Seven trials of the same intervention on a standardised scale. Switching to random effects moves the pooled estimate by 0.0009 — from 0.5151 to 0.5160 — and widens the interval by 78.2%, from 0.2599 to 0.4631. The estimate barely moves; how much you should trust it changes completely.
7 studies · random effects
0.5160 (95% CI 0.2844 to 0.7475)
z = 4.3677, p = 1.26e-5. The other model gives 0.5151 with an interval 78.2% narrower. τ² = 0.05000, so between-study variance is adding to every study’s own uncertainty.
Pooled effect
0.5160
SE 0.1181
I²
58.0%
CI 2.9–81.8%
τ²
0.05000
τ = 0.2236
Largest weight
23.3%
half the pool in 3
The prediction interval for the next study is -0.1341 to 1.1661 — 2.81× the width of the confidence interval above. It crosses no effect while the confidence interval does not: the average is positive, the next trial need not be.
Every study and what it is worth
Each study’s effect, interval and share of the pooled estimate
Study
Effect
95% CI
SE
Weight
Trial A
0.6100
0.4140 to 0.8060
0.1000
23.26%
Trial B
0.3700
0.0956 to 0.6444
0.1400
20.05%
Trial C
0.0700
-0.3024 to 0.4424
0.1900
16.21%
Trial D
0.9400
0.5088 to 1.3712
0.2200
14.18%
Trial E
0.2100
-0.3388 to 0.7588
0.2800
10.87%
Trial F
0.9100
0.2632 to 1.5568
0.3300
8.78%
Trial G
0.7900
0.0060 to 1.5740
0.4000
6.65%
Pooled
0.5160
0.2844 to 0.7475
0.1181
100%
Drop each study in turn
The pooled estimate recomputed with each study left out
Omitted
Pooled without it
Shift
I² without it
Trial A
0.4993
-0.0167
60.6%
Trial B
0.5562
+0.0402
61.2%
Trial C
0.5912
+0.0752
37.8%
Trial D
0.4444
-0.0716
50.9%
Trial E
0.5548
+0.0388
61.6%
Trial F
0.4781
-0.0379
60.9%
Trial G
0.4971
-0.0189
63.8%
A pooled result that changes materially when one study leaves is a result about that study.
On the seven trials shipped here, switching from fixed effect to random effects moves the pooled estimate by 0.0009 — from 0.5151 to 0.5160 — and widens the interval by 78.2%. The answer barely moves; how much you should trust it changes completely. The prediction interval for the next trial is wider still, and it crosses no effect while the confidence interval does not.
Inverse-variance pooling under both the fixed-effect and random-effects models
DerSimonian–Laird τ², with the two models shown against each other rather than one at a time
Cochran’s Q, I² and a confidence interval for I², which most calculators omit
A prediction interval for where the next study lands, not just where the average is
Per-study weights in percent, and how many studies hold half the pool
Leave-one-out influence: the pooled effect recomputed with each study dropped
Log-scale entry for odds ratios, risk ratios and hazard ratios, read back on the ratio scale
Both models Prediction interval Leave-one-out Weights in full
Pooling cannot repair biased inputs. It averages them.
Updated 13 September 2026 · Works in any browser, no installation
A meta-analysis is a weighted average in which precise studies count for more. The weight is one over the variance, which is the only weighting that minimises the variance of the result. Weighting by sample size instead is common and wrong: two trials of the same size can have very different precision. The one real decision is whether the studies are estimating a single true effect or a distribution of true effects — and that decision changes the interval far more than it changes the estimate.
At a glance
Formula shown
Fixed effect: wᵢ = 1/SEᵢ², pooled θ = Σwᵢyᵢ / Σwᵢ, SE = 1/√(Σwᵢ). Random effects replaces the weight with 1/(SEᵢ² + τ²), where the DerSimonian–Laird estimate is τ² = max(0, (Q − k + 1) / (Σwᵢ − Σwᵢ²/Σwᵢ)) and Q = Σwᵢ(yᵢ − θ_fixed)². Because τ² adds the same amount to every study’s variance, it pulls all the weights toward equality — which is why the biggest study loses share the moment heterogeneity is admitted.
Scenario support
Combining randomised trials of the same treatment, pooling effect sizes across replications in psychology or education, aggregating diagnostic accuracy studies, summarising a literature for a guideline or systematic review, and checking whether a headline pooled result survives dropping its largest contributor.
Educational estimate
Planning support from the values you enter — not professional advice.
The estimate barely moves. The interval moves 78%
The fixed-versus-random choice is usually presented as a choice of answer. On the shipped preset it is almost entirely a choice of how wide to draw the uncertainty.
Fixed effect gives 0.5151 with an interval of 0.3851 to 0.6450. Random effects gives 0.5160 with 0.2844 to 0.7475.
The point estimate moved by 0.0009. The interval got 78.2% wider. Anyone reading only the pooled number would see no difference at all between the two analyses.
The weights tell you why. Under fixed effect the most precise trial owns 43.94% of the pool and two trials hold half of it. Under random effects τ² = 0.0500 is added to every variance, the weights flatten, and the same trial drops to 23.26%.
Which is the real consequence: random effects listens harder to the small studies. That is right when the studies genuinely differ and wrong when the small ones are simply worse.
A confidence interval is not a forecast
This is the number most meta-analyses leave out, and on heterogeneous data it changes the conclusion.
The random-effects interval here is 0.2844 to 0.7475. That is where theaverage true effect sits.
The prediction interval is −0.1341 to 1.1661. That is where the next trial’s true effect sits — 2.81× as wide, and it crosses zero.
So the honest summary is: on average this works, and the next trial may find nothing. Both sentences come from the same data, and only the first survives if you report the confidence interval alone.
It needs at least three studies because it is built on a t distribution with k − 2 degrees of freedom, and with k = 3 that multiplier is large enough that the interval is rarely worth much. It is most useful from about eight studies upward.
Who is actually being averaged
A pooled effect is a claim about a body of evidence, and it is worth knowing how much of that body is doing the work.
Switch to the dominant-study preset. One precise trial takes 90.51% of the fixed-effect pool. Four other trials share the remaining 9.49% between them.
Under random effects that same trial holds 39.69%, and the pooled estimate moves from 0.2552 to 0.4426 — a shift larger than the difference between most of the individual studies.
Neither answer is a mistake. They answer different questions: fixed effect asks “what is the common effect?” and random effects asks “what is the average of the effects?”.
The weight column is printed for exactly this reason. A meta-analysis where one study holds most of the weight is a report on that study with error bars borrowed from its neighbours.
Odds ratios must be pooled in logs
This is the most common arithmetic error in a hand-built meta-analysis, and it is silent.
An odds ratio of 4 and an odds ratio of 0.25 are the same size of effect in opposite directions. Their arithmetic mean is 2.125, which claims a substantial effect where there is none.
In logs they are +1.386 and −1.386 and they average to exactly zero, which exponentiates back to an odds ratio of 1.
The same applies to risk ratios and hazard ratios, and to their confidence intervals: a ratio interval is symmetric in logs, never on the natural scale.
The shipped log-odds preset pools to exactly 0.6500, with an interval of 0.5310 to 0.7957 once the ratio scale is switched on. Risk differences, mean differences and standardised mean differences need no transform and go in as they are.
Drop each study and see what survives
Leave-one-out is the cheapest robustness check there is, and it catches the failure mode that matters most.
Every row refits the whole analysis without that study. If a conclusion depends on a single trial, this is where it shows.
On the shipped preset the largest shift is 0.0752, from dropping the trial with the smallest effect, and I² falls from 58.0% to 37.8% when it leaves — that one study is a substantial part of the disagreement.
A shift that changes the sign, or crosses a decision threshold, is not a sensitivity analysis result. It is the finding.
It will not detect a bias shared by every study. If all seven trials over-report for the same reason, dropping any one of them changes nothing.
What pooling cannot do
Meta-analysis has a reputation as the top of the evidence hierarchy. That is only true when the inputs deserve it.
It cannot fix bias. Averaging twenty biased studies gives a very precise estimate of the bias. The interval shrinks; the error does not.
It cannot see the studies that were never published. Check that separately with Egger’s test and a funnel plot.
It cannot tell you whether the studies belong together. I² measures how much they disagree, not whether pooling them was a sensible idea in the first place — that is a judgement about populations, doses and outcomes, made before any arithmetic.
And τ² is badly estimated below about ten studies. DerSimonian–Laird is the standard, and at small k it is known to run low, which makes the random-effects interval narrower than it should be. Treat a random-effects analysis of five studies as indicative.
Reporting a pooled result
Five items, and the first two are what make the result checkable by someone else.
Give the model and say why. “Random effects, because the trials used different doses” is a reason; “random effects because I² was high” is choosing the model from the data.
Give k, τ² and I² with its interval. I² alone is not enough — on these seven trials it is 58.0% with an interval from 2.9% to 81.8%.
Give the prediction interval when k allows it. It is the number a reader planning the next study actually needs.
Give the per-study weights. One number for the pool hides whether it rests on one trial or twenty.
And report the funnel-asymmetry test separately, rather than folding it into the pooled estimate.
Sources and methodology
References for inverse-variance pooling, τ² and prediction intervals.
Method. Both models are computed in the same pass so they can be compared rather than chosen in advance: the fixed-effect pool uses weights of 1/SE², and the random-effects pool adds the DerSimonian–Laird τ² to every variance before inverting. Cochran’s Q is formed against the fixed-effect estimate, which is what makes τ² and I² consistent with the weights shown in the table. The prediction interval follows Higgins, Thompson and Spiegelhalter, on k − 2 degrees of freedom, and is withheld below three studies rather than printed with a meaningless multiplier. The suite asserts that the weights always total 100%, that the random-effects standard error is never below the fixed-effect one, that the pooled estimate always lies inside the range of the studies, that rescaling every effect and standard error leaves Q and I² untouched while multiplying the pool, and that the prediction interval always contains the confidence interval. That engine is verified on every change against 134 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
I²Heterogeneity across studies: I² with a confidence interval, Cochran's Q, τ², H and every study's share of Q.
Egger's TestEgger's regression test for funnel-plot asymmetry, with the intercept, its interval, the plotted regression and I² beside it.
Hedges' gBias-corrected standardised mean difference using the exact gamma correction, with Cohen's d beside it and a confidence interval.
Effect SizeCohen d, Hedges g and the overlap between groups, with a sample-size control that moves the p-value while leaving the effect size fixed — the same d gives t = 1.29 at n=30 and 23.57 at n=10,000.
Odds RatioOdds ratio, relative risk, risk difference and number needed to treat from one 2x2 table — because an odds ratio of 6.00 can describe a relative risk of 1.50.
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.
An educational tool, not a substitute for a protocol-driven systematic review. Pooling cannot repair bias in the studies being pooled — averaging biased inputs produces a precise estimate of the bias. The DerSimonian–Laird τ² is known to run low below about ten studies, which makes random-effects intervals narrower than they should be at small k.
Launched the meta-analysis calculator with fixed-effect and random-effects pooling computed in the same pass, so the two models are compared rather than chosen in advance.
Added the DerSimonian-Laird tau-squared and showed its consequence on the shipped preset: the pooled estimate moves 0.0009 while the interval widens 78.2%.
Added a Higgins-Thompson-Spiegelhalter prediction interval, withheld below three studies rather than printed with a meaningless multiplier.
Added per-study weight percentages and a leave-one-out table, because a pooled result resting on one study is a result about that study.
Added log-scale entry for odds, risk and hazard ratios, with the shipped preset pooling to exactly 0.6500.
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