Descriptive Statistics: Central Tendency, Spread and Percentages

Describing a geographical dataset

A list of measurements can contain more detail than we can easily interpret. Measures of central tendency summarise its centre: what is a typical value? Measures of spread describe how much the values vary. Both are useful when comparing places: two rivers could have similar typical pebble sizes but very different variation in those sizes.

Consider these pebble diameters, in centimetres, already arranged in order:

$2,\ 3,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 20$

Choosing a typical value

The mean shares the total equally between all observations:

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

The diameters total 77 cm. Dividing by the 11 pebbles gives a mean diameter of 7 cm. The mean uses every measurement, so an unusually large or small value can pull it towards that extreme.

The median is the middle value in an ordered list. There are 11 diameters, so the sixth value, 6 cm, is the median. If there were an even number of observations, we would add the two middle values and divide by two. The median depends on position rather than the size of every value, making it useful when unusually large or small observations would distort the mean.

The mode is the most frequent value. Here it is 3 cm, which occurs twice. A dataset can have more than one mode, or no single value occurring more frequently than the others. The mode can also describe categories, such as the most frequently recorded land use, for which calculating a mean would make no sense.

When measurements are grouped into intervals, we can identify a modal class rather than an exact mode. It is the interval with the greatest frequency; grouping does not reveal which individual value within it occurs most often.

Measuring variation: range and inter-quartile range

The range describes the full distance from the smallest value to the greatest:

$$\text{Range}=\text{greatest value}-\text{smallest value}$$

For the pebble diameters, the range is $20-2=18\text{ cm}$. It is quick to calculate, but depends entirely on two observations. The unusually large 20 cm pebble therefore has a strong influence.

Quartiles divide ordered data into four parts. The lower quartile, $Q_1$, marks approximately the first quarter of the data; the upper quartile, $Q_3$, marks approximately three quarters. The is the second quartile.

One method for finding quartiles in an individual list is to find the median of each half. For an odd number of values, leave the overall median out of both halves. In our list, the lower half is $2,3,3,4,5$, giving $Q_1=3\text{ cm}$. The upper half is $$, giving .

The inter-quartile range, or IQR, measures the spread of the middle 50%:

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

Here it is $9-3=6\text{ cm}$. Unlike the range, it does not depend directly on the smallest and greatest values. Use an average alongside a measure of spread: the median and IQR together describe both a typical size and how closely the central measurements cluster around it. A smaller IQR indicates less variation in the middle half of the data, not necessarily a smaller typical value.

Cumulative frequency: keeping a running total

Frequency tells us how many observations fall into a category or interval. Cumulative frequency adds each frequency to all those before it.

This constructed table represents journey times for 20 people; it is an illustration, not survey evidence.

The cumulative frequency of 12 at 20 minutes means that 12 people had journey times of 20 minutes or less. The final cumulative frequency equals the total number of observations. The modal class is $10<t\leq20$ minutes, because its frequency of 8 is the greatest.

A cumulative frequency graph places the measured variable on the horizontal axis and cumulative frequency on the vertical axis. For grouped data, plot each running total at its upper class boundary: here, the points are $(10,4)$, $(20,12)$, $(30,18)$ and $(40,20)$, starting at . Joining these points gives a way to estimate values within the intervals.

Data for Estimating quartiles from cumulative frequency

To estimate quartiles from the graph, use one quarter, one half and three quarters of the total frequency. With 20 observations, these levels are 5, 10 and 15. From each level on the vertical axis, move horizontally to the plotted line, then vertically down to the time axis. The reading is a journey time, not a frequency.

For example, the median is read at cumulative frequency 10. On this straight-line construction it gives approximately 17.5 minutes. The lower and upper quartiles are approximately 11.25 and 25 minutes, so the estimated IQR is $25-11.25=13.75\text{ minutes}$. These are estimates because grouping hides the original measurements; the straight lines assume observations are evenly distributed within each interval.

Percentiles describe position, not percentage change

Percentiles divide ordered data into 100 parts. The 25th percentile is the lower quartile, the 50th is the median and the 75th is the upper quartile.

A journey time at the 90th percentile has approximately 90% of journey times at or below it, leaving approximately 10% above it. It does not mean that the journey takes 90 minutes or that its duration has increased by 90%.

For a cumulative frequency graph containing $N$ observations, locate the $p$th percentile at cumulative frequency:

$$\frac{p}{100}\times N$$

In the table, the 90th-percentile level is $0.90\times20=18$. The cumulative frequency reaches 18 at 30 minutes, so the graph gives a 90th-percentile estimate of 30 minutes.

Percentage increase and decrease

An absolute change tells us how many people, millimetres or other units have been added or lost. A percentage change compares that change with the starting value, allowing changes in differently sized places to be compared.

$$\text{Percentage increase}=\frac{\text{new value}-\text{original value}}{\text{original value}}\times100$$ $$\text{Percentage decrease}=\frac{\text{original value}-\text{new value}}{\text{original value}}\times100$$

Calculating a percentage increase

In a village, the population increases from an original value of 120 people to a new value of 150 people. Calculate the percentage increase.

$$\text{Percentage increase} = \frac{\text{new value} - \text{original value}}{\text{original value}} \times 100$$ $$\frac{150 - 120}{120} \times 100 = 25\,\text{\%}$$

The result means the extra population is one quarter of the village's original population. Reversing the change would not give the same percentage: a decrease from 150 to 120 people is measured against 150, so it is 20%. The starting value determines the comparison.