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Risk Modelling and Survival Analysis · Graduation and graduation tests

Serial Correlation and Smoothness Tests in Graduation

Updated 11 October 2026 · Fact-checked

The serial correlation test checks whether standardised deviations between crude and graduated rates cluster in runs of the same sign (clumping). The smoothness test checks that third differences of graduated rates are small and change progressively. Fit and smoothness are tested separately. A good graduation must pass both kinds of check.

Understand Serial Correlation and Smoothness Tests

A graduation replaces noisy crude rates with smooth rates. Two things must hold. The graduated rates must stay close to the data (adherence). They must also be smooth. Tests of fit and tests of smoothness check these two aims separately.

The chi-squared, signs and runs tests look at the deviations between crude and graduated rates. They can all be passed while the deviations are still clumped. Clumping means several positive deviations in a row, then several negative ones. That pattern shows the graduated curve sits above the data in one age range and below it in another. The curve is then biased locally even if the overall fit looks fine.

The serial correlation test targets this. You take the individual standardised deviations in age order and measure their correlation with the next age's deviation (lag 1). If the deviations are independent, this correlation is close to zero. A large positive value means clumping. You compare the statistic with a normal critical value, one-tailed, because clumping gives positive correlation.

The smoothness test is different. It does not use the data at all. It looks only at the graduated rates. A smooth curve is well approximated locally by a low-degree polynomial, so its third differences should be small and change gradually. You compute the third differences of the graduated rates and judge them against the size of the rates. There is no fixed critical value, so this is a judgement.

Use the tests together. Graduating by parametric formula or by a standard table usually gives smoothness automatically. Graduating by smoothing methods may need the smoothness check. Both need the serial correlation test to catch clumping.

Key rules to remember

Individual standardised deviation
z_x = (θ_x − E_x μ̂_x) ÷ √(E_x μ̂_x)
θ_x is actual deaths, E_x μ̂_x is expected deaths under the graduated rate. Under the null hypothesis z_x is approximately N(0,1).
Mean of deviations
z̄ = (1/m) Σ z_x
m is the number of ages. Sum over all m ages.
Lag-1 serial correlation coefficient
r₁ = [ (1/(m−1)) Σ_{x=1}^{m−1} (z_x − z̄)(z_{x+1} − z̄) ] ÷ [ (1/m) Σ_{x=1}^{m} (z_x − z̄)² ]
Numerator has m−1 terms, denominator has m terms. If the question gives a formula, use that version.
Test statistic
√m × r₁ ~ N(0,1) approximately under H₀
Large m is needed for the approximation. Reject independence if the value is too large and positive (one-tailed). Check the question for the form to use.
Third difference
Δ³μ_x = μ_{x+3} − 3μ_{x+2} + 3μ_{x+1} − μ_x
Built by repeated differencing: Δμ_x = μ_{x+1} − μ_x, then Δ² = Δ(Δμ), then Δ³.
Smoothness criterion
Δ³μ̂_x small relative to μ̂_x and changing progressively with x
A judgement test with no tabulated critical value.

How to solve Serial Correlation and Smoothness Tests questions

Decide first which test is asked for. Serial correlation needs crude data and expected deaths. Smoothness needs only the graduated rates.

  1. 1Identify the test: clumping of deviations means serial correlation. Smoothness of graduated rates means third differences.
  2. 2For serial correlation, compute z_x for each age. Use actual deaths minus expected deaths, divided by √(expected deaths), unless z_x values are given.
  3. 3Find z̄ and the sum of (z_x − z̄)². If z̄ is about 0, state that and simplify.
  4. 4Compute the lag-1 products (z_x − z̄)(z_{x+1} − z̄) for x = 1 to m−1 and add them. Divide by m−1 for the numerator. Divide the squared sum by m for the denominator. Then r₁ = numerator ÷ denominator.
  5. 5Compute √m × r₁. State H₀: deviations are independent. Compare with the one-tailed normal critical value (1.645 at 5%). Conclude in words about clumping.
  6. 6For smoothness, list the graduated rates in age order. Compute first, second, then third differences, taking care with signs.
  7. 7Compare the third differences with the rates. Check whether they are small and move progressively. State your conclusion, and note any sudden jump.
  8. 8Finish with a comment: what the result means for the graduation, and whether another graduation or method should be considered.

Quickest way: Shortcut for serial correlation and third differences

When to use it: Under time pressure in a numerical question with a small table of ages.

  1. Write the z_x values in a column. Check whether their sum is close to 0; if so, treat z̄ as 0 only if the question allows it, and say so.
  2. Make one column for z_x² and one for z_x × z_{x+1}. Add both columns.
  3. Remember the counts: m−1 products, m squares. Missing this is the usual slip.
  4. For differences, subtract in a ladder: write Δ, then Δ², then Δ³ in neighbouring columns. Check that third differences sum to μ_{x+3} − 3μ_{x+2} + 3μ_{x+1} − μ_x for one entry.
  5. Look at the sign pattern of the z_x before the arithmetic. Long runs of the same sign suggest clumping and give you a sense check for the sign of r₁.

Common mistakes in Serial Correlation and Smoothness Tests

  • Dividing both parts of r₁ by the same number

    The numerator has m−1 products and the denominator has m squares, and students use m or m−1 for both.

    Fix: Use m−1 for the lag-1 products and m for the squares, unless the question's formula says otherwise.

  • Using a two-tailed critical value

    Students default to ±1.96 for any normal test.

    Fix: Clumping gives positive correlation, so use a one-tailed test: reject only if √m × r₁ is too large and positive, as stated in the question or your working.

  • Forgetting to standardise the deviations

    Raw differences between actual and expected deaths are easier to compute.

    Fix: Divide by √(expected deaths) to get z_x. State this in your answer.

  • Using the wrong sign pattern in third differences

    The coefficients 1, −3, 3, −1 are applied in the wrong order.

    Fix: Compute differences stepwise: first, second, then third. Check one third difference with the coefficient formula.

  • Claiming a smoothness test has a critical value

    Other graduation tests give a p-value or table value.

    Fix: Say that smoothness is judged by third differences being small and progressive. Give a comparison with the size of the rates.

  • Concluding that a graduation is good because it passes serial correlation

    Students treat one test as complete.

    Fix: State that serial correlation tests only clumping. Fit, bias and smoothness need their own tests.

Worked examples

Example 1

A graduation covers six consecutive ages. The individual standardised deviations in age order are 1.2, 0.8, 0.5, −0.4, −0.9, −1.2. Carry out a lag-1 serial correlation test at the 5% level (one-tailed, critical value 1.645) for clumping of deviations.

Show the solution
  1. H₀: the deviations are independent. H₁: positive serial correlation (clumping). m = 6.
  2. Mean: 1.2 + 0.8 + 0.5 − 0.4 − 0.9 − 1.2 = 0, so z̄ = 0.
  3. Sum of squares: 1.44 + 0.64 + 0.25 + 0.16 + 0.81 + 1.44 = 4.74. Denominator = 4.74 ÷ 6 = 0.79.
  4. Lag-1 products: 1.2 × 0.8 = 0.96; 0.8 × 0.5 = 0.40; 0.5 × (−0.4) = −0.20; (−0.4) × (−0.9) = 0.36; (−0.9) × (−1.2) = 1.08. Sum = 2.60.
  5. Numerator = 2.60 ÷ 5 = 0.52. So r₁ = 0.52 ÷ 0.79 = 0.658.
  6. Test statistic = √6 × 0.658 = 2.449 × 0.658 = 1.612.
  7. 1.612 < 1.645, so we do not reject H₀ at 5%.

Answer: The statistic is 1.612, just below 1.645, so there is no significant evidence of clumping at the 5% level. It is borderline, and the runs of three positive then three negative deviations look clumped. With only six ages the test has little power, so you would also look at the pattern of signs.

Example 2

Graduated mortality rates for ages 50 to 55 are 0.0040, 0.0044, 0.0049, 0.0056, 0.0063, 0.0071. Compute the third differences and comment on smoothness.

Show the solution
  1. First differences: 0.0004, 0.0005, 0.0007, 0.0007, 0.0008.
  2. Second differences: 0.0001, 0.0002, 0.0000, 0.0001.
  3. Third differences: 0.0002 − 0.0001 = 0.0001; 0.0000 − 0.0002 = −0.0002; 0.0001 − 0.0000 = 0.0001.
  4. Check the first with the formula: 0.0056 − 3 × 0.0049 + 3 × 0.0044 − 0.0040 = 0.0056 − 0.0147 + 0.0132 − 0.0040 = 0.0001.
  5. Compare with the rates, which lie between 0.0040 and 0.0071. The third differences are at most 0.0002 in size, a small fraction of the rates.

Answer: The third differences are 0.0001, −0.0002 and 0.0001. They are small relative to the rates, so the graduation is reasonably smooth. The small sign changes are of the order of rounding at 4 decimal places, but you would note that they are not fully progressive.

Exam tips

  • Read the question for which test is wanted. A smoothness question gives only the graduated rates, and a serial correlation question gives deviations or deaths.
  • Show the formula for r₁ and the counts (m−1 and m) before substituting. Method marks are often given for these.
  • Always finish with a conclusion in words: reject or not, and what it says about clumping or smoothness.
  • In written answers, explain why the chi-squared and signs tests can miss clumping. This is a frequent theory question.
  • For smoothness, give the third-difference calculation and a comparison with the rates. Do not invent a critical value.

Practice questions from Graduation and graduation tests

Serial Correlation and Smoothness Tests in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Serial Correlation and Smoothness Tests: frequently asked questions

What is clumping of deviations in graduation?

Clumping means that deviations between crude and graduated rates run in groups of the same sign over neighbouring ages. It shows that the graduated curve is above or below the data over a range of ages. Tests such as chi-squared can pass even when this happens.

Why do we use third differences to test smoothness?

A smooth curve is close to a low-degree polynomial over short age ranges. For such a curve, third differences are small. Large or erratic third differences suggest the graduated rates are not smooth.

Is the serial correlation test one-tailed or two-tailed?

It is normally one-tailed, because clumping produces positive correlation between neighbouring deviations. You reject independence if the statistic is large and positive. Follow the question if it states otherwise.

Does the smoothness test use the crude data?

No. It uses only the graduated rates. This is why it is separate from the tests of fit, which compare graduated rates with the data.