Practical examples of effect size calculations for power analysis

Starting with real examples of effect size calculations for power analysis

Most people learn the formulas for Cohen’s d, odds ratios, or correlation coefficients, but freeze when they have to plug in real numbers. So let’s flip the script and start with concrete examples of effect size calculations for power analysis, then generalize from there.

We’ll walk through several realistic scenarios:

• A clinical trial comparing blood pressure between two treatments
• A psychology experiment with a pre/post design
• An education study comparing pass rates
• A public health survey using correlations
• An A/B test using conversion rates
• A logistic regression with odds ratios
• A repeated-measures design with within-subject correlations
• A trend over time in a longitudinal study

Each one shows how to turn the data you actually have (means, SDs, proportions, correlations) into the effect size your power analysis needs.

Example of effect size calculation: two-group mean difference (Cohen’s d)

Imagine a randomized clinical trial comparing a new blood pressure medication to standard care. You want to plan the sample size before running the trial.

Pilot or past data (systolic blood pressure in mmHg):

• Control group mean: 140
• Treatment group mean: 132
• Pooled standard deviation: 15

Cohen’s d for two independent means is:

\[ d = \frac{M_1 - M_2}{SD_{pooled}} \]

Plug in the numbers:

\[ d = \frac{140 - 132}{15} = \frac{8}{15} \approx 0.53 \]

So the effect size is about d = 0.53, a moderate effect.

You can now use this d in a power analysis tool (for example, G*Power or the pwr package in R) to estimate sample size for 80–90% power at your chosen alpha level.

This is one of the clearest examples of effect size calculations for power analysis because most people already understand mean differences and standard deviations.

Pre/post example of effect size calculation (paired d)

Now shift to a within-subject design: a psychology experiment where participants complete a stress scale before and after an intervention.

Pilot data (stress score, 0–40 scale):

• Pre-intervention mean: 28
• Post-intervention mean: 22
• Standard deviation of change scores: 9

For paired data, Cohen’s d for the mean change is often calculated as:

\[ d = \frac{M_{pre} - M_{post}}{SD_{change}} \]

Compute it:

\[ d = \frac{28 - 22}{9} = \frac{6}{9} \approx 0.67 \]

So you have d ≈ 0.67, a moderate-to-large effect.

Because within-subject designs are more efficient (they reduce error variance by accounting for individual differences), this effect size often translates into a smaller required sample size for the same power. In your power analysis software, you’d select a paired t-test, specify d = 0.67, your alpha (often 0.05), and your desired power.

Again, this is a concrete example of effect size calculations for power analysis that uses numbers most researchers already have from a pilot or small feasibility study.

Proportion-based examples of effect size calculations for power analysis

Many modern studies don’t compare means; they compare proportions. Think pass/fail, yes/no, clicked/didn’t click, recovered/didn’t recover. In these cases, your effect size often starts as a difference in proportions.

Education study: difference in pass rates

Suppose you’re evaluating a new teaching method and want to plan a study around exam pass rates.

From previous semesters:

• Traditional teaching pass rate: 70% (0.70)
• New method preliminary pass rate: 82% (0.82)

The raw difference in proportions is:

\[ \Delta p = p_1 - p_2 = 0.82 - 0.70 = 0.12 \]

You can use this 12 percentage-point difference directly in a power analysis for comparing two proportions. Many tools take p1, p2, alpha, and desired power and return the required total sample size.

If you prefer a standardized effect size (Cohen’s h for proportions), you use:

\[ h = 2 \cdot \arcsin(\sqrt{p_1}) - 2 \cdot \arcsin(\sqrt{p_2}) \]

Compute step by step:

• \( \sqrt{0.82} \approx 0.905 \), arcsin ≈ 1.14
• \( \sqrt{0.70} \approx 0.837 \), arcsin ≈ 0.99

Then:

\[ h = 2(1.14 - 0.99) = 2(0.15) = 0.30 \]

So Cohen’s h ≈ 0.30, usually interpreted as a small-to-moderate effect. You can feed this h into software that uses standardized effect sizes.

A/B test example: conversion rates in a product experiment

A tech team wants to compare two website designs. From past experiments:

• Baseline conversion rate: 4.0% (0.040)
• Expected improved rate: 4.8% (0.048)

Difference in proportions:

\[ \Delta p = 0.048 - 0.040 = 0.008 \]

That’s a 0.8 percentage-point increase. Small in absolute terms, but maybe very meaningful in revenue.

To express this as Cohen’s h:

• \( \sqrt{0.048} \approx 0.219 \), arcsin ≈ 0.221
• \( \sqrt{0.040} \approx 0.200 \), arcsin ≈ 0.202

\[ h = 2(0.221 - 0.202) = 2(0.019) = 0.038 \]

So h ≈ 0.04, a tiny standardized effect. This is a classic case where online experiments need large sample sizes to detect small but financially important effects. It’s one of the best examples of effect size calculations for power analysis in industry because it shows how small improvements can still justify big studies.

Correlation-based example of effect size calculation

Suppose you’re planning a public health survey to investigate the association between physical activity (minutes per week) and depression scores.

Past studies, such as those summarized by the National Institutes of Health, often report correlations in the range of r = -0.20 to -0.30 between physical activity and depressive symptoms.

Let’s say you want to detect r = -0.25.

The effect size here is the correlation coefficient:

• Effect size: r = -0.25

Power analysis tools for correlations typically ask for:

• Expected correlation (r)
• Alpha level
• Desired power

You enter r = -0.25 directly. You don’t need to transform it unless the software uses Fisher’s z. If it does, it will usually handle the transformation internally.

This kind of correlation-based scenario is a straightforward example of effect size calculations for power analysis in observational research.

Logistic regression example of effect size: odds ratios

In medical and epidemiological research, logistic regression is everywhere. Here, effect sizes are often expressed as odds ratios (OR).

Imagine you’re planning a study on smoking and a binary health outcome, like presence or absence of a particular disease. Prior literature, such as resources at CDC, might suggest an odds ratio around 1.8 for current smokers vs. non-smokers.

For a power analysis of logistic regression, you often need:

• The odds ratio you want to detect
• The prevalence of the predictor (e.g., proportion of smokers)
• The baseline probability of the outcome in the reference group

Say:

• Expected OR for smoking: 1.8
• Proportion of smokers in your population: 30% (0.30)
• Baseline disease probability in non-smokers: 10% (0.10)

The effect size is specified as OR = 1.8. Many power tools for logistic regression (e.g., in R or PASS) take OR directly. Others use the log odds ratio:

\[ \beta = \ln(OR) = \ln(1.8) \approx 0.588 \]

So your effect size on the logit scale is β ≈ 0.59.

This is a real-world example of effect size calculations for power analysis that connects regression coefficients to an interpretable quantity (odds ratios) you often see in medical journals.

Repeated-measures example: within-subject correlation and effect size

Power analysis for repeated-measures designs often needs two pieces:

• The mean difference you care about
• The within-subject correlation between time points

Consider a study tracking weight loss at baseline, 3 months, and 6 months after a lifestyle program. You might have pilot data from a small feasibility study.

Pilot results:

• Mean weight at baseline: 210 lb
• Mean weight at 6 months: 198 lb
• Standard deviation of baseline weights: 30 lb
• Standard deviation of 6-month weights: 28 lb
• Correlation between baseline and 6-month weight: r = 0.85

To get a standardized effect size for the baseline-to-6-month change, you can estimate the SD of the difference scores using:

\[ SD_{diff} = \sqrt{SD_1^2 + SD_2^2 - 2rSD_1SD_2} \]

Plug in:

\[ SD_{diff} = \sqrt{30^2 + 28^2 - 2(0.85)(30)(28)} \]
\[ = \sqrt{900 + 784 - 2(0.85)(840)} \]
\[ = \sqrt{1684 - 1428} = \sqrt{256} = 16 \]

Mean difference:

\[ \Delta M = 210 - 198 = 12 \text{ lb} \]

Standardized effect size (paired d):

\[ d = \frac{12}{16} = 0.75 \]

So your effect size for the baseline-to-6-month change is d = 0.75, and the within-subject correlation r = 0.85 is also used in many repeated-measures power calculations.

This is one of the more nuanced examples of effect size calculations for power analysis because it shows how correlation between time points affects the variance of change scores.

Trend over time example: effect size for a linear slope

Longitudinal studies often want to detect a trend rather than just a before/after difference.

Suppose you’re studying average daily screen time among teenagers, collected every year from age 13 to 18. You want to know if screen time is increasing over cohorts, and you’re planning a study for 2025.

From national surveys (for instance, technology use data summarized by organizations like Pew Research Center), you might estimate that average screen time has been increasing by about 0.3 hours per day per year in recent cohorts, with a standard deviation of yearly screen time around 2 hours.

A rough standardized effect size for the slope can be approximated by:

\[ f^2 = \frac{R^2}{1 - R^2} \]

where R² is the proportion of variance in screen time explained by year.

If you estimate that year explains about 5% of the variance (R² = 0.05), then:

\[ f^2 = \frac{0.05}{1 - 0.05} = \frac{0.05}{0.95} \approx 0.053 \]

So your effect size is f² ≈ 0.05, a small-to-moderate effect in regression terms.

In a power analysis for linear regression with a single predictor (year), you’d use f² = 0.05, your number of planned observations, alpha, and desired power.

This trend-based scenario is a more advanced example of effect size calculations for power analysis, but it matches how many social science and public health studies are actually analyzed.

How to choose realistic effect sizes from real-world data

All these examples of effect size calculations for power analysis hinge on one thing: choosing an effect size that isn’t fantasy. A few practical strategies help:

Use prior studies.

• Look at meta-analyses or large trials in your field.
• For health topics, sources like NIH, CDC, and Mayo Clinic often summarize effect sizes or at least provide enough numbers (means, SDs, proportions) to compute them.

Use pilot data.

• Even a small pilot can give you rough estimates of means, SDs, and correlations.
• Be conservative: if your pilot suggests d = 0.6, consider planning around d = 0.4–0.5 to avoid underpowering the main study.

Translate to what matters.

• In medicine, an odds ratio of 1.3 may be modest statistically but meaningful if the disease is severe.
• In product analytics, a 0.5 percentage-point lift in conversion can be huge in revenue terms.

Check sensitivity.

• Run power analyses for a range of effect sizes—small, medium, and large relative to your context.
• This gives you a sense of how sample size requirements explode as effect sizes shrink.

When you look back at the best examples of effect size calculations for power analysis—clinical trials, education interventions, A/B tests—the common pattern is this: they start with realistic differences pulled from prior evidence, not wishful thinking.

Frequently asked questions about examples of effect size calculations for power analysis

Q1. Can you give a simple example of converting raw data to an effect size for power analysis?
Yes. Suppose two groups have means 50 and 55 with a pooled SD of 10. Cohen’s d is (55 − 50) / 10 = 0.5. You then use d = 0.5 in a power analysis for a two-sample t-test.

Q2. What are common examples of effect size calculations for power analysis in clinical research?
Common examples include mean differences in lab values (e.g., blood pressure, cholesterol), odds ratios for treatment vs. control (e.g., risk of hospitalization), and hazard ratios in survival analysis. These are usually derived from past trials or observational studies and then converted into standardized measures like d, OR, or log(OR) for planning.

Q3. How do I handle small effects, like in A/B tests, when calculating effect size?
For small effects in A/B tests, you typically work with differences in proportions (e.g., 4.0% vs. 4.5% conversion) and possibly Cohen’s h. Even if h is tiny (say 0.02–0.05), that’s still a valid effect size; it just means you’ll need a very large sample. This is where sensitivity analyses and realistic traffic estimates matter.

Q4. Is there an example of using effect size from a meta-analysis for power analysis?
Yes. Suppose a meta-analysis of school-based reading interventions reports an average standardized mean difference of d = 0.35. You can use d = 0.35 as the target effect size in your power analysis for a new trial, perhaps adjusting slightly downward if you expect more heterogeneity or a less controlled setting.

Q5. Do I always need a standardized effect size like Cohen’s d, or can I use raw differences?
It depends on your software. Some tools accept raw inputs, like two proportions or two means plus SDs. Others expect standardized metrics like d, h, f², or r. The examples of effect size calculations for power analysis in this guide show both approaches; the key is to match your input to what the software expects.

Effect size is the bridge between real-world quantities and the abstract world of power analysis. When you ground your planning in realistic mean differences, proportions, correlations, and odds ratios—like the real examples above—you end up with studies that are neither underpowered nor wildly oversized, but sized to answer the questions you actually care about.