Hypothesis Test Calculator (t, z, chi-square)

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two sample t test calculator

Two-sample t test calculator (Welch and pooled)

Type the mean, standard deviation and size of each group, or paste the raw data, and the "t, 2 means" tab does the rest. The default is Welch, which does not require equal variances; the pooled t is one click away. The result brings the written hypothesis, the p-value, the rejection region on the chart and a sentence ready for your report.

Welch, pooled and what the difference between them teaches

The two-sample t test checks whether the population means of two groups differ, with H0: μ₁ − μ₂ = 0 (or another value δ₀, if you want to test a minimum difference). The statistic is t = (x̄₁ − x̄₂ − δ₀) / SE. In the pooled t, SE = √(sp²(1/n₁ + 1/n₂)) uses a combined variance sp² and has n₁ + n₂ − 2 degrees of freedom. In Welch, SE = √(s₁²/n₁ + s₂²/n₂) and the degrees of freedom come from the Welch-Satterthwaite formula, almost always fractional. That is why the Welch df tends to be smaller.
The classic NIST/SEMATECH example compares the fuel economy of 249 American cars (mean 20.14458 mpg, s = 6.41470) with 79 Japanese ones (30.48101 mpg, s = 6.10771). The pooled t gives −12.6206 with 326 df, and the two-sided 5% critical value is ±1.9673, so H0 falls by a wide margin. Welch gives −12.946 with 136.9 df. Both conclude the same because the variances are similar; when groups differ in size and variance, such as 12 and 15 observations with s = 9.1 and 14.3, Welch (t = 1.258, df = 23.95, p = 0.2206) and pooled (t = 1.198, df = 25, p = 0.2422) already part ways.
Beyond the p-value, look at the effect size and the confidence interval of the difference. Cohen d divides the difference by the pooled standard deviation: for the cars it is −1.63 (large); in the 12 and 15 observation example, 0.46 (small to medium), with a 95% CI of −3.65 to 15.05, which includes zero. If the same subjects were measured twice, the right test is the paired one, and for three or more groups ANOVA avoids the accumulated error of several t tests.

Checked examples

US vs Japanese cars, pooled t (NIST)

Input: n 249, x̄ 20,14458, s 6,41470 × n 79, x̄ 30,48101, s 6,10771
Expected output: t = −12,6206, gl = 326, crítico ±1,9673

Rejects H0 at 5%.

Same cars, Welch

Input: mesmos dados / same data
Expected output: t = −12,946, gl = 136,87

p of the order of 1.6e-25.

Two small groups, Welch

Input: n 12, x̄ 84,2, s 9,1 × n 15, x̄ 78,5, s 14,3
Expected output: t = 1,258, gl = 23,95, p = 0,2206

Fails to reject H0; d = 0.46.

Same groups, pooled

Input: mesmos dados / same data
Expected output: t = 1,198, gl = 25, p = 0,2422

95% CI of the difference: −4.10 to 15.50.

Student sleep data, Welch

Input: 10 valores por remédio / 10 values per drug
Expected output: t = −1,8608, gl = 17,776, p = 0,0794

Same data in the paired t: p = 0.0028.

Full tool FAQ

How do I interpret the p-value?

The p-value is the probability of observing a statistic as extreme as your sample, or more, assuming H0 is true. It is not the probability that H0 is true. With p = 0.0794 and α = 0.05, the data are not "unlikely enough" to reject H0; with p = 0.0028, they are. Very small p-values appear in scientific notation: 2.5e-7 means 0.00000025.

Why say "fail to reject H0" instead of "accept H0"?

z test or t test for one mean?

Welch or pooled t for two means?

One-sided or two-sided test?

Chi-square goodness of fit or independence? And what if expected counts are small?

When should I use the paired t instead of the two-sample t?

What are Cohen d, Cramér V and η²?

The ANOVA was significant. Now what?

Does my data need to be normal? Can I trust the calculations?

Frequently asked questions

Welch or pooled, which one?

When in doubt, Welch. It loses almost no power when variances are equal and protects the p-value when they are not. The pooled version is only justified with clearly equal variances and similar sizes.

Can I paste the data instead of typing the means?

Yes. Switch to "Raw data" and paste each group into its box, using new lines, semicolons, commas or spaces and decimal commas. The calculator computes n, mean and standard deviation and also runs the Levene test.

How do I test a minimum difference, not just zero?

In "Advanced options", enter the difference under H0 (δ₀). With δ₀ = 5 and one-sided right, for example, the alternative hypothesis becomes μ₁ − μ₂ > 5.

Next steps

Data summary
Group	Size (n)	Mean	s
Group 1	10	0.75	1.789
Group 2	10	2.33	2.0022

Data summary

Group

Size (n)

Mean

Group 1

0.75

1.789

Group 2

2.33

2.0022

Two-sample t test calculator (Welch and pooled)

Welch, pooled and what the difference between them teaches

The two-sample t test checks whether the population means of two groups differ, with H0: μ₁ − μ₂ = 0 (or another value δ₀, if you want to test a minimum difference). The statistic is t = (x̄₁ − x̄₂ − δ₀) / SE. In the pooled t, SE = √(sp²(1/n₁ + 1/n₂)) uses a combined variance sp² and has n₁ + n₂ − 2 degrees of freedom. In Welch, SE = √(s₁²/n₁ + s₂²/n₂) and the degrees of freedom come from the Welch-Satterthwaite formula, almost always fractional. That is why the Welch df tends to be smaller.

The classic NIST/SEMATECH example compares the fuel economy of 249 American cars (mean 20.14458 mpg, s = 6.41470) with 79 Japanese ones (30.48101 mpg, s = 6.10771). The pooled t gives −12.6206 with 326 df, and the two-sided 5% critical value is ±1.9673, so H0 falls by a wide margin. Welch gives −12.946 with 136.9 df. Both conclude the same because the variances are similar; when groups differ in size and variance, such as 12 and 15 observations with s = 9.1 and 14.3, Welch (t = 1.258, df = 23.95, p = 0.2206) and pooled (t = 1.198, df = 25, p = 0.2422) already part ways.

Beyond the p-value, look at the effect size and the confidence interval of the difference. Cohen d divides the difference by the pooled standard deviation: for the cars it is −1.63 (large); in the 12 and 15 observation example, 0.46 (small to medium), with a 95% CI of −3.65 to 15.05, which includes zero. If the same subjects were measured twice, the right test is the paired one, and for three or more groups ANOVA avoids the accumulated error of several t tests.

Checked examples

US vs Japanese cars, pooled t (NIST)

Input: n 249, x̄ 20,14458, s 6,41470 × n 79, x̄ 30,48101, s 6,10771
Expected output: t = −12,6206, gl = 326, crítico ±1,9673

Rejects H0 at 5%.

Same cars, Welch

Input: mesmos dados / same data
Expected output: t = −12,946, gl = 136,87

p of the order of 1.6e-25.

Two small groups, Welch

Input: n 12, x̄ 84,2, s 9,1 × n 15, x̄ 78,5, s 14,3
Expected output: t = 1,258, gl = 23,95, p = 0,2206

Fails to reject H0; d = 0.46.

Same groups, pooled

Input: mesmos dados / same data
Expected output: t = 1,198, gl = 25, p = 0,2422

95% CI of the difference: −4.10 to 15.50.

Student sleep data, Welch

Input: 10 valores por remédio / 10 values per drug
Expected output: t = −1,8608, gl = 17,776, p = 0,0794

Same data in the paired t: p = 0.0028.

Full tool FAQ

How do I interpret the p-value?

Why say "fail to reject H0" instead of "accept H0"?

z test or t test for one mean?

Welch or pooled t for two means?

One-sided or two-sided test?

Chi-square goodness of fit or independence? And what if expected counts are small?

When should I use the paired t instead of the two-sample t?

What are Cohen d, Cramér V and η²?

The ANOVA was significant. Now what?

Does my data need to be normal? Can I trust the calculations?

Frequently asked questions

Welch or pooled, which one?

Can I paste the data instead of typing the means?

How do I test a minimum difference, not just zero?

In "Advanced options", enter the difference under H0 (δ₀). With δ₀ = 5 and one-sided right, for example, the alternative hypothesis becomes μ₁ − μ₂ > 5.

Data summary
Group	Size (n)	Mean	s
Group 1	10	0.75	1.789
Group 2	10	2.33	2.0022

Data summary

Group

Size (n)

Mean

Group 1

0.75

1.789

Group 2

2.33

2.0022