Appendix 4 - Ratios, percentages and hazard scales
Use Appendix 4 numerical skills to compare quantities, interpret change and understand hazard probabilities. Learn to check the denominator, units and scale before deciding what a number means.
Units, ratios and proportions
A number without its unit can be ambiguous: rainfall of 20 mm is not the same quantity as a river level of 20 cm. Convert measurements to compatible units before combining them. There are 100 cm in 1 m and 1000 m in 1 km. Because area has two dimensions, 1 km² is m². A 2 km by 3 km rectangle is 6 km², or 6,000,000 m².
A ratio compares quantities in order. If a sample contains 12 rounded and 8 angular pebbles, rounded:angular is 12:8, simplified to 3:2. The rounded proportion of the whole is 12/(12+8) = 0.6. Three rounded pebbles for every two angular pebbles is not the same as three fifths being angular: preserve the category order.
Ratios can compare different units, such as people per square kilometre, but state them clearly. A density of 500 people/km² is a population divided by an area; it is not a population count. A change in area boundary can change density even if no one moves. When a calculation produces an implausible result, check conversions and the denominator before inventing a geographical explanation.
Percentages and changes in shares
A percentage expresses a proportion out of 100. If 24 of 80 respondents use a bus, the bus share is . Compare percentages only when you know their denominators: 30% of a small village can be fewer people than 5% of a city.
The original value is the baseline. A rise from 40 to 50 is .
A fall from 50 back to 40 is 20%, because the baseline is now 50. Equal numerical changes need not have equal percentage changes. If the baseline is zero, this ordinary percentage-change formula cannot be used: report the absolute increase and explain the starting point instead.
Distinguish percentage points from percentage change. If the share of journeys by bus rises from 20% to 30%, it increases by 10 percentage points. Relative to the original 20%, it increases by 50%. A headline saying only “up 10%” hides which calculation was intended. A share can increase even when its count falls, if the overall total falls faster.
Frequency and annual hazard chance
Frequency is how often an event occurs in a stated interval or dataset. Probability describes the chance of an outcome under stated conditions. An event described as having a 1:200 annual chance has probability , or 0.5%, in a particular year under the model's assumptions.
“One in 200 years” is often used as shorthand for a return period, but it is not a timetable. A flood can happen in consecutive years; a previous event does not use up the next year's chance. A long-run average interval does not promise a flood-free gap. Nor does a 0.5% annual chance mean only 0.5% chance during a whole lifetime: exposure accumulates across years. That longer-period calculation requires assumptions about whether annual chances remain constant and independent.
Hazard probabilities may change as rainfall, land use, sea level or defences change. They are estimates from evidence/models, not exact guarantees. A map showing probability also needs information about exposure and vulnerability before it can describe likely human losses. Two sites can face the same flood chance but very different consequences.
Logarithmic scales and orders of magnitude
On an ordinary linear scale, equal spacing represents equal additions, such as 10, 20, 30. On a base-ten logarithmic scale, equal spacing represents equal multiplication: 1, 10, 100, 1000. From 10 to 100 is a tenfold increase, one order of magnitude; from 10 to 1000 is a hundredfold increase, two orders. A graph can therefore show a large range without squeezing its smaller values into the baseline.
Earthquake magnitude illustrates why the scale matters. In the traditional Richter local-magnitude comparison, one whole magnitude unit corresponds to ten times the measured wave amplitude under the scale's correction rules. A difference of two units corresponds to times amplitude, not twice. Energy is a different quantity: approximately 32 times per magnitude unit, so do not label the tenfold amplitude factor “ten times energy”. Modern reporting often uses moment magnitude rather than the original local Richter scale; neither is a direct count of deaths or building damage.
Always read the quantity and key. A straight line on a logarithmic vertical axis can represent multiplicative growth, not a constant annual numerical addition. Positive values are required for an ordinary logarithmic axis: zero cannot simply be plotted at the first tick. When comparing a numerical claim, state the factor, units, source date and uncertainty that the evidence supports.