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When two trials measure the same underlying construct with different instruments — a 20-point anxiety scale in one study, a 40-point anxiety scale in another — their raw mean differences can’t be averaged together in a meta-analysis. The standardized mean difference (SMD) solves this by re-expressing each study’s effect in units of that study’s own standard deviation rather than the original scale, so results measured on different instruments become directly comparable and poolable. In practice, “SMD” in a meta-analysis almost always means one specific statistic: Hedges’ (adjusted) g, not the raw Cohen’s d most researchers learn first. This guide covers why that distinction exists, the exact formula and small-sample correction behind it, when SMD is the right effect measure versus the plain mean difference (MD), and what Cochrane’s own guidance says about interpreting a pooled SMD correctly.
Why meta-analysis needs a standardized effect measure at all
Most outcomes in a systematic review are continuous — a symptom score, a test result, a rating scale — and different trials of the same intervention frequently measure that outcome with different instruments, different scale ranges, or different units. The Cochrane Handbook is explicit that SMD is the effect measure for exactly this situation: it is used “when the studies all assess the same outcome, but measure it in a variety of ways” (Handbook §6.5.1.2), such as several different validated scales for the same underlying condition. If every included trial used the identical instrument and scale, the plain mean difference (MD) — the raw, unstandardized difference in means, still in the original units — is the better choice, because it keeps a directly interpretable unit instead of trading it away for comparability. SMD is a deliberate trade: it buys the ability to pool studies that would otherwise be incomparable, at the cost of losing the original measurement scale.
Cochrane’s SMD is Hedges’ g, not raw Cohen’s d
This is the detail most explanations of “standardized mean difference” skip, and it matters for anyone reading or conducting a Cochrane review: the Cochrane Handbook states directly that “the particular definition of SMD used in Cochrane reviews is the effect size known in social science as Hedges’ (adjusted) g” (§6.5.1.2) — not the more commonly taught Cohen’s d. Both statistics divide a mean difference by a pooled standard deviation drawn from both groups’ outcome data (which is also what distinguishes SMD/Hedges’ g from Glass’s delta, a related standardized measure that divides by only one group’s SD, typically the control group’s). The difference between d and g is a small-sample bias correction, covered in the next section. Software and published reviews don’t always flag which version of “SMD” they’re reporting, so it’s worth checking — see How to Interpret Cohen’s d for when the distinction between d and g matters for interpretation, and Cohen’s d Effect Size: Calculation & Interpretation Guide for the single-study calculation this guide builds on.
The formula: Cohen’s d, the correction factor, and Hedges’ g
Cohen’s d for two independent groups is the mean difference divided by the pooled standard deviation:
d = (M₁ − M₂) / SDpooled, where SDpooled = √[((n₁−1)SD₁² + (n₂−1)SD₂²) / (n₁+n₂−2)]
Cohen’s d has a known, predictable upward bias in small samples — it systematically overstates the true effect. Hedges’ g removes that bias with an exact correction factor, J:
g = d × J, where J = 1 − 3 / (4df − 1) and df = n₁ + n₂ − 2
J is always slightly less than 1, so g is always slightly smaller in magnitude than d. The correction shrinks toward negligible as sample size grows — by roughly n=20 per group it barely moves the estimate — which is why How to Interpret Cohen’s d recommends reporting g by default whenever either group has fewer than about 20 observations, and reporting it by default in a meta-analysis regardless of sample size, since study-level bias otherwise accumulates and skews the pooled estimate once dozens of small trials are combined.
Worked example (illustrative, not a real published study)
The numbers below are constructed to demonstrate the arithmetic only — they are not drawn from any real trial or dataset.
Study A (larger trial): treatment group mean 12, SD 4, n₁=30; control group mean 15, SD 5, n₂=30.
- SDpooled = √[(29×4² + 29×5²) / 58] = √[(464 + 725) / 58] = √20.50 ≈ 4.53
- d = (12 − 15) / 4.53 ≈ −0.66
- df = 58, J = 1 − 3/(4×58−1) = 1 − 3/231 ≈ 0.987
- g = −0.66 × 0.987 ≈ −0.65
Study B (smaller trial, same intervention, different scale range): treatment group mean 22, SD 6, n₁=10; control group mean 27, SD 7, n₂=10.
- SDpooled = √[(9×6² + 9×7²) / 18] = √[(324 + 441) / 18] = √42.50 ≈ 6.52
- d = (22 − 27) / 6.52 ≈ −0.77
- df = 18, J = 1 − 3/(4×18−1) = 1 − 3/71 ≈ 0.958
- g = −0.77 × 0.958 ≈ −0.73
Two things to notice. First, the correction moved Study B’s estimate more than Study A’s, exactly as expected — the bias in d grows as sample size shrinks. Second, because both studies’ results are now expressed in pooled-SD units rather than their original (different) scales, they can be combined in a single pooled estimate. That pooling step itself — weighting each study’s g by the inverse of its variance and combining under a fixed-effect or random-effects model — is what meta-analysis software performs; it is not done by hand in practice, and this guide doesn’t attempt to reproduce that calculation here.
SMD vs. mean difference (MD): choosing the right effect measure
The choice isn’t about which statistic is more sophisticated — it’s about whether the included studies share a common scale:
- Use MD (mean difference) when every included study measured the outcome on the same scale, in the same units — for example, blood pressure in mmHg across all trials. MD keeps the pooled result in a directly interpretable, real-world unit.
- Use SMD (Hedges’ g) when studies measured a shared underlying construct with different instruments or scales — different depression inventories, different pain scales, different lab assays reported on different units. Standardizing is the only way to combine them at all.
Mixing MD and SMD studies within a single pooled estimate isn’t meaningful — if any subset of studies used different scales, that subset (or the whole outcome) needs SMD.
Interpreting a pooled SMD: Cochrane’s own cautions
The Cochrane Handbook (§6.5.1.2) flags real limitations that are easy to skip past when a pooled SMD looks like a clean, single number:
- It assumes SD differences reflect scale differences, not real population variability. If one trial’s participants genuinely vary more than another’s (for reasons unrelated to the measurement instrument — a broader inclusion criterion, for instance), that real variability gets folded into the standardization and can distort the comparison.
- It’s harder to interpret clinically. A pooled SMD of 0.5 doesn’t map onto any specific real-world unit the way a mean difference in mmHg or points on a named scale does; readers need a benchmark (see How to Interpret Cohen’s d on why 0.2/0.5/0.8 is field-dependent, not universal) or a back-transformation to a familiar scale to make it meaningful to a clinical or policy audience.
- Terminology matters. Cochrane specifically recommends using the term “SMD” rather than the more generic “effect size” when reporting this statistic, to avoid confusion with the broader, looser use of that phrase elsewhere in the literature.
Where this shows up in meta-analysis software
You won’t typically calculate g by hand across a full set of included studies — meta-analysis software does the per-study standardization and the pooling together. RevMan (Cochrane’s own review-authoring tool) documents the underlying formulae in its Web Knowledge Base under “Statistical algorithms and calculations used in Review Manager.” In R, the metafor package’s escalc() function and the meta package both compute standardized mean differences with the small-sample correction applied, alongside the heterogeneity statistics (Q, I², τ²) needed to decide whether a fixed-effect or random-effects pooling model is appropriate. Whichever tool is used, it’s worth confirming in its documentation whether the “SMD” it reports is Hedges’ g (the small-sample-corrected version) or plain Cohen’s d, since the two are not interchangeable inputs to the same pooled estimate.
Frequently asked questions
Is Hedges’ g the same as Cohen’s d?
No, though they’re closely related. Hedges’ g applies a small-sample bias-correction factor (J) to Cohen’s d. The two converge as sample size grows — by around n=20 per group the difference is negligible — but for smaller studies g is the more accurate, less biased estimate.
Why does Cochrane define SMD as Hedges’ g instead of Cohen’s d?
Because meta-analyses routinely include small trials, and Cohen’s d’s small-sample bias accumulates and skews a pooled estimate once many such studies are combined. Hedges’ g removes that bias at the individual-study level before pooling, which is why the Cochrane Handbook specifies it as the version of SMD used in Cochrane reviews.
When should a meta-analysis use SMD instead of the raw mean difference (MD)?
When the included studies measured the same underlying outcome with different instruments or scales. If every study used an identical scale and unit, MD is preferable — it preserves the original, directly interpretable unit instead of trading it away for standardization.
Does the Hedges’ g correction matter for large studies?
Only marginally. The correction factor J approaches 1 as degrees of freedom increase, so in trials with roughly 20 or more participants per group, g and d are numerically very close. The correction matters most in meta-analyses that pool many small studies, where uncorrected bias in d can accumulate across the pooled estimate.
Can Cohen’s d and Hedges’ g be mixed across studies in the same meta-analysis?
Not without correcting them to a common basis first. Since g is simply d multiplied by a known correction factor, a study reported only as d can be converted to g if its group sample sizes are known — but pooling a mix of uncorrected d values and corrected g values without reconciling them first would introduce exactly the small-sample bias the correction exists to remove.
