Which t test should I use?

One sample if you are testing a single group’s mean against a known value. Two independent samples if the groups are different people or units. Paired if the same subjects were measured twice, or if each observation in one list has a designated partner in the other.

What does a p-value below 0.05 actually mean?

That if the null hypothesis were true, a result at least this extreme would occur less than 5% of the time. It does not mean there is a 5% chance the null is true, and it says nothing about the size or importance of the effect.

Should I use Welch or pooled for two samples?

Welch, unless you have good reason to believe the variances are equal. Welch costs almost nothing in power and protects you when that assumption is wrong. With equal sample sizes the two standard errors coincide, so the practical difference is small.

Why do my degrees of freedom come out as 7.82 instead of 8?

That is the Welch–Satterthwaite value, and it is not a rounding error. Unequal group variances produce a fractional number of degrees of freedom, which is always at or below the pooled n₁ + n₂ − 2.

My result is not significant. Does that mean there is no effect?

No. It means the study could not detect one. With a small sample, a real and important effect can easily produce a non-significant p-value — which is why the confidence interval is worth reading alongside it.

t-test Calculator — Free Online Calculator

What Is the t-test Calculator?

A t test is a significance test for a difference between means, or between one mean and a hypothesised value. It is the workhorse of experimental and observational statistics.

The test works by reducing the observed difference to a standardised statistic and asking how often that statistic, or something more extreme, would arise by chance if the null hypothesis held.

How Does the t-test Calculator Work?

The standard error of the difference depends on the design: s/√n for one sample, √(s₁²/n₁ + s₂²/n₂) for two independent samples, or s_d/√n for paired data. Dividing the difference by that standard error gives t.

How many degrees of freedom remain depends on how the variance was estimated. One sample and paired designs lose one; two samples lose two under pooling, or land on the fractional Welch value when variances differ.

The p-value is the area of the t distribution beyond the statistic in the tail(s) your hypothesis specifies. Comparing it to alpha gives the decision, and the confidence interval expresses the same evidence as a range for the difference.

t-test Calculator Formula & Variables

The core mathematical equation utilized by this calculator is expressed as:

t = \frac{\Delta}{\mathrm{SE}} \qquad p = \Prob\left(T \ge |t|\right) \qquad \mathrm{CI} = \Delta \pm t_{\alpha/2}\,\mathrm{SE}

Variable Definitions

Symbol	Variable Meaning & Units
Δ	difference between the sample mean(s) and the null value
SE	standard error of that difference
t*	critical t for the chosen confidence level and degrees of freedom
α	significance level — 0.05 by convention
df	degrees of freedom available to the estimate

The t statistic is scaled by the standard error, which is why sample size and variability both matter. The p-value is the tail area beyond that statistic on the t distribution, and the confidence interval is the same information expressed as a range — one that excludes zero exactly when the test is significant.

How to Use the t-test Calculator

Select the design that matches your data collection. This is the decision that matters most; the rest is arithmetic.
Set the alternative hypothesis before you look at the result. Two-sided is the default because it is the test that cannot be accused of hindsight.
Leave the variance option at Welch unless equal variances were part of the study design from the outset.
Read three things together: the p-value, the confidence interval and Cohen’s d. The first says whether, the second says how precisely, the third says how much.

Step-by-Step Example Calculation

Two groups: 28.6 (n=5, SD 2.07) against 24.4 (n=5, SD 2.42)

Input Values:

mode:two_sample

direction:two_sided

alpha:0.05

variance:welch

mean1:28.6

sd1:2.0702

n1:5

mean2:24.4

sd2:2.4166

n2:5

Worked Steps: Welch df is 7.82, the two-sided p-value is about 0.0017, and the 95% interval for the difference runs from roughly 1.94 to 6.46 — clear of zero, so the null is rejected.

Understanding Your Result

Rejecting H₀ means the data are inconsistent with the difference being zero (or with the null value). It does not prove the alternative and does not establish importance.

The critical value is the boundary: 2.131 for a two-sided 5% test at 20 degrees of freedom, and larger for smaller df. If |t| exceeds it, the result is significant.

A confidence interval excluding zero is the same conclusion as p < alpha. Read the width to see how much the study actually pinned down.

The decision is about evidence, not about whether an effect exists. A significant result with a tiny effect means the effect is real but negligible; a non-significant result with a huge effect means the study was too small.

Factors That Affect the Result

Sample size, which enters twice: it shrinks the standard error and it raises the degrees of freedom, so more data strengthens the test on both fronts.
Within-group variability. Large SDs inflate the standard error and push t towards zero.
Pairing. Within-subject designs remove subject-level variation and are frequently far more sensitive than comparing independent groups.
The effect size in the population, which is the only term that reflects what is actually there.

When Should You Use This Calculator?

Comparing a measured mean against a target or specification value.
Comparing two independent groups on a continuous outcome.
Comparing paired measurements — before and after, matched subjects, twin or sibling comparisons.
Any situation where the population standard deviation is unknown and must be estimated from the data.

Assumptions & Limitations

Observations must be independent within groups. Cluster sampling, repeated measures on the same units and matched pairs all violate this unless handled explicitly.
The outcome should be approximately normal within each group, or n should be large enough for the central limit theorem to help. With small n and skewed data, the t approximation can be poor.
The test assumes the difference being tested is the difference you care about. Testing several outcomes and reporting only the significant one inflates the false positive rate.
For a two-sample test, Welch’s degrees of freedom are themselves an approximation, accurate for all but the smallest samples.

Frequently Asked Questions

Calculation Accuracy & Reference Note

Tail probabilities use an accurate incomplete-beta implementation, and the critical values are inverse-transformed with bisection refinement — both far more precise than the displayed digits.

Standard Reference: Student’s t test (1908); Welch (1947) degrees of freedom; Cohen (1988) effect sizes.