Measures of Central Tendency and Spread

Summarising geographical measurements

Geographers often collect many measurements, such as rainfall totals, river depths or pedestrian waiting times. A summary helps us compare datasets without listing every observation. Two different questions matter: what is typical, and how variable are the measurements?

Measures of central tendency describe a dataset’s centre. Measures of spread describe how widely its values vary. A typical value alone cannot tell us whether observations are closely clustered or very different from one another.

Mean, median and mode describe different kinds of typical value

The mean shares the total equally between all observations:

$$\text{mean}=\frac{\text{sum of values}}{\text{number of values}}$$

The median is the middle of an ordered dataset. For an odd number of observations, select the single middle value. For an even number, find the mean of the two middle values. The mode is the most frequent value. A dataset can have more than one mode, or no uniquely most frequent value.

Consider a set of eight rainfall measurements, in millimetres, already arranged in order:

2, 3, 3, 4, 6, 6, 7, 9

Their total is 40 mm, so the mean is 5 mm. The middle values are 4 mm and 6 mm, giving a median of 5 mm. Both 3 mm and 6 mm occur twice, more often than any other value, so there are two modes.

The appropriate measure depends on what you want to show. The mean uses every numerical observation, but an unusually large or small value can pull it away from the main cluster. The median is less affected by extreme values, so it can better represent a typical observation when the data are unevenly distributed. The mode identifies the most common observation and can also summarise categories, such as the most common land use; numerical means cannot meaningfully average category names.

Grouped data and the modal class

Sometimes measurements are summarised in intervals rather than recorded individually. The table below is a constructed illustration of waiting times for 40 pedestrians at a crossing. Here, $t$ is waiting time in seconds.

Frequency is the number of observations in each interval. The modal class is $20\leq t<30$ seconds because it contains the most pedestrians: 16. This tells us the most common interval, not the most common exact waiting time. Grouping has hidden the individual measurements.

Range and quartiles describe spread

The range measures the distance from the smallest observation to the largest:

$$\text{range}=\text{largest value}-\text{smallest value}$$

For the rainfall measurements, the range is $9-2=7$ mm. It is quick to calculate, but it depends only on the two extremes. One unusual observation can therefore make the range much larger.

Quartiles divide ordered data into four parts. The lower quartile, , marks approximately one-quarter of the observations; the marks halfway; and the upper quartile, , marks approximately three-quarters. The , or IQR, measures the width of the middle half:

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

To find quartiles from a list of individual values, first put them in order and divide them into lower and upper halves. For an even number of values, each half contains half the observations. For an odd number, leave the single middle value out of both halves. Find the median of the lower half for $Q_1$ and the median of the upper half for $Q_3$.

For the rainfall data, the lower half is 2, 3, 3, 4, giving $Q_1=3$ mm. The upper half is 6, 6, 7, 9, giving $Q_3=6.5$ mm. The IQR is therefore $$ mm.

Because it focuses on the middle 50%, the IQR is less affected by extreme observations than the range. A smaller IQR means the middle half of the measurements are more closely clustered. It does not mean that every observation is close to the median.

Building a cumulative frequency graph

Cumulative frequency is a running total. In the waiting-time table, the first two frequencies add to 12. This means that 12 pedestrians waited less than 20 seconds. Adding the next frequency gives 28 pedestrians waiting less than 30 seconds. The final cumulative frequency is 40, the total number of observations.

To plot grouped continuous measurements, put waiting time on the horizontal axis and cumulative frequency on the vertical axis. Plot each cumulative total against its class’s upper boundary: for example, plot 12 against 20 seconds. The total for the entire interval has been reached at that boundary, not at its midpoint.

Add the starting point at the lowest class boundary with cumulative frequency zero. Join successive points with straight lines to make the graph below. A cumulative frequency graph cannot decrease: adding more observations cannot reduce the running total.

Data for Estimating median and quartiles from cumulative frequency

Reading the median and quartiles from the graph

Let $n$ be the total number of observations. On a cumulative frequency graph, locate the lower quartile at , the at , and the upper quartile at on the vertical axis. From each position, move horizontally to the graph, then vertically down to read the measurement.

For the 40 waiting times, these cumulative frequencies are 10, 20 and 30. Reading the straight-line graph gives approximately:

• $Q_1=17.5$ seconds;
• median $=25$ seconds;
• $Q_3=32.5$ seconds.

The IQR is therefore about 15 seconds. This means the middle half of the waiting times extend from about 17.5 to 32.5 seconds. The median indicates that approximately half the pedestrians waited less than 25 seconds and half waited longer.

These readings are estimates, because the table does not show where individual observations fall within each interval. Straight lines treat the observations as evenly spread within each interval. The same plotting and reading method transfers to real grouped fieldwork data, but this constructed graph is not evidence about an actual crossing.

Comparing datasets meaningfully

Use a measure of central tendency alongside a measure of spread. A lower median waiting time indicates a shorter typical wait; a smaller IQR indicates less variation among the middle half of waits. These are different conclusions. Two crossings could have the same median but quite different IQRs, so neither statistic should replace the other.

Central tendency

• Mean: sum of values ÷ number of values; uses every observation and is affected by extremes.
• Median: middle of ordered data; average the two middle values when there is an even number.
• Mode: most frequent value or category.
• Modal class: interval with the highest frequency.