Statistical Power Calculator (t-Test Power)

Estimate the Power of a Planned Two-Group Study

Statistical Power

Formula

\[ \delta = d\sqrt{\frac{n}{2}}, \qquad \nu = 2n - 2 \\ \lambda = 1 - \beta = P\!\left(T_{\nu,\,\delta} > t_{\alpha/2}\right) + P\!\left(T_{\nu,\,\delta} < -t_{\alpha/2}\right) \]

Where:

\(d\) — effect size (Cohen's d), the expected difference between group means divided by the common standard deviation
\(n\) — sample size per group
\(\nu\) — degrees of freedom of the independent-samples t test
\(\delta\) — noncentrality parameter, which scales the effect by the precision of the study
\(T_{\nu,\delta}\) — a random variable following the noncentral t distribution with \(\nu\) degrees of freedom and noncentrality \(\delta\)
\(t_{\alpha/2}\) — the critical value of the (central) t distribution for a two-sided test at level \(\alpha\)

The formula gives the probability that a two-sided independent-samples t test rejects the null hypothesis when the true standardized difference between the groups is \(d\). For a one-sided test, the second term is dropped and \(t_{\alpha/2}\) is replaced by the one-sided critical value \(t_{\alpha}\).

Statistical Power – Calculation Example

Example: a researcher plans a study comparing two groups with 64 participants each, expects a medium effect of \(d = 0.5\), and will use a two-sided t test at \(\alpha = 0.05\).

\[ \delta = 0.5\sqrt{\frac{64}{2}} = 0.5\sqrt{32} \approx 2.828 \]

\[ \nu = 2 \times 64 - 2 = 126 \]

The two-sided critical value at \(\alpha = 0.05\) with 126 degrees of freedom is \(t_{0.025} \approx 1.979\). Evaluating the noncentral t distribution with \(\nu = 126\) and \(\delta = 2.828\):

\[ 1 - \beta \approx P(T > 1.979) + P(T < -1.979) \approx 0.801 \]

Result: the estimated power is about 80.1%, and the probability of a Type II error is \(\beta \approx 19.9%\).

How to use the calculator

Enter the expected standardized difference in Effect size (Cohen's d). If you have pilot data or published means and standard deviations, the Cohen's d calculator can compute this value for you.
Enter the planned number of observations in each arm in Sample size per group. Both groups are assumed to have the same size.
Set the Significance level (α), most commonly 0.05.
Choose the Test direction: two-sided if either group could come out ahead, one-sided only if the direction of the effect was predicted before the data were collected.
Read the estimated power, the Type II error probability, the degrees of freedom, the noncentrality parameter, and a plain-language interpretation of the results.

What statistical power means

Statistical power is the probability that a study will detect a real effect when one exists — that is, the chance of obtaining a statistically significant result if the true difference between the groups equals the effect size you specified. Its complement \(\beta\) is the probability of a Type II error: missing a genuine effect because the sample was too small, the effect smaller than expected, or the measurements too noisy.

Power is conventionally targeted at 80% or higher. A study with 80% power that tests a true effect of the assumed size will find it about four times out of five; the remaining 20% of replications would produce a non-significant result despite the effect being real. Low-powered studies are problematic in two ways: they often miss real effects, and the significant results they do produce tend to exaggerate the effect size.

Power depends on three quantities, and the calculator lets you explore how they interact. Larger effects are easier to detect, larger samples shrink the standard error and increase power, and stricter significance thresholds (smaller α) reduce power. Increasing the sample size is usually the only lever a researcher controls directly: with \(d = 0.5\) at \(\alpha = 0.05\), 64 per group yields about 80% power, but 26 per group yields only about 42%.

The interpretation line tells you the estimated probability of rejecting the null hypothesis under your assumptions. Treat it as a planning estimate, not a guarantee: the true power depends on the actual effect size, which is never known exactly in advance.

Assumptions and limitations

Equal, independent groups: the calculation assumes two independent samples of equal size. For paired designs, the effective sample size and standard deviation differ, and this tool will overestimate or underestimate power.
Normality and equal variances: the result assumes the observations are normally distributed in each group and that the two populations share a common variance, as the standard independent-samples t test does. The t test is fairly robust to moderate departures from normality when \(n\geq 30\) per group.
Effect size must be specified honestly: power is only as good as the \(d\) you enter. Overly optimistic assumptions inflate the estimated power, so it is wise to compute power for a range of plausible effect sizes.
One-sided tests require pre-registration: choosing a one-sided test after looking at the data inflates the effective false-positive rate. Only select "One-sided" if the direction was fixed in advance.
Designed for means, not proportions or survival: this calculator addresses the two-group t test. For comparing two proportions, use the two-proportion z-test calculator; for a single mean against a known standard, see the z-test for a mean calculator.

Frequently Asked Questions

How can I calculate statistical power by hand?

For a quick approximation, treat the t test as a z test for large samples: compute the noncentrality parameter \( \delta = d\sqrt{n/2} \), then estimate power as \( P(Z > z_{\alpha/2} - \delta) \) for a two-sided test, where \( z_{0.025} \approx 1.96 \). This works well when \( n \geq 50 \) per group. For small samples you need the noncentral t distribution, which requires statistical software or printed tables.

What sample size do I need for 80% power?

A useful rule of thumb for a two-sided test at \( \alpha = 0.05 \) is \( n \approx 16/d^2 \) per group. For example, a medium effect of \( d = 0.5 \) needs about \( 16/0.25 = 64 \) participants per group. For 90% power, use \( n \approx 21/d^2 \). Halving the required sample size requires doubling the effect size, so small effects demand very large studies.

What is considered a good statistical power value?

80% is the conventional minimum, and many fields now encourage 90% or more. Power below 50% means a real effect is more likely to be missed than found. High power also protects against exaggerated findings: studies that barely reach significance tend to overestimate the true effect size. Since power depends on the effect size you assume, report the power calculation alongside the assumed \( d \) so readers can judge whether it was realistic.

What do small, medium, and large effect sizes mean?

Cohen's rough benchmarks are \( d = 0.2 \) (small), \( d = 0.5 \) (medium), and \( d = 0.8 \) (large). A small effect is hard to distinguish from noise without big samples: detecting \( d = 0.2 \) at 80% power takes about 394 participants per group. These labels are field-specific conventions, not universal truths — in psychology, many published effects fall near \( d = 0.4 \), so plan around realistic values from prior literature or pilot data.

Can I use this calculator after collecting my data to get 'observed power'?

Post-hoc power computed from your observed effect size and p-value adds no new information: it is mathematically locked to your p-value, so a non-significant result always implies low observed power. It cannot tell you whether a real effect exists. This calculator is designed for planning — deciding a sample size before the study. After data collection, interpret the effect size and confidence interval instead.

What if my two groups have unequal sample sizes?

This calculator assumes equal groups, but you can approximate unequal designs using the harmonic mean \( n_h = 2n_1n_2/(n_1+n_2) \) as the per-group sample size. Because the harmonic mean is dominated by the smaller group, an imbalance always costs power: 100 and 50 participants behave roughly like 67 per group. For an exact calculation with unequal \( n \), the noncentrality parameter becomes \( \delta = d\sqrt{n_1n_2/(n_1+n_2)} \).

How are Type I and Type II errors related to power?

A Type I error (rate \( \alpha \)) is a false positive: declaring an effect when none exists. A Type II error (rate \( \beta \)) is a false negative: missing a real effect. Power is simply \( 1 - \beta \), the chance of correctly detecting the effect. The two error rates trade off: lowering \( \alpha \) from 0.05 to 0.01 makes false positives rarer but reduces power, so you need a larger sample to compensate.

Statistical Power Calculator