The story of the two districts: ratios, percentages and why raw numbers lie

Asha works in a provincial health office. On Monday morning her director slides two reports across the desk. In the hills district, the clinic recorded 120 cases of a stomach illness this month. In the river district, it recorded 200. “Send the extra medicine to the river district,” says the director. “They are clearly worse.”

Asha pauses. She has learned to ask a question before she answers a number: out of how many people? It is a small habit, and it is the beginning of nearly all useful maths.

The question that changes everything

She looks up the populations. The hills district has 40,000 people. The river district has 100,000. Now she can compare fairly. In the hills, 120 cases among 40,000 people is 3 cases for every 1,000 people. In the river district, 200 cases among 100,000 is 2 cases for every 1,000 people.

The district with fewer cases has the higher rate. The extra medicine, it turns out, belongs in the hills. A raw count had pointed in the wrong direction, because the two districts were not the same size.

What a ratio really is

A ratio compares two quantities by dividing one by the other. “3 per 1,000” is a ratio. So is “one teacher for every 25 students”, “5 litres of water per minute” and “1 centimetre on the map for every 50,000 centimetres on the ground”. Each tells you how much of one thing comes with a given amount of another.

• Per capita means per person. Income per capita, or cases per capita, lets you compare places of different sizes.
• A scale such as 1 to 50,000 is a ratio. It explains how a map shrinks the world.
• Speed is a ratio of distance to time. Density is a ratio of mass to volume, or people to area.

Percentages: ratios dressed in a hundred

A percentage is a ratio where the second number is 100. “15 percent” means 15 out of every 100. Asha uses it constantly, because it is a language that everyone in a meeting understands.

Here is how she checks a claim. A clinic reports that 3 of its 20 patients missed a follow-up visit, and another says 30 of 400 did. The first number, 3, is much smaller, so it seems better. Turn both into percentages. Three out of 20 is 15 percent. Thirty out of 400 is 7.5 percent. The second clinic is, in fact, twice as good at keeping patients.

The trap: percent change versus percentage points

A month later, a newspaper reports that vaccination coverage rose from 10 percent to 15 percent. Two reporters write two headlines. One says coverage “rose by 5 percent”. The other says it “rose by 50 percent”. Both are correct, and they mean different things.

• Percentage points compare the two percentages directly: 15 minus 10 is 5 percentage points.
• Percent change compares the new value with the old one: an increase of 5 on a base of 10 is a 50 percent increase.

When someone says a number rose by a percent, ask whether they mean points or relative change. The difference can turn a modest improvement into a dramatic-sounding one.

Percent of what? The base matters

Asha keeps a second habit. When she sees a percentage, she looks for the base. If a district’s school enrolment “fell by 20 percent”, was that 20 percent of 50 pupils or of 5,000? A 20 percent change of a tiny group can be one family moving away. The same percentage of a large group is a crisis.

Percentages also cannot be added carelessly. A price that rises 10 percent and then falls 10 percent does not return to where it began. Start at 100: up 10 percent gives 110. Down 10 percent from 110 removes 11, leaving 99.

Where this shows up in AI, maps and 3D

• Model accuracy is a percentage. A model that is “95 percent accurate” on a test where 95 percent of the cases are easy tells you very little. Ask: 95 percent of what?
• Map scale is a ratio, and so is image resolution: a pixel covers a fixed number of metres on the ground.
• Heat maps and choropleth maps colour areas by a rate, not a count. A map of raw counts mostly shows where people live.
• A 3D model’s scale, for example 1 unit equals 1 metre, is a ratio. Get it wrong and a building appears the size of a toy.
• Cost and performance: a faster model that is 10 percent cheaper per query is a different claim from one that is 10 percentage points more accurate.

The formulas behind the story

Asha’s habits are three short formulas. Write them once, and you can check any report in seconds.

In the story: the hills district gives (120 ÷ 40,000) × 1,000 = 3 per 1,000, and the river district gives (200 ÷ 100,000) × 1,000 = 2 per 1,000. The smaller count has the larger rate.

In the story: coverage moved from 10% to 15%. The change in points is 15 − 10 = 5 points. The percent change is (15 − 10) ÷ 10 × 100 = 50%. Same data, two honest numbers with different meanings.

In the story: 100 × 1.10 × 0.90 = 99. Percentages multiply. They do not cancel.

Try it yourself

1. 01Find a news report that gives two counts, for example cases, crimes or accidents in two places.
2. 02Look up the populations, and turn each count into a rate per 1,000 or 100,000.
3. 03Check whether the ranking changes.
4. 04Find a percentage in the same report, and work out the base it applies to.

A short checklist for any number

• Is it a count or a rate?
• If it is a percentage, a percentage of what?
• If it changed, is the change in points or in percent?
• Is the comparison fair: same period, same group size, same definition?
• Would the answer change if I used a different base?

Frequently asked questions

Why do people use per 1,000 or per 100,000 instead of percentages?

Because many events are rare. “0.003 percent” is hard to picture, while “3 per 100,000” is easier. They are the same ratio in different clothes.

Is a bigger percentage always worse or better?

No. It depends on what is being measured and on the base. Fifteen percent unemployment is serious, but 15 percent growth in clean energy is welcome.

How do I turn a ratio into a percentage?

Divide the first number by the second, and multiply by 100. For 3 out of 20, that is 3 divided by 20, which is 0.15, which is 15 percent.

What is a base rate?

It is how common something is to begin with. Ignoring it is a classic way to misread numbers, and the next story in this path shows how.

Asha’s director sent the medicine to the hills. Nothing about the decision needed advanced mathematics. It needed one question, asked at the right moment: compared with what? Most good analysis starts there.