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T-Test Calculator

Q: Raw data or summary statistics?

Either: give sample_a / sample_b as lists, or mean, SD (with n − 1) and n for each group. In paired mode the summary values describe the differences a − b.

Q: Why Welch instead of the classic pooled t-test?

Welch's test keeps the nominal error rate when the variances or sample sizes differ and loses almost nothing when they are equal; its degrees of freedom are fractional. R's t.test uses it by default.

Q: Which p-value should I report?

The two-sided p unless the direction was fixed before seeing the data; the one-sided p given here is for the alternative in the observed direction and is half the two-sided value.

Runs a one-sample, Welch two-sample (unequal variances) or paired Student t-test from raw samples or summary statistics (mean, SD, n), returning t, degrees of freedom, two- and one-sided p-values, the confidence interval and the decision at the chosen alpha.

Test type (optional)Which t-test to run. Sample A values (optional)Raw observations of the first (or only) sample. Alternatively give mean_a, sd_a and n_a. Sample B values (optional)Raw observations of the second sample (two_sample) or the paired partner values in the same order (paired). Alternatively give mean_b, sd_b and n_b. Mean of A (optional)Summary alternative to sample_a: sample mean of A (in paired mode: mean of the differences a − b). SD of A (optional)Sample standard deviation (n − 1) of A, or of the paired differences. n of A (optional)Number of observations in A, or number of pairs. Mean of B (optional)Summary alternative to sample_b (two_sample only): sample mean of B. SD of B (optional)Sample standard deviation (n − 1) of B. n of B (optional)Number of observations in B. Hypothesised mean (μ0) (optional)Null-hypothesis value: the population mean (one_sample) or the mean difference (two_sample, paired). Usually 0 for differences. Significance level (α) (optional)Significance level as a probability (0.05 = 5 %), not a percentage. Confidence intervals and critical values use level 1 − alpha.

When to use

You want to test whether a sample mean differs from a hypothesised value, whether two independent groups have different means, or whether paired before/after measurements changed.

Do not use when: The data are proportions or counts (use proportion-z-test or chi-square-test), you have three or more groups (ANOVA), or you already have t and df and only need the p-value (use p-value).

Formula

one_sample: t = (x̄ − μ0) / (s / √n), df = n − 1. paired: same on the differences d = a − b. two_sample (Welch): t = (x̄a − x̄b − μ0) / √(sa²/na + sb²/nb), df = (sa²/na + sb²/nb)² / ((sa²/na)²/(na − 1) + (sb²/nb)²/(nb − 1)). p = P(|T_df| ≥ |t|); CI = estimate ± t(1 − alpha/2, df) × SE

The two-sample test is Welch's version, which does not assume equal variances and is recommended as the default (the pooled-variance Student test is not offered). p-values come from the regularized incomplete beta function; the test assumes approximately normal data or n large enough for the central limit theorem.

Inputs

Parameter	Type	Required	Description
`mode`	enum: one_sample \| two_sample \| paired	default two_sample	Which t-test to run.
`sample_a`	number_list	no	Raw observations of the first (or only) sample. Alternatively give mean_a, sd_a and n_a.
`sample_b`	number_list	no	Raw observations of the second sample (two_sample) or the paired partner values in the same order (paired). Alternatively give mean_b, sd_b and n_b.
`mean_a`	number	no	Summary alternative to sample_a: sample mean of A (in paired mode: mean of the differences a − b).
`sd_a`	number	no	Sample standard deviation (n − 1) of A, or of the paired differences. Range: > 0
`n_a`	integer	no	Number of observations in A, or number of pairs. Range: ≥ 2
`mean_b`	number	no	Summary alternative to sample_b (two_sample only): sample mean of B.
`sd_b`	number	no	Sample standard deviation (n − 1) of B. Range: > 0
`n_b`	integer	no	Number of observations in B. Range: ≥ 2
`hypothesized_mean`	number	default 0	Null-hypothesis value: the population mean (one_sample) or the mean difference (two_sample, paired). Usually 0 for differences.
`alpha`	number	default 0.05	Significance level as a probability (0.05 = 5 %), not a percentage. Confidence intervals and critical values use level 1 − alpha. Range: > 0

Outputs

Output	Type	Description
`test`	string	Which t-test was run and on how many observations.
`mean_a`	number	Sample mean of A (paired mode: mean of the differences).
`mean_b`	number	Sample mean of B (two_sample only).
`mean_difference`	number	mean_a − hypothesized_mean (one_sample), mean_a − mean_b (two_sample) or mean of a − b (paired).
`standard_error`	number	Standard error of the estimated difference.
`t_statistic`	number	(estimated difference − hypothesized_mean) / standard_error.
`degrees_of_freedom`	number	n − 1 for one-sample and paired tests; Welch–Satterthwaite approximation (fractional) for two samples.
`p_value_two_sided`	number	P(\|T\| ≥ \|t\|) under H0.
`p_value_one_sided`	number	Half the two-sided p: for the alternative in the direction of the observed difference.
`t_critical`	number	t quantile at 1 − alpha/2 for the degrees of freedom.
`ci_lower`	number	Lower limit of the 100·(1 − alpha) % confidence interval for the mean (one_sample) or the mean difference.
`ci_upper`	number	Upper limit of the same interval.
`significant`	boolean	true when p_value_two_sided < alpha.
`decision`	string	Plain-language conclusion of the two-sided test.

Example

Welch: 5.1, 4.9, 5.6, 5.8, 6.0 vs 4.2, 4.8, 4.4, 4.6, 4.5: {"mode":"two_sample","sample_a":[5.1,4.9,5.6,5.8,6],"sample_b":[4.2,4.8,4.4,4.6,4.5]} → {"mean_a":5.48,"mean_b":4.5,"mean_difference":0.98,"standard_error":0.2311,"t_statistic":4.2409,"degrees_of_freedom":5.7504,"p_value_two_sided":0.005979,"p_value_one_sided":0.002989,"t_critical":2.4729,"ci_lower":0.4086,"ci_upper":1.5514,"significant":true}

Paired: Student's sleep data (drug 2 minus drug 1, n = 10): {"mode":"paired","sample_a":[1.9,0.8,1.1,0.1,-0.1,4.4,5.5,1.6,4.6,3.4],"sample_b":[0.7,-1.6,-0.2,-1.2,-0.1,3.4,3.7,0.8,0,2]} → {"mean_difference":1.58,"standard_error":0.389,"t_statistic":4.0621,"degrees_of_freedom":9,"p_value_two_sided":0.002833,"ci_lower":0.7001,"ci_upper":2.4599,"significant":true}

GET https://tttkmbb.com/api/v1/calculate/t-test?mode=two_sample&sample_a=5.1%2C4.9%2C5.6%2C5.8%2C6&sample_b=4.2%2C4.8%2C4.4%2C4.6%2C4.5

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/t-test (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/t-test · Markdown: https://tttkmbb.com/statistics/t-test.md · JSON definition: https://tttkmbb.com/statistics/t-test.json
MCP: server https://tttkmbb.com/mcp, tool calculate_t_test
OpenAPI operationId: calculate_t_test
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

FAQ

Raw data or summary statistics?

Either: give sample_a / sample_b as lists, or mean, SD (with n − 1) and n for each group. In paired mode the summary values describe the differences a − b.

Why Welch instead of the classic pooled t-test?

Welch's test keeps the nominal error rate when the variances or sample sizes differ and loses almost nothing when they are equal; its degrees of freedom are fractional. R's t.test uses it by default.

Which p-value should I report?

The two-sided p unless the direction was fixed before seeing the data; the one-sided p given here is for the alternative in the observed direction and is half the two-sided value.

Related calculators

P-Value Calculator — p-value for a t, z or chi-square statistic you already have.
Cohen's d Effect Size Calculator — Effect size of the difference between the two groups.
Confidence Interval Calculator — z-based interval for a single mean.