4.4(a) - Describing geographical distributions

Mean, median and mode

A distribution describes the values taken by a variable across observations. A measure of central tendency summarises a typical or central value, but different measures answer different questions. Before calculating, establish what each observation represents, its units and whether cases are comparable. Averaging erosion rates measured over unequal time windows may conceal a change in conditions.

The arithmetic mean is the sum of values divided by the number of observations. It uses every value, so an extreme observation can move it substantially. The median is the middle value after sorting; with an even number of observations it is the mean of the two central values. The mode is the most frequent value or category. A dataset may have more than one mode, or no unique mode. The mode can describe a nominal category such as the most common land use, whereas adding land-use labels to calculate a mean would have no meaning.

Worked even-sized sample

Illustrative travel times: 2, 4, 4, 6, 8, 10, 12 and 18 minutes. Total the eight values, then locate the central pair and the most frequent observation:

The longest journey pulls the mean above the median, while the mode identifies the most frequent observation.

Worked odd-sized sample

Five pebble long axes are 7, 9, 9, 13 and 22 centimetres. Find the mean and the third ordered observation:

The median is one observation when the sample size is odd; it is not the mean of positions two and three.

Choose a measure in relation to the enquiry. The median may describe a typical household income when a few very high incomes affect the mean. The mean remains useful when total resource demand matters. Neither should replace the underlying distribution. A mean of grouped data estimated using class midpoints is an estimate because exact values within each class are unknown; do not label it exact. Similarly, averaging two district means without considering their population sizes can misrepresent the combined population.

Range and interquartile range

Dispersion describes how spread out observations are. Two places can have the same mean but very different distributions. The range is the maximum minus the minimum, so it describes the full observed span. It is easy to calculate but depends only on two values: an unusual observation can change it greatly, and a larger sample has more opportunities to contain an extreme value.

The interquartile range, or IQR, describes the spread of the middle half of the ordered data. It is the upper quartile minus the lower quartile. Quartiles divide the ordered distribution into quarters, but calculation conventions differ, especially for small samples. State and apply the given convention; software and examination questions may use a particular method. A numerical difference caused solely by a legitimate convention is not evidence of a different underlying distribution.

Worked quartiles from ordered halves

Use travel times 2, 4, 4, 6, 8, 10, 12 and 18 minutes. Take the median of each half:

The IQR excludes the outer quarters, so it describes a different span from the full range.

Worked second ordered distribution

Slope angles are 2, 5, 7, 9, 12, 14, 18 and 23 degrees. Split the ordered values into two groups of four, then calculate:

Retain the angle unit: dispersion describes the measured variable rather than a dimensionless count.

The IQR is less sensitive to individual extreme values than the range, which can make it useful for skewed geographical data. That resistance also discards information about the tails. If an enquiry concerns emergency access for the longest journeys, focusing only on the middle half would conceal relevant disadvantage. Select a statistic because of what it reveals, while explaining what it omits.

Standard deviation

Standard deviation summarises dispersion around the mean using every observation. First calculate each deviation from the mean. Squaring prevents positive and negative deviations from cancelling. Average the squared deviations to obtain a variance, then take its square root so standard deviation has the original variable's units. A standard deviation of two metres measures spread in metres; variance would be in square metres.

The formula depends on whether the listed cases are the entire population being described or a sample used to estimate a wider population's variation. For a population, divide the sum of squared deviations by $n$. For the usual sample estimate, divide by $n-1$. This correction accounts for estimating the mean from the same sample; it gives a larger value for the same list. A question or source must state the convention when it matters.