Basic Number Skills, Scale, and Percentages

Working with numbers and units

Geographical numbers describe different things: a population is a count of people, a river's width is a length, and a woodland's area measures the surface it covers. A number needs its context and, where appropriate, its unit to make its meaning clear. A distance of 500 metres is very different from 500 kilometres.

Equivalent measurements can be expressed using different units:

• $1\text{ m}=100\text{ cm}$
• $1\text{ km}=1\,000\text{ m}$

To convert metres to kilometres, divide by 1,000; to convert kilometres to metres, multiply by 1,000. For example, 1,500 m is 1.5 km. Converting to a larger unit gives a smaller numerical value, but the actual distance stays the same.

Quantities must use compatible units before they can be compared or combined. When calculating a rectangular area, for example, express both its length and width in metres if the answer is to be in square metres.

Using scale to find ground distances

A map scale links distance on a map to distance on the ground. A ratio scale of means one unit on the map represents 25,000 of the on the ground. Therefore, 1 cm represents 25,000 cm, which is 250 m. At , 1 cm represents 500 m.

For a straight distance, measure between the points with a ruler, then multiply by the ground distance represented by each centimetre. At $1:25\,000$, a map distance of 4 cm represents $4\times250=1\,000\text{ m}$, or 1 km.

For a curved route, divide it into short, nearly straight sections. Measure each section, add their lengths, then apply the scale. Alternatively, mark successive sections along the edge of a piece of paper, pivoting it at each bend, and compare the total with the map's scale bar. Following the route measures its length rather than the straight-line distance between its endpoints.

On an actual map, use its stated scale or scale bar: do not assume every map uses the same scale. These methods transfer to real maps, but ruler measurements and approximating curves introduce some uncertainty.

Calculating area and converting square units

Area measures how much surface a shape covers. For a rectangle, multiply length by width. For a triangle, multiply the base by its perpendicular height and halve the result:

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

An irregular area can be divided into simpler shapes. Calculate their areas and add them, taking care not to overlap the shapes. An approximation to an irregular boundary gives an estimated area rather than an exact measurement.

Consider a rectangular area measuring 4 cm by 2 cm on a map at $1:25\,000$. Its ground dimensions are 1,000 m by 500 m. Its area is therefore $1\,000\times500=500\,000\text{ m}^2$, or $$.

Area conversions involve two dimensions. A square measuring 1 km by 1 km measures 1,000 m by 1,000 m, so:

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

Similarly, $1\text{ m}^2=10\,000\text{ cm}^2$. To convert square metres to square kilometres, divide by 1,000,000, not by 1,000. This is why converting both lengths before multiplying is a useful way to calculate ground area from a map.

Comparing a part with the whole

Geographical data often describe how a total is divided: for example, the share of people living in urban areas or the share of surveyed journeys made by bus. A proportion compares one part with the whole it belongs to.

$$\text{Proportion}=\frac{\text{part}}{\text{whole}}$$

A proportion can be written as a fraction, a decimal or a percentage. These are different ways of expressing the same share. For example, $\frac{20}{100}=0.2=20\%$.

In a survey of 100 journeys, suppose 20 are made by bus and 80 are made on foot. The proportion made by bus is $\frac{20}{100}$, or one fifth. The denominator includes all the journeys, not just those made on foot.

Comparing two quantities using a ratio

A ratio compares two quantities directly, rather than comparing one part with the whole. Using the same survey, the ratio of bus journeys to walking journeys is $20:80$.

Dividing both sides by 20 simplifies this to $1:4$. This means that for every one bus journey there are four walking journeys. Simplifying changes the numbers used to express the comparison, but not the relationship between them.

Order matters. Bus journeys to walking journeys is $1:4$, whereas walking journeys to bus journeys is $4:1$. Also notice that the bus journeys make up one of the five total parts in $1:4$, so their proportion is , not .

Distinguishing magnitude from frequency

Magnitude describes size; frequency describes how often something occurs. These answer different geographical questions.

For earthquakes, magnitude describes the size of an event. Frequency describes how often earthquakes occur. An area could experience frequent earthquakes without every earthquake having a large magnitude. Counting events does not, by itself, tell you their size.

In survey data, frequency can also mean the number of observations in a category. In the journey survey, the frequency of bus journeys is 20. Its percentage is 20% because there are 100 journeys altogether. A frequency is a count; a percentage expresses that count relative to a total.

Expressing an amount as a percentage

A percentage expresses a proportion out of 100. Divide the part by the whole, then multiply by 100:

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

This makes shares easier to compare when totals differ. The same number of bus journeys would represent a larger share of a small survey than of a large survey.

For example, suppose 25 of 360 surveyed homes have a particular characteristic. Their percentage is $\frac{25}{360}\times100=6.944\ldots\%$, which rounds to 7% to the nearest whole number. This means approximately seven out of every 100 homes in the survey have that characteristic; it does not mean that only seven homes were counted.

Measuring percentage increase and decrease

Percentage change compares a change with the original value. First find the difference between the final and original values, then express that difference as a percentage of the original:

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

A positive result shows an increase. A negative result shows a decrease.

For example, suppose the number of recorded incidents falls from 25 in one year to 21 in the next. The change is $21-25=-4$ incidents. The percentage change is $\frac{-4}{25}\times100=-16\%$: a .

If the number instead rises from 25 to 30, the change is five incidents and the percentage change is $\frac{5}{25}\times100=20\%$: a 20% increase.

The original value provides the starting point for the comparison. This is why reversing a change does not generally give the same percentage: a rise from 25 to 30 is 20%, but a fall from 30 to 25 is about 16.7%. The difference is five in each case, but it is compared with a different starting value.

Understanding and finding percentiles

A percentile describes a value's position within data ordered from smallest to largest. The 70th percentile is a threshold with approximately 70% of observations at or below it and approximately 30% above it. Exact proportions can differ in small datasets or where values are repeated.

A percentage and a percentile answer different questions. A rainfall total that is 70% of another total is a comparison of amounts. A rainfall total at the 70th percentile is relatively high within a particular set of rainfall totals; it is not necessarily 70% of the largest total.

To determine a percentile, first put the measurements in ascending order, then find the required position. There are several calculation conventions. One method uses:

$$\text{Position}=\frac{p}{100}(n+1)$$

Here, $p$ is the requested percentile and $n$ is the number of observations. When this gives a whole-number position, use the value at that position. When the position is fractional, round the position down and up to the neighbouring whole numbers, then average the two measurements at those positions. Under this convention, take their mean even when the fractional position is not exactly halfway between them. This is one method, not a uniquely prescribed OCR rule; follow a stated method when one is supplied.

For example, nine daily rainfall totals, in millimetres, are already ordered:

$2,\ 4,\ 6,\ 8,\ 10,\ 12,\ 15,\ 18,\ 24$.

Using this method, the 70th-percentile position is $\frac{70}{100}(9+1)=7$. The seventh value is 15 mm, so the 70th percentile is 15 mm. The calculation gives a position first; the percentile itself is a rainfall measurement. It indicates relatively high rainfall within these nine observations, not within all places or all years.

For the 25th percentile of the same dataset, the position is $\frac{25}{100}(9+1)=2.5$. Rounding down and up gives positions 2 and 3, whose rainfall totals are 4 mm and 6 mm. Their mean is $\frac{4+6}{2}=5\text{ mm}$, so the under this convention. A percentile can therefore lie between recorded measurements rather than being one of them. It marks the lower-quarter threshold of this rainfall dataset.

The 50th percentile is the median. The 25th and 75th percentiles correspond to the lower and upper quartiles. Whether a high percentile is desirable depends on what is being measured: a high rainfall percentile has a different meaning from a high percentile for journey time.

Units, scale and area

• $1\text{ m}=100\text{ cm}$; $1\text{ km}=1\,000\text{ m}$.
• $1\text{ m}^2=10\,000\text{ cm}^2$; .