Understanding statistical power is crucial for designing effective research studies. This calculator helps you determine the statistical power of a two-sample t-test, given the means, standard deviations, sample size per group, and significance level. Statistical power is the probability that a study will detect an effect when there actually is an effect to be detected. In simpler terms, it's the likelihood of avoiding a Type II error (a false negative).
What is Statistical Power?
Statistical power is the probability of correctly rejecting a false null hypothesis. A study with high power has a good chance of detecting a true effect. Conversely, a study with low power might fail to find a real effect, leading to a Type II error. Researchers typically aim for a power of 0.80 (80%) or higher, meaning there's an 80% chance of detecting a true effect if one exists.
Key Components of Power Analysis:
Effect Size: This quantifies the magnitude of the difference or relationship you expect to find. A larger effect size is easier to detect, requiring less power. In this calculator, the effect size is derived from the difference in means and the pooled standard deviation (Cohen's d).
Sample Size (n): The number of observations or participants in each group. Larger sample sizes generally lead to higher power, as they provide more information and reduce sampling error.
Significance Level (Alpha, α): This is the probability of making a Type I error (false positive), typically set at 0.05. It's the threshold for statistical significance. A smaller alpha (e.g., 0.01) makes it harder to reject the null hypothesis, thus reducing power.
Power (1 – β): The probability of correctly rejecting the null hypothesis when it is false. Beta (β) is the probability of a Type II error.
How to Use This Calculator:
Input the following parameters for your two groups:
Mean of Group 1 (μ1): The average value for your first group.
Standard Deviation of Group 1 (σ1): The variability within your first group.
Mean of Group 2 (μ2): The average value for your second group.
Standard Deviation of Group 2 (σ2): The variability within your second group.
Sample Size per Group (n): The number of participants or observations in each group. This calculator assumes equal sample sizes for both groups.
Significance Level (Alpha): Choose a common alpha level (e.g., 0.05, 0.01, 0.10) for your hypothesis test.
The calculator will then estimate the statistical power for a two-tailed independent samples t-test.
Example Scenario:
Imagine a study comparing the effectiveness of two different teaching methods on student test scores. Researchers hypothesize that Method A will result in higher scores than Method B.
Mean of Group 1 (Method A): 75
Standard Deviation of Group 1 (Method A): 10
Mean of Group 2 (Method B): 70
Standard Deviation of Group 2 (Method B): 12
Sample Size per Group (n): 30 students per method
Significance Level (Alpha): 0.05
Using these inputs, the calculator would estimate the power of the study to detect a difference of this magnitude. With these values, the calculated power is approximately 0.418 (41.8%). This suggests that the study has only a 41.8% chance of detecting a true difference of 5 points between the methods, given the variability and sample size. This is generally considered low, prompting researchers to consider increasing the sample size to achieve the desired 80% power.
Limitations:
This calculator uses a Z-approximation for power calculation, which is generally accurate for larger sample sizes (typically n > 30 per group). For very small sample sizes, the approximation may be less precise. It also assumes equal sample sizes and a two-tailed independent samples t-test. For more complex designs or specific distributions, specialized statistical software is recommended.
0.05 (5%)
0.01 (1%)
0.10 (10%)
Calculated Power:
Enter values and click 'Calculate Power'.
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// Approximation for erf(x)
function erf(x) {
// constants
var a1 = 0.254829592;
var a2 = -0.284496736;
var a3 = 1.421413741;
var a4 = -1.453152027;
var a5 = 1.061405429;
var p = 0.3275911;
// Save the sign of x
var sign = 1;
if (x < 0) {
sign = -1;
x = -x;
}
// A&S formula 7.1.26
var t = 1.0 / (1.0 + p * x);
var y = 1.0 – (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * Math.exp(-x * x);
return sign * y;
}
// Standard Normal CDF (Cumulative Distribution Function)
function norm_cdf(x) {
return 0.5 * (1 + erf(x / Math.sqrt(2)));
}
// Function to get Z-critical value for a two-tailed test based on alpha
function getZCritical(alpha) {
if (alpha === 0.05) {
return 1.96; // For 0.05 two-tailed (0.025 in each tail)
} else if (alpha === 0.01) {
return 2.576; // For 0.01 two-tailed (0.005 in each tail)
} else if (alpha === 0.10) {
return 1.645; // For 0.10 two-tailed (0.05 in each tail)
}
return 1.96; // Default to 0.05 if something unexpected happens
}
function calculatePower() {
var mean1 = parseFloat(document.getElementById("mean1").value);
var sd1 = parseFloat(document.getElementById("sd1").value);
var mean2 = parseFloat(document.getElementById("mean2").value);
var sd2 = parseFloat(document.getElementById("sd2").value);
var n = parseInt(document.getElementById("sampleSize").value);
var alpha = parseFloat(document.getElementById("alpha").value);
var resultDiv = document.getElementById("result");
// Input validation
if (isNaN(mean1) || isNaN(sd1) || isNaN(mean2) || isNaN(sd2) || isNaN(n) || isNaN(alpha)) {
resultDiv.innerHTML = "Please enter valid numbers for all fields.";
return;
}
if (sd1 <= 0 || sd2 <= 0) {
resultDiv.innerHTML = "Standard deviations must be positive.";
return;
}
if (n < 2) {
resultDiv.innerHTML = "Sample size per group must be at least 2.";
return;
}
if (alpha = 1) {
resultDiv.innerHTML = "Significance level (Alpha) must be between 0 and 1.";
return;
}
// 1. Calculate Pooled Standard Deviation (Sp)
// Assuming sd1 and sd2 are sample standard deviations
var pooledVariance = (((n – 1) * sd1 * sd1) + ((n – 1) * sd2 * sd2)) / (2 * n – 2);
var Sp = Math.sqrt(pooledVariance);
// 2. Calculate Cohen's d (Effect Size)
var effectSize = Math.abs(mean1 – mean2) / Sp;
// 3. Calculate Non-Centrality Parameter (NCP_Z) for Z-approximation
// For a two-sample t-test with equal n
var NCP_Z = effectSize * Math.sqrt(n / 2);
// 4. Determine Critical Z-value (Z_crit) for a two-tailed test
var Z_crit = getZCritical(alpha);
// 5. Calculate Power using Z-approximation for two-tailed test
// Power = P(Z > Z_crit – NCP_Z) + P(Z < -Z_crit – NCP_Z)
// This is equivalent to norm_cdf(NCP_Z – Z_crit) + norm_cdf(-NCP_Z – Z_crit)
var power = norm_cdf(NCP_Z – Z_crit) + norm_cdf(-NCP_Z – Z_crit);
// Format result
var powerPercentage = (power * 100).toFixed(2);
resultDiv.innerHTML = "The estimated statistical power is: " + powerPercentage + "%";
}