Observed against expected, with the expected count shown stratum by stratum.
Observed against what the reference rates predict
Four age strata, 45 deaths observed against 32.35 expected. SMR 1.391036 with an exact interval from 1.0146 to 1.8613 — the excess is real, and the lower limit sits only just above 1.
45 observed, 32.3500 expected across 4 strata
SMR = 1.391036
Exact 95% interval 1.0146 to 1.8613, p = 0.026142. That is an excess of 12.65 events against the reference population. The interval excludes 1.
SMR
1.391036
observed ÷ expected
Expected
32.3500
from the reference rates
95% interval
1.015 – 1.861
exact Poisson
Excess events
12.65
observed minus expected
Each stratum’s reference rate, person-time and expected event count
Stratum
Reference rate
Your person-time
Expected events
Share of expected
1
0.000800
10,000
8.0000
24.7%
2
0.002100
5,000
10.5000
32.5%
3
0.005500
1,500
8.2500
25.5%
4
0.014000
400
5.6000
17.3%
Total
16,900
32.3500
100%
The expected count is weighted by your cohort’s person-time, not the reference population’s. That is what makes indirect standardisation work — and what makes two SMRs from cohorts with different age structures non-comparable to each other, even when both are compared against the same reference.
Exact Poisson interval on the observed count SMR × 100 is the conventional presentation Two SMRs are each comparable to 1, not to each other
What this tool shows
Five deaths observed against 2.07 expected gives a normal-approximation p-value of 0.0417 — significant — and an exact 95% interval of 0.7843 to 5.6369, which includes 1. The two standard methods contradict each other at small counts, which is exactly where SMRs are usually computed. The tool reports both and builds the expected count stratum by stratum, because that denominator is where every real difficulty with an SMR lives.
The expected count built from stratum-specific reference rates and your own person-time
SMR with an exact Poisson confidence interval on the observed count
The normal-approximation p-value alongside it, so the disagreement is visible
Excess or deficit events in absolute terms, not only as a ratio
Each stratum’s contribution to the expected total, as a share
Why two SMRs are each comparable to 1 and not to each other
Indirect standardisation Exact Poisson CI Per-stratum expected Excess events
Each SMR is comparable to 1. Two SMRs are not comparable to each other.
Updated 12 September 2026 · Works in any browser, no installation
A standardised mortality ratio is the deaths observed in a cohort divided by the deaths the reference population’s rates would predict for that same cohort. An SMR of 1.39 means 39% more deaths than expected. Because the expected count uses the study cohort’s own age structure, the comparison to the reference is fair — and a comparison between two different cohorts’ SMRs is not.
At a glance
Formula shown
Expected = Σ (reference rate in stratum i × your person-time in stratum i). SMR = observed ÷ expected. The exact 95% interval divides the exact Poisson limits for the observed count by the expected: lower = χ²(0.975, 2O)/2 ÷ E and upper = χ²(0.025, 2(O+1))/2 ÷ E. The expected count is treated as fixed, which is the standard assumption and is why only the observed count carries uncertainty.
Scenario support
Occupational cohort mortality studies, comparing a workforce or a region against national rates, hospital and surgeon outcome monitoring, insurance and actuarial experience studies, and any setting where a group’s event count must be judged against what its own age and sex mix would predict.
Educational estimate
Planning support from the values you enter — not professional advice.
Two SMRs are each comparable to the reference, not to each other
This is the defining property of indirect standardisation and the most commonly violated one. It follows directly from how the expected count is built.
The expected count weights the reference rates by YOUR cohort’s person-time. Two cohorts with different age structures therefore use different weights, so their SMRs are ratios against differently-constructed baselines.
Which means two cohorts with identical stratum-specific mortality can report different SMRs, purely because one is older than the other. Nothing about their actual risk differs.
“Industry A has an SMR of 1.4 and industry B has 1.2, so A is more dangerous” does not follow. It follows only if the two workforces have the same age distribution, which they almost never do.
Directly standardised rates are the comparable alternative, because they apply each cohort’s own rates to one common standard population — the opposite direction of weighting, and the one that makes cohorts comparable at the cost of needing stable stratum-specific rates in every cohort.
At small counts the p-value and the interval disagree
The tool’s third preset is not a contrived example. It is a small occupational cohort of the kind SMRs exist for, and the two standard methods reach opposite conclusions about it.
Five observed against 2.07 expected: the normal approximation gives z = (5 − 2.07)/√2.07 = 2.0366 and p = 0.0417. Significant at the conventional level.
The exact Poisson interval on the same data runs 0.7843 to 5.6369. It includes 1, which by the usual reading means the cohort is not distinguishable from the reference.
The interval is the one to believe. The normal approximation to a Poisson count of five is poor, and it is anti-conservative here — it is claiming more evidence than the data contains.
The two converge as counts grow. The first preset, with 45 observed, has p = 0.0261 and an interval of 1.0146 to 1.8613 — both saying the same thing, because forty-five events is enough for the approximation to work.
Everything difficult about an SMR is in the expected count
The division is trivial. Choosing what to divide by is the entire analysis, and the tool prints every stratum’s contribution for that reason.
Strata must be fine enough that the rate is genuinely constant within each. Mortality rises steeply with age, so five-year bands are conventional and twenty-year bands smuggle in exactly the confounding standardisation exists to remove.
The reference population must be the right one. National rates against a regional cohort import every regional difference — deprivation, ethnicity, urbanicity — into the SMR as if it were the exposure.
The calendar period has to match. Mortality rates fall over time, so 1990 reference rates applied to 2020 person-time manufacture an apparent deficit out of nothing but progress.
And the person-time must be the time actually at risk, which is the same requirement as for any incidence rate and is where most data errors enter.
The per-stratum table makes a dominant stratum visible. When one age band supplies most of the expected count, the whole SMR is effectively a statement about that band, and the reference rate for it had better be right.
The healthy worker effect pulls occupational SMRs below 1
Occupational cohorts routinely report SMRs below 1 for all-cause mortality even in genuinely hazardous industries, and it is a selection effect rather than a protective one.
People well enough to hold a job are healthier than the general population, which includes the permanently sick and disabled. The comparison is not like for like.
The effect is strongest early in follow-up and fades over decades, as the initial selection wears off — which means an SMR computed over a short follow-up is more biased than one over a long one.
An all-cause SMR of 0.85 in a hazardous industry can conceal a real excess, because the baseline it is compared against is too high.
Cause-specific SMRs are the usual response. The healthy worker effect acts most strongly on cardiovascular and general mortality, so a cause-specific excess can stand out against an all-cause deficit in the same cohort.
Comparing against another working population avoids it entirely, at the cost of a reference whose own rates are less stable.
Screening many groups will find excesses that are not there
SMRs are often computed for every region, hospital, occupation or facility in a dataset, and then the high ones are investigated. That procedure has a false-positive rate nobody usually states.
Testing 100 groups at the 5% level produces about five apparent excesses by chance, even when every group has exactly the reference rate.
Small groups produce the most extreme SMRs in both directions, because a small expected count makes the ratio volatile. Ranking by SMR therefore ranks partly by smallness.
A funnel plot is the standard remedy — plotting each SMR against its expected count with control limits that widen as the count falls, so a small unit is not flagged for the same SMR that would flag a large one.
Shrinkage estimators are the other standard remedy, pulling small-group SMRs toward the overall mean by an amount that depends on how little data supports them.
A single pre-specified SMR needs none of this. The problem arises entirely from computing many and looking at the largest.
Reporting an SMR
Four things, and the first two are what make the ratio reconstructable.
Give observed and expected separately. An SMR of 2.0 from 4 observed and 2 expected and one from 400 and 200 are not the same finding, and the ratio alone hides which you have.
Name the reference population and the calendar period. The SMR is a statement about that comparison and means nothing without it.
Give the exact confidence interval, not only a p-value. They disagree below about ten events, and the interval carries the magnitude as well as the evidence.
And say how many SMRs you computed. One pre-specified comparison and the largest of fifty are different claims, and only the first means what a p-value says it means.
Sources and methodology
References for standardised mortality ratios and their intervals.
Method. The expected count is accumulated stratum by stratum from the reference rates and the cohort’s own person-time, and each stratum’s contribution is printed, so a dominant band is visible rather than buried in a total. The confidence interval is the exact Garwood interval on the observed count divided by the expected, treating the expected count as fixed — the standard assumption. Both the exact interval and the normal-approximation p-value are computed, which is what makes the small-count disagreement a measurement rather than a caution: 5 observed against 2.07 expected gives p = 0.0417 and an interval of 0.7843 to 5.6369, while 45 against 32.35 gives p = 0.0261 and 1.0146 to 1.8613, where the two agree. A zero or negative expected count, a negative observed count and mismatched stratum lists all return no result rather than a number. That engine is verified on every change against 115 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Incidence RateEvents per person-time with the exact Poisson interval, including at zero events: the exact two-sided upper limit is 3.688879 per unit of person-time, not the 3 the one-sided rule of three gives.
Poisson DistributionPoisson probabilities with a dispersion test against your own variance — because a Poisson forces variance to equal the mean, and real count data usually does not, which is exactly where the tail goes wrong.
Relative RiskRisk ratio and odds ratio from one table with the divergence between them plotted: they agree to half a percent at a 1% baseline, and at an 80% baseline the odds ratio is exactly half the risk ratio.
Attributable RiskRisk difference, attributable fraction and population attributable fraction, which rank risk factors in opposite orders: RR 9 at 1% exposure gives a PAF of 7.41%, RR 1.2 at 80% gives 13.79%.
Confidence IntervalIntervals for a mean or a proportion using t at every sample size and Wilson rather than the textbook Wald formula — with both methods shown, because Wald returns [0,0] at zero successes.
p-valueA p-value from a t or z statistic, one- or two-tailed — with a panel that holds an effect fixed and grows the sample, so you can watch significance appear from nothing but n.
An educational tool, not medical or occupational-health advice. An SMR is only as good as the reference rates and strata behind its expected count, and two SMRs from cohorts with different age structures cannot be compared with each other — occupational cohorts in particular run below 1 through the healthy worker effect rather than through any protective exposure.
Published an SMR tool that builds the expected count stratum by stratum from reference rates and the cohort's own person-time, and prints each stratum's share — because that denominator is where every real difficulty with an SMR lives.
Found and shipped a disagreement between the two standard methods at the counts SMRs are usually computed at: 5 observed against 2.07 expected gives a normal-approximation p of 0.0417, which is significant, and an exact Poisson interval of 0.7843 to 5.6369, which includes 1. The interval is the one to trust; the approximation is anti-conservative there.
Showed the two converging where they should: 45 observed against 32.35 expected gives p = 0.0261 and an interval of 1.0146 to 1.8613, both saying the same thing.
Recorded the property that makes SMRs so often misused — the expected count is weighted by each cohort's OWN age structure, so two SMRs are each comparable to 1 and not to each other, and two cohorts with identical stratum-specific mortality can report different SMRs.
Verified that the SMR interval is exactly the Poisson interval on the observed count rescaled, at six different observed counts, and that expected events scale exactly linearly with person-time.
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