How to Calculate The Confidence Interval for Cohen's D

What is Cohen's d?

Cohen's d is a standardized measure of effect size that compares the difference between two group means to the pooled standard deviation of the groups. It helps researchers understand the practical significance of their findings beyond just statistical significance.

The formula for Cohen's d is:

Where:

• M₁ = Mean of group 1
• M₂ = Mean of group 2
• SD_pooled = Pooled standard deviation of the two groups

Cohen (1988) suggested benchmarks for interpreting effect sizes:

• Small effect: d = 0.2
• Medium effect: d = 0.5
• Large effect: d = 0.8

Confidence Interval Formula

The confidence interval for Cohen's d can be calculated using the following formula:

Where:

• CI = Confidence interval
• d = Cohen's d effect size
• t_α/2,df = Critical t-value from t-distribution table
• SE_d = Standard error of Cohen's d
• df = Degrees of freedom (n₁ + n₂ - 2)

The standard error of Cohen's d is calculated as:

Where n₁ and n₂ are the sample sizes of the two groups.

How to Calculate

1. Calculate Cohen's d using the formula: d = (M₁ - M₂) / SD_pooled
2. Calculate the degrees of freedom: df = n₁ + n₂ - 2
3. Calculate the standard error of d: SE_d = √[(n₁ + n₂)/(n₁n₂) + d²/2(n₁ + n₂)]
4. Look up the critical t-value from a t-distribution table with df degrees of freedom and α/2 significance level
5. Calculate the confidence interval: CI = d ± t × SE_d

Example Calculation

Let's calculate the 95% confidence interval for Cohen's d using the following data:

1. Calculate pooled standard deviation:
   SD_pooled = √[((n₁ - 1) × SD₁² + (n₂ - 1) × SD₂²) / (n₁ + n₂ - 2)]
   = √[((29 × 8²) + (24 × 7²)) / (53)]
   = √[(232 × 64 + 192 × 49) / 53]
   = √[(14,592 + 9,408) / 53]
   = √(23,999.6 / 53)
   ≈ √452.826
   ≈ 21.28
2. Calculate Cohen's d:
   d = (55 - 48) / 21.28 ≈ 0.329
3. Calculate degrees of freedom: df = 30 + 25 - 2 = 53
4. Calculate standard error of d:
   SE_d = √[(55/750) + (0.329²/106)]
   ≈ √[0.0747 + 0.00098]
   ≈ √0.07568
   ≈ 0.275
5. Look up critical t-value (α = 0.05, df = 53): t ≈ 2.006
6. Calculate confidence interval:
   CI = 0.329 ± 2.006 × 0.275
   = 0.329 ± 0.551
   ≈ (0.329 - 0.551, 0.329 + 0.551)
   ≈ (-0.222, 0.880)

The 95% confidence interval for Cohen's d is approximately -0.222 to 0.880. This means we are 95% confident that the true effect size lies within this range.

Interpretation

The confidence interval for Cohen's d provides several important insights:

• The width of the interval indicates the precision of the estimate. A narrower interval suggests more precise measurement.
• If the interval includes zero, it suggests the effect may not be statistically significant.
• The position of the interval relative to Cohen's benchmarks (0.2, 0.5, 0.8) helps assess the practical significance.

In our example, the interval spans from -0.222 to 0.880, which includes zero, suggesting the effect might not be statistically significant. However, the upper bound of 0.880 indicates a possible large effect if the true value is at the higher end of the interval.