below represents journey times for an imaginary group of 80 pupils, measured in minutes. The relevant cumulative frequencies are $80/4=20$, $80/2=40$ and $3\times80/4=60$.

Data for Estimating quartiles from cumulative frequency

Tracing horizontally from 20 to the graph and then down gives 12 minutes, the lower quartile. Repeating this from 40 gives a median of 18 minutes, and from 60 gives an upper quartile of 26 minutes. The IQR is $26-12=14$ minutes, so the middle half of journey times spans an interval 14 minutes wide.

These readings are estimates when the graph represents grouped data: the individual measurements within each class are unknown, and the line or curve assumes how they are spread within the intervals. Reading from a graph also introduces limited precision. The same tracing method transfers to any cumulative frequency graph.

Drawing and reading a box plot

A box plot places five summary values on a numerical scale, in this order:

minimum — lower quartile — median — upper quartile — maximum.

To draw one, first choose a labelled scale covering the full range. Draw a box from the lower quartile to the upper quartile, with a line inside it at the median. Draw whiskers from the box to the minimum and maximum, and mark the endpoints. Position everything using the numerical scale rather than spacing the five values equally.

For a constructed journey-time distribution with a minimum of 5 minutes, lower quartile of 12 minutes, median of 18 minutes, upper quartile of 26 minutes and maximum of 40 minutes, the box extends from 12 to 26. The median line is at 18, and the whiskers reach 5 and 40.

The box’s width represents the IQR, $26-12=14$ minutes; the distance between the whisker endpoints represents the range, $40-5=35$ minutes.

Each successive section contains approximately a quarter of the observations, but the sections need not have equal lengths. A long section means those observations occupy a wider interval of values. The median need not be halfway across the box. A box plot summarises a distribution; it does not show every individual measurement or reveal its mode.

Comparing distributions in context

When comparing box plots, use the medians to compare typical values and the IQRs to compare the spread of the middle half. Draw the plots against the same numerical scale so that their positions and lengths can be compared directly.

Suppose two groups have journey-time medians of 18 and 22 minutes, and IQRs of 14 and 8 minutes. The first group has a lower median, suggesting a shorter typical journey. The second has a smaller IQR, so its middle 50% of journey times are more consistent. Neither statement means that every pupil in one group has a shorter journey than every pupil in the other.

The same reasoning works with other summary measures. For example, recorded distances travelled in a ten-minute interval have medians of 7.1 cm for snails and 9.7 cm for slugs. Their ranges are 3.1 cm and 4.5 cm respectively. The slugs have the higher median distance, suggesting they typically travelled further during the interval. The snails have the smaller range, so their recorded distances were less spread out.

Recognising the limits of a conclusion

Summary values make large data sets easier to understand, but hide detail. A higher median does not establish that every observation is higher, and a smaller IQR does not rule out extreme values.

If the data come from a sample, consider whether conclusions can reasonably extend to the population. Measuring the heights of only five pupils may give an unreliable picture of a whole school. Measuring only older pupils can bias the result towards greater heights. A careful conclusion distinguishes what the recorded data show from what they suggest about the wider population.

Quartiles and spread

$Q_1$, median and $Q_3$ mark approximately 25%, 50% and 75% through an ordered distribution.

$$\mathrm{IQR}=Q_3-Q_1$$

IQR measures the spread of the middle 50%; range measures maximum minus minimum. IQR is less affected by extreme values.

From a cumulative frequency graph

Find total frequency $N$. Read horizontally to the curve, then down to the data axis from:

• $N/4$ for $Q_1$
• $N/2$ for the median
• $3N/4$ for $Q_3$.

Grouped-data readings are estimates because individual values within classes are unknown.

Box plots

Use minimum, $Q_1$, median, $Q_3$, maximum.

Box: $Q_1$ to $Q_3$. Internal line: median. Whiskers: minimum and maximum. Draw against a labelled numerical scale.

Comparisons and limitations

Compare median with median for typical value and IQR with IQR for middle-half spread. Quote values and interpret them in context.

Mean uses all values but is affected by extremes. Median is less affected by extremes. Mode may be absent or unrepresentative. Range depends only on the endpoints.

Summary values hide detail. A small or biased sample may not represent its population.