What is the difference between Welch's and Student's t-test?

Student's t-test assumes equal variances in both groups (pooled variance). Welch's t-test does NOT assume equal variances and uses the Welch-Satterthwaite formula for degrees of freedom. Welch's test is recommended as the default because it is more robust and performs well even when variances are equal.

When should I use a paired t-test instead of an independent two-sample t-test?

Use a paired t-test when observations are naturally matched or repeated on the same subjects (before/after, left/right, matched pairs). Use an independent two-sample t-test when the two groups are completely separate with no pairing relationship.

How is the pooled standard deviation calculated for Cohen's d?

For the two-sample Cohen's d, the pooled SD is: s_pooled = sqrt(((n1-1)*s1^2 + (n2-1)*s2^2) / (n1+n2-2)). This weights each group's variance by its degrees of freedom. Cohen's d = (mean1 - mean2) / s_pooled.

What are the Welch-Satterthwaite degrees of freedom?

The Welch-Satterthwaite formula estimates effective degrees of freedom when variances are unequal: df = (s1^2/n1 + s2^2/n2)^2 / ((s1^2/n1)^2/(n1-1) + (s2^2/n2)^2/(n2-1)). This is typically not an integer and results in a more conservative test than assuming equal variances.

Why is a paired t-test more powerful than an independent t-test?

The paired t-test removes between-subject variability by analyzing differences within pairs. This typically produces a smaller standard error and larger t-statistic, making it easier to detect true effects. The tradeoff is that you need matched/repeated data.

What happens if my paired samples have different lengths?

Paired samples MUST have the same number of observations because each measurement in group 1 is paired with exactly one measurement in group 2. If lengths differ, you cannot perform a paired t-test -- use an independent two-sample t-test instead.

Can I use a paired t-test with more than two time points?

The paired t-test compares exactly two conditions. For 3+ time points or conditions, use repeated measures ANOVA. However, you can use pairwise paired t-tests between specific pairs with a Bonferroni correction for multiple comparisons.

All Tools

T-Test Calculator

Perform one-sample, independent two-sample (Welch's), and paired t-tests with step-by-step solutions and Python code.

One-Sample T-Test

Input Type

Data Values (comma, space, or newline separated)

Hypothesized Mean (μ₀)

Tail Type

Sample Data:

Enter your data and click Run T-Test to see results.

Quick Reference:

One-Sample: Compare sample mean to a known value
Two-Sample: Compare means of two independent groups
Paired: Compare matched/before-after data

Significance Levels:

p < 0.05: Significant (reject H₀)
p < 0.01: Very significant
p < 0.001: Highly significant

Frequently Asked Questions

What are the assumptions of a one-sample t-test?

The one-sample t-test assumes: (1) data are continuous, (2) observations are independent, (3) the sample is randomly selected, and (4) the population is approximately normally distributed (less critical for n > 30 by CLT). The test is robust to moderate violations of normality.

When should I use a one-sample t-test vs a z-test?

Use a t-test when the population standard deviation is unknown (almost always in practice). Use a z-test only when the population standard deviation is known. For large samples (n > 100), the results are virtually identical since the t-distribution converges to the normal.

How do I interpret Cohen's d effect size?

Cohen's d measures the standardized difference between means. Guidelines: |d| = 0.2 is small, |d| = 0.5 is medium, |d| = 0.8 is large. However, context matters -- a small effect size in medicine can still be clinically significant.

What does the confidence interval for the mean tell me?

The 95% confidence interval provides a range of plausible values for the true population mean. If the hypothesized value falls outside this interval, the test will be significant at the 5% level. It gives more information than just a p-value.

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Frequently Asked Questions