Risk Modelling and Survival Analysis · Graduation and graduation tests

Crude vs Graduated Estimates of Mortality Explained

Updated 11 October 2026 · Fact-checked

Crude mortality rates are raw estimates from observed deaths and exposure at each age. They are irregular because of random sampling variation. Graduation smooths them so they reflect the underlying true pattern. You must balance smoothness against adherence to data: too smooth ignores the data, too close keeps the random noise.

Understand Crude vs Graduated Estimates of Mortality

A crude rate is the raw estimate you get from data. At each age x, you divide observed deaths by the central exposed to risk, giving the crude estimate of the force of mortality, or by the initial exposed to risk, giving the crude estimate of the mortality rate q. Each age is estimated separately from its own data.

Crude rates are irregular because each one is a random variable. Deaths at each age are random, and the number of lives at each age is limited. So the crude rate at age x is the true rate plus sampling error. Ages with small exposure are the noisiest. The result is a series that jumps up and down from age to age.

The true underlying mortality is expected to be smooth. Mortality changes gradually with age, apart from special features such as the accident hump in young adult ages. So we graduate: we fit a smooth curve or adjust the crude rates to get graduated estimates that are our best view of the true rates.

The aims of graduation are to: remove random sampling errors so the underlying pattern shows; produce a smooth set of rates suitable for calculating premiums and reserves; and use information from neighbouring ages to improve the estimate at each age. Smooth rates also make sense in practice, since a premium that jumps between adjacent ages is hard to justify.

There is a trade-off. Smoothness means the graduated rates follow a regular pattern. Adherence to data means they stay close to the crude rates. A graduation that is too smooth may be biased and miss real features of the data. A graduation that follows the data too closely keeps the random noise. Good graduation finds a sensible balance, and graduation tests check both aspects.

Key rules to remember

Crude central rate: m̂x = θx ÷ Ex^c; θx = observed deaths aged x; Ex^c = central exposed to risk. This estimates the force of mortality μ, assumed constant over the year.
Crude initial rate: q̂x = θx ÷ Ex; Ex = initial exposed to risk. Use when exposure is measured at the start of the year of age.
Crude estimate as a random variable: crude rate = true rate + random error; Explains irregularity. Variance falls as exposure rises, so large groups give smoother crude rates.
Graduated estimates: graduated rate at x = fitted value from a smooth function of age; Written with a ˚ over the symbol, such as q̊x or μ̊x, to distinguish from crude estimates.

How to solve Crude vs Graduated Estimates of Mortality questions

Use this approach for any question that asks you to explain, compare or discuss crude and graduated rates.

1Identify what is crude and what is graduated. Crude rates come straight from deaths divided by exposure. Graduated rates have been smoothed or fitted.
2State why the crude rates vary: random sampling error, small exposure at some ages, and each age estimated independently.
3State the assumption that true mortality is smooth in age, and mention any real features, such as the accident hump, that should not be smoothed away.
4List the aims of graduation: remove random error, give smooth rates, and borrow strength from adjacent ages.
5Discuss the trade-off: smoothness against adherence to data, with the risk of each extreme.
6If numbers are given, compute crude rates, compare with the graduated rates, and comment on which ages differ and by how much.
7Link to checking: goodness of fit tests check adherence; smoothness tests check smoothness.
8Finish with a clear conclusion that answers the exact question asked.

Quickest way: Crude versus graduated in one pass

When to use it: Use for short MCQs or 2 to 4 mark written parts asking for reasons, aims or the trade-off.

Write: crude = raw, random, irregular; graduated = smoothed, best estimate of the true rates.
Give the cause: sampling error from limited exposure.
Give two aims: remove random error, produce smooth rates for practical use.
Give the trade-off: too smooth means poor adherence and bias; too close means noisy and not smooth.
If asked about tests, say goodness of fit tests check adherence and smoothness tests check smoothness.

Common mistakes in Crude vs Graduated Estimates of Mortality

Saying crude rates are irregular because the data are wrong or contain errors.
Students link irregularity to data quality rather than randomness.
Fix: Say the irregularity comes from random variation in the number of deaths, even with perfect data. Small exposure makes it worse.
Claiming graduated rates are always better because they are smoother.
Smoothness sounds like improvement.
Fix: Explain that over-smoothing can introduce bias and hide real features. A good graduation balances smoothness and adherence.
Saying the aim of graduation is to match the crude rates exactly.
Confusing graduation with fitting data perfectly.
Fix: Exact matching keeps the random error. The aim is to estimate the underlying true rates, so some departure from crude rates is expected.
Mixing up central and initial exposed to risk when computing crude rates.
Both are called exposure and appear in similar formulas.
Fix: Use θx ÷ Ex^c for the central rate and θx ÷ Ex for the initial rate. Check which one the question gives.
Forgetting that graduation uses neighbouring ages.
Students think each age is graduated independently, as in crude estimation.
Fix: State that the smooth curve links ages, so information from nearby ages improves the estimate at each age.
Smoothing away genuine features such as the accident hump.
Assuming all bumps are noise.
Fix: Note that mortality is smooth in general, but real patterns can exist. The graduation method should allow for them where the data support it.

Worked examples

Example 1

A mortality investigation records the following. Age 60: 12 deaths, central exposed to risk 1,500. Age 61: 9 deaths, central exposed to risk 1,200. Age 62: 20 deaths, central exposed to risk 1,600. (a) Calculate the crude central rates. (b) Explain why the pattern is irregular and what graduation would do.

Show the solution

Use m̂x = θx ÷ Ex^c.
Age 60: 12 ÷ 1,500 = 0.0080.
Age 61: 9 ÷ 1,200 = 0.0075.
Age 62: 20 ÷ 1,600 = 0.0125.
The rate falls from age 60 to 61 and then rises sharply to age 62. Mortality is expected to rise steadily with age at these ages, so the fall and the jump are not plausible as true rates.
The cause is random variation in deaths with limited exposure. Each age is estimated separately, so errors are not smoothed out.
Graduation would fit a smooth increasing curve through the three ages, using information from all of them. The age 61 rate would rise and the age 62 rate would fall relative to the crude values.

Answer: Crude central rates: 0.0080, 0.0075 and 0.0125. The irregularity is due to random sampling error. Graduation gives a smooth rising pattern.

Example 2

Discuss the trade-off between smoothness and adherence to data when graduating crude mortality rates.

Show the solution

Define the terms. Adherence to data means the graduated rates stay close to the crude rates. Smoothness means the graduated rates change gradually and regularly with age.
Explain the pull in each direction. Perfect adherence means using the crude rates themselves, which are irregular because of random error.
Perfect smoothness could be a very simple curve, such as a straight line, which ignores real shape in the data and so is biased.
State the consequence of too much smoothness: poor fit and biased estimates, so the rates do not represent the experience.
State the consequence of too much adherence: the rates keep random noise and are not smooth, so they are unsuitable for pricing and reserving.
Describe the solution: choose a method and a number of parameters, or a degree of smoothing, that gives a good fit while staying smooth. Then test it: goodness of fit tests for adherence, smoothness tests for smoothness.

Answer: Smoothness and adherence pull in opposite directions. Too smooth means bias and poor fit; too close to the data means noise. A good graduation balances both and is checked with fit and smoothness tests.

Exam tips

Always give the reason for irregularity as random sampling variation, and mention small exposure.
When asked to discuss the trade-off, name both extremes and the consequence of each. Marks are for both sides.
Show the crude rate formula and check whether the question uses central or initial exposure.
Link the topic to testing. Mention that fit tests check adherence and smoothness tests check smoothness.
For numerical comments, point to specific ages where the crude pattern is implausible.

Practice questions from Graduation and graduation tests

Crude vs Graduated Estimates of Mortality: frequently asked questions

Why are crude mortality rates not smooth?

Each crude rate is based on a limited number of deaths, which vary randomly. Each age is also estimated on its own. So the crude rates contain random error and jump around even if true mortality is smooth.

What is the purpose of graduating mortality rates?

Graduation removes random error, gives smooth rates and uses information from nearby ages. The smooth rates are better estimates of the true mortality and are suitable for premiums, reserves and other calculations.

What is the difference between smoothness and adherence to data?

Smoothness means rates change gradually with age. Adherence means the graduated rates stay close to the crude rates. Improving one usually weakens the other, so graduation seeks a balance.

Do graduated rates have to equal crude rates at each age?

No. Graduated rates are estimates of the underlying true rates, so they differ from the crude rates by some amount. The differences should be small and not follow a pattern, which is what graduation tests examine.