# How can Cohen's d < 1.96 ever be statistically significant?

I am having trouble understanding the statistical significance of Cohen's d.

My reasoning goes like this: To conclude that experimental group differs from control group significantly (p < 0.05), the experimental group's mean must be 1.96z below or above the control group's mean. In this case the Cohen's d will also be about 2, right? If Cohen's d is lower than 2, also the z difference between groups will be lower than that, which means the means from two groups won't be statistically different from each other anymore.

• Cohen's $d$ is determined using the standard deviation (typically of the available sample). Hypothesis testing is performed using the standard error $\frac{sd}{\sqrt{N}}$. The former assesses how big of a standardized difference you observed. The latter is used to determine the probability that you obtained a sample with the characteristics you did from a population of interest. Jan 22, 2018 at 10:48
A Cohen's $$d$$ is a standardized measure of difference. However, it is not weighted by the sample size. To go from a Cohen's $$d$$ to a $$t$$ statistics, the former must be mulitplied by $$\sqrt{\tilde{n}/2}$$ in which $$\tilde{n}$$ is the harmonic mean of the sample sizes. Conversely, the $$t$$ statistic is an effect size magnified by the sample size. The larger the sample, the bigger $$t$$ will end up. Hence, you could for example have a $$t$$ statistic of 1.96 (borderline significant) obtained from a sample of two groups, each containing two million participants. The resulting Cohen's $$d$$ is in that case a mere 0.00196! (yet, borderline significant!).