Ratio, proportion, rate, and the specific measures built from them
Nearly every number an epidemiologist reports is one of three basic frequency measures. Telling them apart is mostly about one question: is the numerator part of the denominator?
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Ratio
Compares any two values. The numerator does not have to be part of the denominator. You could compare apples to oranges.
ExampleDeath-to-case ratio: deaths ÷ new cases of a disease
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Proportion
Compares a part to the whole. The numerator is always included in the denominator. Often shown as a percentage.
ExampleCase-fatality rate: deaths from a disease ÷ everyone who had that disease
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Rate
Measures how frequently something happens in a population over a specific span of time, putting frequency in the context of population size and time.
Example70 new cases of a disease per 1,000 people per year
Fuzzy vocabulary, on purpose: Epidemiologists often call something a "rate" even when it's technically a proportion (like attack rate or prevalence "rate") because the language has stuck around for over a century. Don't let the name fool you: check whether time is actually part of the calculation.
Incidence: How Fast New Cases Appear
Incidence Proportion (Attack Rate)
New cases during a time periodPopulation at risk at the start
The probability that someone in the group develops disease during that window. In outbreaks, this is called the attack rate.
Incidence Rate (Person-Time)
New cases during a time periodTotal person-time observed
Used when people are followed for different lengths of time. It accounts for the fact that someone watched for 6 months contributes less "at-risk time" than someone watched for 3 years.
🧮 Worked Example: Adding Up Person-Time
A 12-month study follows 5 people to see who picks up a new infection. Each person counts only while they're still at risk: stop their clock the month they get sick, or when they leave the study.
Person
What happened
Months at risk
Ana
Healthy the whole year
12
Ben
Got sick in month 4
4
Cy
Moved away after month 6
6
Dee
Got sick in month 9
9
Eli
Already sick when the study began
0 (left out)
Total
2 new cases (Ben, Dee)
31
Incidence rate = 2 new cases ÷ 31 person-months = 0.065, or about 6.5 per 100 person-months. The three classic mistakes: giving Ben and Dee the full 12 months, counting Eli (he was never at risk, so he's neither a new case nor at-risk time), and dividing by 5 people instead of 31 person-months. Rates are usually scaled to a round number (per 100, per 1,000, per 100,000) so they're easy to read and compare; rarer events get the bigger base.
Which population goes on the bottom? For a rate over a whole year, use the average (midyear) population, since people are born, die and move during the year. Estimate it as (population at the start + population at the end) ÷ 2. A town of 137,000 on January 1 with 2,400 births and 2,000 deaths ends the year at 137,400, so its average population is 137,200; 538 new cases that year is 538 ÷ 137,200 × 10,000 ≈ 39 per 10,000 person-years.
🤧 Worked Example: The Sunnyvale Flu Watch
The school nurse tracks a class of 40 students through the month of January. On January 1, 2 students are already out sick with the flu (from before winter break). During January, 6 more students develop new flu symptoms. By January 31, some have recovered, and on that exact day, 3 students are currently sick.
Measure
Calculation
Result
Incidence proportion (attack rate)
6 new cases ÷ 38 students not already sick on Jan 1
≈ 15.8%
Point prevalence (on Jan 31)
3 currently sick ÷ 40 total students
7.5%
Period prevalence (all of January)
8 sick at any point (2 preexisting + 6 new) ÷ 40 total students
20%
Why these three numbers all differ: Incidence only counts new cases and only counts students who could still get sick. Point prevalence is a single-day snapshot of everyone currently ill, new or not. Period prevalence adds up everyone who was ever sick at any moment during the whole month, which is why it's the largest number here.
Prevalence: How Many Are Currently Affected
Point Prevalence
All current cases on one specific dateTotal population on that date
A single-moment snapshot: includes both new and pre-existing cases.
Period Prevalence
All cases present at any time in a periodTotal population during that period
A wider window: anyone ill at any point counts, even if they've since recovered.
Easy to mix up on purpose: Incidence = NEW cases only. Point prevalence = a snapshot of NOW, one specific day. Period prevalence = ANYONE sick during the WHOLE window, old or new. The math is simple fractions: the trap is picking the wrong one.
How incidence and prevalence connect: for a steady situation, prevalence ≈ incidence × average duration. Think of a bathtub: new cases pour in (incidence) and leave by recovery or death (duration). If incidence rises and duration stays the same, prevalence rises. A treatment that keeps people alive longer without curing them also raises prevalence, while a quick cure lowers it.
Quick check: click a measure, then click the scenario that fits it:
Measure
Scenario
0 of 3 matched
Mortality: Measuring Deaths
Measure
What it divides
Worked Example
Crude death rate
All deaths ÷ total population
45 deaths in a town of 30,000 → 1.5 per 1,000
Cause-specific death rate
Deaths from one specific cause ÷ total population
Deaths from heart disease ÷ population
Case-fatality rate
Deaths from a disease ÷ people who had that disease
4 deaths among 200 diagnosed → 2%
Mortality rate vs. case-fatality rate, with the same numbers: a city of 50,000 has 1,000 flu cases in January and 50 flu deaths. The flu mortality rate is 50 ÷ 50,000 × 1,000 = 1 per 1,000 (everyone in the city on the bottom). The case-fatality rate is 50 ÷ 1,000 = 5% (only the sick on the bottom).
Crude vs. age-adjusted rates: a crude rate uses the whole population as it is. Two places with different age mixes can't be compared fairly that way: an older population has more cancer and heart disease deaths just because of age. Age adjustment (age standardization) recalculates each place's rate as if both had the same standard age mix, so the comparison is fair.
Counting lost years State/Nats:Years of potential life lost (YPLL) adds up how early people died: each death before a cutoff age (often 75) counts (75 − age at death), so a death at 20 counts 55 years. It gives extra weight to deaths of young people. A disability-adjusted life year (DALY) is one lost year of healthy life: years of life lost to early death plus years lived with disability (each weighted from 0 for full health to 1 for death).
Percent change vs. percentage points: if a rate falls from 20% to 15%, it dropped 5 percentage points, but the percent change is (15 − 20) ÷ 20 × 100 = −25%. Percent change = (new − old) ÷ old × 100.
A death rate is a rate; a case-fatality rate is a proportion. The crude death rate's denominator is the whole population (most of whom never had the disease); the case-fatality rate's denominator is only the people who actually got sick.
Growth Rate & Doubling Time State/Nats
When an outbreak grows exponentially, N(t) = N0 × ert, where r is the growth rate per day. Two ways to get the doubling time Td:
From a graph or table: find how long it takes to go from any value to twice that value.
From two counts: r = ln(N ÷ N0) ÷ t, then Td = ln 2 ÷ r ≈ 0.693 ÷ r. Example: 120 active infections grow to 540 in 10 days. r = ln(4.5) ÷ 10 ≈ 0.150 per day, so Td ≈ 0.693 ÷ 0.150 ≈ 4.6 days.
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Q3Of 40 students, 6 develop new flu symptoms this month, out of 38 who weren't already sick. What is the incidence proportion?
6 ÷ 38 ≈ 15.8%. Incidence proportion (attack rate) only counts new cases divided by the population still at risk.
Q4Why is period prevalence usually a bigger number than point prevalence for the same group and disease?
Period prevalence counts everyone who was sick at any point during the whole time window, while point prevalence only counts who is sick on one single day, so period prevalence can only be equal to or larger than any single day's point prevalence.
Q5Out of 200 people diagnosed with a disease, 4 die from it. What is the case-fatality rate, and is it a true rate or a proportion?
4 ÷ 200 = 2%. It's technically a proportion (the numerator, deaths, is a subset of the denominator, people with the disease), even though "rate" is in its name.