Geographical Measurements and Numerical Calculations

Numbers need geographical meaning

Geographical numbers describe things such as distances, areas, populations and traffic counts. A number becomes useful when you know what it measures and, where needed, its unit. A river width of 12 m is different from a river depth of 12 cm.

Counts usually use whole numbers: a traffic survey might record 40 vehicles. Measurements can use decimals because a distance or area need not be a whole number of metres or kilometres. Large quantities may be expressed in millions: 2.5 million means 2,500,000. Keeping the meaning and size of each number clear helps you judge whether a calculation is sensible.

Converting units without changing the quantity

A unit conversion changes how a quantity is expressed, not its actual size. Since $1\text{ km}=1,000\text{ m}$, a distance of 2.4 km is 2,400 m. Multiply by 1,000 when converting kilometres to metres; divide by 1,000 when converting metres to kilometres. Similarly, $1\text{ m}=100\text{ cm}$.

Area conversions work differently from length conversions because area has two dimensions. A square measuring 1 km along each side measures 1,000 m along each side, so its area is $1,000\times1,000=1,000,000\text{ m}^2$.

Therefore:

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

To convert m² to km², divide by 1,000,000. To convert km² to m², multiply by 1,000,000. The same principle means that $1\text{ m}^2=10,000\text{ cm}^2$.

Using map scale to find ground distances

A ratio scale of 1:25,000 means that one unit on the map represents 25,000 of the same unit on the ground. Thus, 1 cm represents 25,000 cm, which is 250 m. A map distance of 4 cm therefore represents $4\times250=1,000\text{ m}$, or 1 km.

Map and ground distances are directly proportional at a fixed scale. Doubling the map distance doubles the ground distance. At 1:10,000, however, 1 cm represents 100 m, so the same 4 cm represents only 400 m. Always establish the scale before calculating.

A scale bar offers another way to convert distances: compare the measured distance with the length representing a stated ground distance. If a map has been enlarged or reduced, use its reproduced scale bar rather than assuming that the original printed ratio still applies.

Calculating and estimating area

For a rectangle, multiply its length by its width. For a triangle, multiply the base by the perpendicular height and halve the result:

$$A_{\text{rectangle}}=l\times w$$ $$A_{\text{triangle}}=\frac{1}{2}bh$$

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

Suppose a rectangular field measures 4 cm by 2 cm on a 1:25,000 map. These lengths represent 1,000 m and 500 m, so its ground area is $1,000\times500=500,000\text{ m}^2$, or 0.5 km². Converting the lengths first makes it clear which units the answer should use.

Real geographical features rarely form perfect rectangles or triangles. You can estimate their areas by fitting simple shapes around their outlines, or dividing them into several shapes and adding the areas. Avoid counting the same area twice. The result is an approximation because the simple shapes do not follow every bend in the boundary.

For example, the Redmires Reservoirs can be approximated by two rectangles, A and C, and a triangle, B. At a map scale of 1:10,000, the measurements and calculations are:

Adding the three areas gives $371,700\text{ m}^2$. Dividing by 1,000,000 gives an estimated total of $0.3717\text{ km}^2$. The detailed decimal answer does not make the shape approximation exact. On another map or a scaled photograph, the same method applies, but you must use that image’s scale and the feature’s own dimensions.

Distinguishing proportion, ratio and magnitude

A proportion compares a part with the whole. Suppose a survey records 20 cyclists and 80 other road users. There are 100 road users altogether, so the proportion who are cyclists is $20/100=0.2$, or 20%.

A ratio compares quantities in a stated order. In the same example, cyclists to other road users is $20:80$, simplified to $1:4$. For every cyclist, there are four other road users. Cyclists to all road users is a different ratio: $20:100=1:5$.

Simplify a ratio by dividing both quantities by the same number. When comparing measurements, first express them in the same units: 500 m to 1 km becomes $500:1,000=1:2$.

Magnitude concerns size. To compare relative sizes, divide one quantity by the other. Suppose one town has 60,000 residents and another has 20,000. The first has three times the population of the second, because $60,000/20,000=3$. This is a multiplicative comparison; the numerical difference is 40,000 residents.

Frequency and fair comparisons

Frequency tells you how often something occurs. In a traffic survey, the frequency of bicycles is the number counted. In a rainfall record, it might be the number of days in a particular rainfall category.

Frequency can be expressed as a proportion of the total:

$$\text{Relative frequency}=\frac{\text{category frequency}}{\text{total frequency}}$$

A five-minute traffic count with 6 bicycles out of 40 recorded transport movements has a bicycle proportion of $6/40=0.15$, or 15%. Percentages help compare surveys with different totals: a count of 6 out of 40 represents a larger share than 6 out of 60. Comparisons of raw event counts also need comparable observation periods; 20 vehicles in five minutes is not directly equivalent to 20 vehicles in an hour.

Percentages: shares and amounts

A percentage expresses a proportion out of 100. To express a part as a percentage of a whole, use:

$$\text{Percentage}=\frac{\text{part}}{\text{whole}}\times100$$

To calculate an amount from a percentage, reverse the process:

$$\text{Amount}=\frac{\text{percentage}}{100}\times\text{total}$$

Suppose 30% of a 250-hectare area is woodland. The woodland covers $30/100\times250=75$ hectares. The percentage describes its share; 75 hectares describes the actual area.

Percentage increases and decreases

Percentage change compares a change with the starting value:

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

Suppose a comparable traffic count rises from 80 vehicles to 100 vehicles. The increase is 20 vehicles, and $20/80\times100=25\%$. A positive result indicates an increase.

If the count falls from 100 vehicles to 80, the signed change is $(80-100)/100\times100=-20\%$: a 20% decrease. Although both changes involve 20 vehicles, the percentages differ because their starting values differ.

A change between two percentages can also be expressed in percentage points. An increase in woodland cover from 20% to 30% is 10 percentage points. Its percentage increase is $10/20\times100=50\%$.

Percentiles locate values within a distribution

A percentile describes where a value lies when data are ordered from smallest to largest. The 80th percentile is a value at or below which approximately 80% of observations lie. It is a measurement value, with the original variable’s unit—not an amount equal to 80% of the maximum.

To locate a percentile, first order the data or use a running count of observations up to each value. Calculate the target number of observations using:

$$\text{Target count}=\frac{p}{100}\times N$$

Here, $p$ is the required percentile and $N$ is the total number of observations. Locate the measurement corresponding to this position. Different calculation conventions can give slightly different answers for small datasets, particularly between two observations.

For a concrete example, suppose 100 daily rainfall measurements are summarised as follows:

For the 80th percentile, the target count is $80/100\times100=80$ days. The table places the 80th-percentile threshold at 12 mm: 80% of the days have rainfall of 12 mm or less, and 20% have more than 12 mm.

The 50th percentile is the median; the 25th and 75th percentiles correspond to the lower and upper quartiles. A high percentile means a relatively high value, but whether that is desirable depends on what is being measured.

Scale, units and area

• At 1:25,000, 1 cm represents 250 m; at 1:10,000, 1 cm represents 100 m.
• Convert map lengths to ground lengths before calculating area.
• Rectangle: $A=lw$. Triangle: $A=\frac{1}{2}bh$, using perpendicular height.