Quantitative Map Skills and Data Interpretation

Numbers need units and context

Geographical numbers describe quantities such as distance, area, river velocity and population. A number becomes useful when you know what it measures, its unit and the place or time it represents. A river velocity of $0.29\text{ m/s}$ means water travels 0.29 metres in one second; it is not a distance of 0.29 metres.

Units let us compare measurements consistently. There are 100 centimetres in one metre and 1,000 metres in one kilometre. To convert metres into kilometres, divide by 1,000; to convert kilometres into metres, multiply by 1,000. Thus 550 m is 0.55 km.

Calculate ground distance using a scale line

A map reduces ground distances so that a large area fits on a page. A scale line shows how much ground distance a length on that particular map represents.

Measure the scale line and the distance you want using the same ruler units. Divide your map distance by the scale-line length, then multiply by the ground distance labelled on the scale line:

$$\text{Ground distance}=\frac{\text{map distance}}{\text{scale-line length}}\times\text{scale-line ground distance}$$

Suppose a scale line labelled 0–1 km measures 4 cm, and the straight-line distance between two points measures 6 cm. The distance is $6\div4\times1=1.5\text{ km}$. This is one and a half scale-line lengths.

For a winding route, follow its bends with a strip of paper or thread, then measure the total length and apply the scale. A straight line between its endpoints would underestimate the route distance.

A ratio scale expresses the same relationship using matching units. At $1:25{,}000$, 1 cm on the map represents 25,000 cm on the ground, or 250 m. At a fixed scale, map distance and ground distance are directly proportional: doubling one doubles the other.

Estimate area using simple shapes

Area measures the surface inside a boundary. An irregular feature can be approximated by rectangles and triangles. Choose shapes that follow the boundary reasonably closely, calculate their areas and add them together. Avoid counting the same surface twice.

$$A_{\text{rectangle}}=\text{length}\times\text{width}$$ $$A_{\text{triangle}}=\frac{1}{2}\times\text{base}\times\text{perpendicular height}$$

The perpendicular height meets the base at a right angle. It need not be a side of the triangle.

Convert the map measurements into ground lengths before using these formulae. For example, at $1:10{,}000$, 1 cm represents 100 m. A rectangle measuring 5.5 cm by 3 cm therefore represents approximately 550 m by 300 m, giving $550\times300=165{,}000\text{ m}^2$.

This approach has been used to estimate the area of Redmires Reservoirs. Reservoirs A and C were approximated as rectangles and B as a triangle:

Adding the three estimates gives $371{,}700\text{ m}^2$. This is an approximation because the simple shapes do not follow every bend in the shorelines. The same method transfers to other mapped features, but its accuracy depends on the measurements and how well the chosen shapes fit.

Area units change in two dimensions

A square kilometre is a square 1,000 m long and 1,000 m wide. Therefore:

$$1\text{ km}^2=1{,}000\times1{,}000=1{,}000{,}000\text{ m}^2$$

To convert m² into km², divide by 1,000,000. The Redmires estimate becomes $371{,}700\div1{,}000{,}000=0.3717\text{ km}^2$. The conversion factor is larger than for length because both dimensions change.

Distinguish ratio, proportion, magnitude and frequency

A ratio compares quantities. A ratio of 21 children to 100 working-age people means 21 children for every 100 people in the comparison group. It does not mean that children form 21% of everyone in those two groups: their proportion would be $21\div(21+100)$, or approximately 17.4%.

A proportion expresses a part of a total. Always identify that total before turning a ratio into a fraction or percentage.

Youth-dependency figures of $21:100$ for Japan and $81:100$ for Nigeria illustrate the importance of matching denominators. Both compare children aged 0–14 with working-age people aged 15–64. Nigeria's figure is $81\div21\approx3.9$ times Japan's. This compares the dependency ratios, not the actual numbers of children in the two countries.

Magnitude concerns how large a quantity is; frequency concerns how often something occurs. In the River Eea table below, for example, the May velocity of 0.21 m/s occurs at two of the six sampled sites. Its frequency is two. That count is different from the magnitude of the velocity itself.

Flood likelihood can be expressed as an annual chance. A 1-in-200 flood, sometimes written as a $1:200$ flood, has a 0.5% chance of happening in any one year. GOV.UK planning guidance uses this equivalence for tidal flooding. The fraction converts to a percentage as $1\div200\times100=0.5\%$. This is an annual probability, not a timetable of one flood every 200 years.

Logarithmic scales multiply rather than add

Some geographical quantities span such a large range that an ordinary scale is inconvenient. A logarithmic scale represents equal multiplying factors with equal steps. For example, equally spaced marks on a base-ten logarithmic axis can represent 1, 10, 100 and 1,000. Each step multiplies the quantity by ten rather than adding the same amount.

The Richter scale is an example. Earthquake magnitude is estimated using the amplitude of ground motion recorded by a seismometer, with corrections for distance. Amplitude means the size of the vibration. A one-unit increase on the Richter scale corresponds to ten times the recorded seismic-wave amplitude.

For example, comparing magnitudes 5, 6 and 7:

Magnitude 7 therefore represents a hundredfold amplitude compared with magnitude 5, not twice the amplitude. A factor of ten is one order of magnitude, so this hundredfold difference is two orders of magnitude. The key is to count the steps and multiply for each one.

Amplitude is not the same quantity as energy released. Each one-unit magnitude increase corresponds to about 30 times as much energy, rather than ten times. Modern scientists usually use moment magnitude for large earthquakes, but the Richter scale remains a useful example of how to interpret a logarithmic scale.

Read spatial and temporal patterns from data

Start with a table's headings. Identify its variables, units, places and dates before looking for a relationship. The following measurements describe river velocity at six sites along the River Eea in Cumbria, at four points in the year.

Reading down one month reveals a spatial pattern. In February, velocity generally increases with distance from the source, rising from 0.15 m/s at 0.5 km to 0.29 m/s at 3 km. The increase is 0.14 m/s. However, velocity remains at 0.20 m/s between the sites at 1.3 km and 1.7 km, so the increase is not continuous at every step.

Reading across one site reveals a temporal pattern. At 3 km, velocity is 0.26 m/s in December, 0.29 m/s in February, 0.24 m/s in May and 0.16 m/s in August. February has the highest recorded value at this site, while August has the lowest.

Data for River Eea velocity in February and August

The graph plots two columns from the table. Read the horizontal axis to locate a site, then compare the heights of the two lines there. August velocity is lower than February velocity at every sampled site. The gap is 0.12 m/s at 0.5 km and 0.13 m/s at 3 km.

A sound conclusion combines a pattern, numerical evidence and an appropriate limit. These measurements support the conclusion that velocity generally increases downstream and is lower in August than February at the sampled sites. They do not, by themselves, establish the cause or show that the same pattern occurs in every year.

Scale and units

• Scale-line method: map distance ÷ scale-line length × labelled ground distance.
• $1:25{,}000$: 1 cm represents 250 m.
• $1\text{ km}=1{,}000\text{ m}$; $1\text{ m}=100\text{ cm}$.