How do you compute Cohen's d for a 3-point mean gap when the pooled SD is 15?

## The definition Cohen's d answers "how far apart are these two groups, measured in units of their own spread?" It is defined as `d = (mean1 - mean2) / s_pooled` With a 3-point gap and a pooled standard deviation of 15, `d = 3 / 15 = 0.2`. The groups' centres sit one fifth of a standard deviation apart. The numerator is the raw effect in the outcome's own units. The denominator converts it into a unitless quantity. That is the whole idea: dividing by spread makes the number comparable across instruments that were never on the same scale. ## The pooled standard deviation The usual denominator combines both samples: `s_pooled = sqrt( ((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2) )` Each group's variance is weighted by its degrees of freedom, the weighted variances are averaged, and the square root is taken. Two consequences worth stating in an interview: - It is a pooled *variance* average, not an average of the two standard deviations. Averaging SDs directly gives a different, wrong number when the groups differ in size or spread. - Pooling presumes the two groups have roughly similar variance. If a treatment changes the spread as well as the centre — a common outcome when an intervention helps some people a great deal and others not at all — pooling blends two genuinely different spreads. Glass's delta sidesteps this by dividing by the control group's SD only, so the denominator is a property of the untreated population and is not disturbed by the treatment. A frequent error is dividing by the **standard error** instead of the standard deviation. The standard error shrinks as the sample grows, so that quantity grows without bound with more data — it is essentially the test statistic, not an effect size. An effect size must not systematically change when you collect more observations; that invariance is what makes it an estimate of a population quantity. ## Interpreting 0.2 Cohen proposed benchmarks for researchers with no field-specific reference point: **0.2 small, 0.5 medium, 0.8 large**. He was explicit that these were conventions offered for want of anything better, not thresholds derived from theory. A more concrete reading of d = 0.2 is distributional. If both groups are roughly normal with equal spread, a d of 0.2 means the distributions overlap substantially: a randomly chosen member of the higher group exceeds a randomly chosen member of the lower group only slightly more often than half the time. About 56% of the time when d = 0.2 — well short of anything a casual observer would notice in an individual case. But "small" is not "unimportant". Two contexts flip the reading: - **Scale.** A d of 0.2 applied to a very large population, on a cheap intervention, can be worth a great deal in aggregate even though no single person notices it. - **Field norms.** Where outcomes are noisy and hard to move, 0.2 may be near the ceiling of what any intervention achieves; where outcomes are precisely measured and strongly manipulable, 0.5 may be unremarkable. Always benchmark against the distribution of effects previously observed on the same outcome, not against the generic conventions. ## Sign, direction and small-sample bias The sign of d follows the order of subtraction, so state which group is which; report the absolute magnitude plus an explicit direction rather than leaving a bare negative number to be misread. Because the pooled SD is itself estimated, d is biased slightly away from zero in small samples — it tends to overstate the population standardized difference. **Hedges' g** multiplies d by a correction factor slightly below one, with the shrinkage larger when the combined sample is small and negligible for large samples. With a few dozen observations per group the correction is worth applying; with thousands it changes nothing you would report. Finally, d is an estimate like any other, so it deserves an interval. A d of 0.2 whose interval runs from 0.15 to 0.25 is a settled small effect; a d of 0.2 whose interval runs from -0.3 to 0.7 is an unsettled question, and the two demand opposite next steps even though the point estimates are identical.