Risk Modelling and Survival Analysis · Graduation and graduation tests

Methods of Graduation: Parametric, Standard Table and Graphical

Updated 11 October 2026 · Fact-checked

Graduation smooths crude mortality rates into a regular set of rates. You can fit a parametric formula such as Gompertz, Makeham or GM(r,s), adjust a standard table by an age shift or linear transformation, or draw a curve by eye. Pick the method, estimate its parameters, then test the fit.

Understand Methods of Graduation

Crude mortality rates, such as q̂x or μ̂x, come from observed data. They jump around from age to age because of random variation. True mortality is expected to change smoothly with age. Graduation replaces the crude rates with smooth rates that still reflect the data. It also lets you borrow strength from neighbouring ages.

There are three methods you must know. Parametric graduation chooses a mathematical formula for μx and estimates its parameters from the data. Graduation by reference to a standard table takes an existing published table and adjusts it to fit your data. Graphical graduation plots the crude rates and draws a smooth curve through them by eye.

In parametric graduation, the Gompertz law is μx = B c^x. The Makeham law adds a constant for age-independent causes of death: μx = A + B c^x. Both are special cases of the GM(r,s) family, which has r polynomial terms plus an exponential of a polynomial with s terms. Gompertz is GM(0,2) and Makeham is GM(1,2). You estimate parameters by maximum likelihood or by minimising a weighted sum of squared differences between crude and fitted rates.

In standard table graduation, you assume your group's mortality has the same shape as the standard table, but at a different level. The simplest version is an age shift: qx = q^s_{x+k}. If k is positive, your group has heavier mortality than the standard, because it behaves like a population that is k years older. A more flexible version is a linear transformation: qx = a + b q^s_{x+k}, or the same form with μ. You choose a, b and k to fit your data.

Graphical graduation is simple and needs no model. The crude rates are usually plotted on a log scale. But it is subjective, hard to reproduce and gives no formal measure of fit. It is rarely used on its own for important work. Whichever method you use, you must then test the graduation (see the graduation tests topics).

Key rules to remember

Gompertz law: μx = B c^x (equivalently ln μx = ln B + x ln c); Two parameters. Same as GM(0,2) with B = e^β0 and c = e^β1.
Makeham law: μx = A + B c^x; Three parameters. Same as GM(1,2). The constant A covers age-independent mortality.
GM(r,s) family: μx = Σ (i = 0 to r−1) αi x^i + exp( Σ (j = 0 to s−1) βj x^j ); r counts the polynomial terms and s counts the exponent terms. Total parameters = r + s.
Age shift to a standard table: qx = q^s_{x+k} or μx = μ^s_{x+k}; One parameter k. If k > 0 your group has heavier mortality than the standard table.
Linear transformation of a standard table: qx = a + b q^s_{x+k} or μx = a + b μ^s_{x+k}; Parameters a, b and optionally k. With a = 0 and b > 1, mortality is heavier than the standard at every age.
Degrees of freedom after graduation: d.f. = number of age groups − number of parameters estimated; Used in the chi-square test. This applies to parametric and standard table graduations, where parameters are fitted to the data.

How to solve Methods of Graduation questions

Use this method for any question on choosing or applying a graduation method.

1Identify the data and the aim. Note whether you are given crude rates, a formula, or a standard table, and whether the question asks you to calculate or to discuss.
2Choose the method. Use a parametric formula if a known law suits the age range. Use a standard table if the data are sparse and a similar population has a reliable table. Use graphical graduation only if the question asks for a quick or exploratory approach.
3Write the model in standard notation, for example μx = A + B c^x or qx = a + b q^s_{x+k}. State what each symbol means.
4Estimate or apply the parameters. For a calculation, substitute the given values at the required age and keep the shift k consistent with x + k.
5Check the result is sensible. Rates must lie between 0 and 1 for qx, be positive for μx and increase with age over adult ages. Compare with the standard table or crude rate to say whether mortality is lighter or heavier.
6For discussion parts, give the advantages and disadvantages: smoothness, number of parameters, reliance on data, subjectivity and ease of testing.
7State that the graduation must be tested for goodness of fit and smoothness. Mention that the degrees of freedom fall by the number of fitted parameters.

Quickest way: Fast route for calculation questions

When to use it: Use this when the question gives a formula or standard table values and asks you to find a graduated rate or a parameter.

Write the formula first. Mark which symbols are known.
For Gompertz or Makeham, take ratios of μ at two ages to remove B. For Gompertz, μ(x+n) ÷ μx = c^n.
For standard table questions, find the row x + k first, then apply a + b × (value).
Compare with the standard value at the same age x to state heavier or lighter.
Check the answer is a valid rate before moving on.

Common mistakes in Methods of Graduation

Using the standard table value at age x instead of x + k.
Students read the same age row out of habit.
Fix: Write the subscript x + k with the actual number, for example 60 + 2 = 62, before looking up the table.
Getting the direction of the age shift wrong.
It is easy to think a positive k means a younger or lighter population.
Fix: If qx = q^s_{x+k} with k > 0, your group at age x has the mortality of the standard at an older age, so it is heavier.
Calling Makeham GM(0,3) or Gompertz GM(1,1).
The r and s roles are mixed up, and s is confused with the exponent degree.
Fix: r is the number of polynomial terms and s is the number of terms in the exponent polynomial. Gompertz is GM(0,2) and Makeham is GM(1,2).
Forgetting to reduce the degrees of freedom for the fitted parameters.
Students treat graduation and testing as separate problems.
Fix: When parameters are estimated from the data, subtract their number from the number of age groups. Do this before reading the chi-square table.
Saying graphical graduation has no disadvantage because it is simple.
Students only remember that it needs no formula.
Fix: State that it is subjective, hard to reproduce and gives no formal fit, so the result still needs testing.
Treating a graduated rate as automatically correct because it is smooth.
Smoothness is confused with accuracy.
Fix: Smoothness and fit are different aims. A graduation can be smooth but far from the data, so you must test both.

Worked examples

Example 1

A Gompertz law μx = B c^x is fitted to a group. Fitted values are μ50 = 0.002 and μ60 = 0.005. (a) Find c. (b) Find μ70. (c) Write the Gompertz law as a GM(r,s) formula and state r and s.

Show the solution

Divide: μ60 ÷ μ50 = c^10 = 0.005 ÷ 0.002 = 2.5.
So c = 2.5^(1/10). ln 2.5 = 0.916291, so ln c = 0.0916291 and c = 1.0960 (to 4 d.p.).
For μ70: μ70 = μ60 × c^10 = 0.005 × 2.5 = 0.0125.
Check with μ50: 0.002 × 2.5² = 0.0125. This agrees.
Take logs: ln μx = ln B + x ln c. So μx = exp(β0 + β1 x) with β0 = ln B and β1 = ln c.
This has no polynomial part, so r = 0, and two exponent terms, so s = 2.

Answer: (a) c ≈ 1.0960. (b) μ70 = 0.0125. (c) Gompertz is GM(0,2): μx = exp(β0 + β1 x), with r = 0 and s = 2.

Example 2

A pension scheme's mortality is graduated by reference to a standard table using qx = 0.0005 + 1.1 q^s_{x+2}. The standard table gives q^s_62 = 0.0120 and q^s_60 = 0.0090. (a) Find the graduated q60. (b) State whether the scheme's mortality at age 60 is heavier or lighter than the standard table, with a reason. (c) Find the expected number of deaths among 2,000 lives aged exactly 60 over one year.

Show the solution

Use x = 60, so x + k = 62 with k = 2.
Substitute: q60 = 0.0005 + 1.1 × 0.0120 = 0.0005 + 0.0132 = 0.0137.
Compare with the standard at the same age: q^s_60 = 0.0090. Since 0.0137 > 0.0090, the graduated rate is higher.
Expected deaths = 2,000 × 0.0137 = 27.4.

Answer: (a) q60 = 0.0137. (b) Heavier, because 0.0137 exceeds the standard rate of 0.0090 at age 60. (c) 27.4 expected deaths.

Exam tips

Know the formulas for Gompertz, Makeham and GM(r,s) exactly, including how r and s are defined. Questions often ask you to identify the GM(r,s) form of a law.
In written answers, name the method and give one advantage and one disadvantage of it. Typical points are number of parameters, data needs, subjectivity and ease of testing.
For standard table questions, show the age x + k row explicitly. This is where marks are gained and lost.
Link graduation to testing. State that the degrees of freedom fall by the number of fitted parameters.
On computer-based Paper B questions, state the model, fit the parameters in R or Excel, and show the fitted rates beside the crude rates. Say what the fit shows.

Practice questions from Graduation and graduation tests

Methods of Graduation: frequently asked questions

What is the difference between parametric and standard table graduation?

Parametric graduation fits a formula such as Gompertz or Makeham directly to your data. Standard table graduation adjusts an existing table, using an age shift or linear transformation, to match your data. The standard table method needs fewer parameters but depends on the standard table being similar to your group.

How do I graduate by reference to a standard table using an age shift?

Set qx = q^s_{x+k} and choose k so the graduated rates fit your data. To find the graduated rate at age x, read the standard table at age x + k. If k is positive, your group has heavier mortality than the standard.

Is the Makeham law the same as GM(1,2)?

Yes. GM(1,2) has one polynomial term, which is a constant, and an exponential of a two-term polynomial. That gives μx = α0 + exp(β0 + β1 x), which is the Makeham form with A = α0, B = e^β0 and c = e^β1.

Do I need to test a graduation after using any of these methods?

Yes. Graduation gives smooth rates, but you must still test whether they fit the data and whether they are smooth. That includes graphical graduation. Use the chi-square and other tests from the graduation tests topics.