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Two-Sample T-Test Calculator

Compare the means of two independent groups with the classic pooled Student t-test.

Test
Data entry
Method
Group 1
Group 2
Alternative hypothesis
Significance level (α)

Student's two-sample t-test (equal variances)

t = -12.621, p = < 0.0001
Degrees of freedom
326
Critical values
-1.967 and 1.967
Difference of means
-10.336
Standard error
0.81901
95% interval
-11.95 to -8.725
Cohen's d
-1.63 (large)

p < 0.0001 < α = 0.05, so reject H0: μ1 = μ2. There is evidence that the two population means differ. The sample means differ by -10.34 (95% interval from -11.95 to -8.725). Cohen's d = -1.63, a large effect.

Student or Welch?

The variances differ by a factor of 1.1 and the group sizes are 249 and 79. Use Welch, which does not assume equal variances.

Methodtdfp-value
Welch-12.946136.87< 0.0001
Student-12.621326< 0.0001

Rejection region and t statistic

Rejection region and t statisticRejection region and t statistic. The two tails is shaded. The t is -12.62.-10-50510t (df = 326)t = -12.62

The shaded area is the rejection region (total area 0.05): values below -1.967 or above 1.967. The dashed line is your statistic.

Step-by-step working

  1. 1Hypotheses
    H0: μ1=μ2H_0:\ \mu_1 = \mu_2H0​: μ1​=μ2​
    ↓
    H1: μ1≠μ2H_1:\ \mu_1 \neq \mu_2H1​: μ1​=μ2​

    two-tailed test at α = 0.05

  2. 2Group summaries
    xˉ1, s1, n1 and xˉ2, s2, n2\bar x_1,\ s_1,\ n_1\ \text{and}\ \bar x_2,\ s_2,\ n_2xˉ1​, s1​, n1​ and xˉ2​, s2​, n2​
    ↓
    20.14, 6.415, 249 and 30.48, 6.108, 7920.14,\ 6.415,\ 249\ \text{and}\ 30.48,\ 6.108,\ 7920.14, 6.415, 249 and 30.48, 6.108, 79
  3. 3Pooled variance
    sp2=(n1−1)s12+(n2−1)s22n1+n2−2s_p^2 = \dfrac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}sp2​=n1​+n2​−2(n1​−1)s12​+(n2​−1)s22​​
    ↓
    sp2=248(6.415)2+78(6.108)2326=40.229s_p^2 = \dfrac{248(6.415)^2 + 78(6.108)^2}{326} = 40.229sp2​=326248(6.415)2+78(6.108)2​=40.229
  4. 4Standard error
    SE=sp2(1n1+1n2)SE = \sqrt{s_p^2\left(\dfrac{1}{n_1}+\dfrac{1}{n_2}\right)}SE=sp2​(n1​1​+n2​1​)​
    ↓
    SE=0.81901SE = 0.81901SE=0.81901
  5. 5Degrees of freedom
    df=n1+n2−2df = n_1 + n_2 - 2df=n1​+n2​−2
    ↓
    df=326df = 326df=326
  6. 6t statistic
    t=xˉ1−xˉ2SEt = \dfrac{\bar x_1 - \bar x_2}{SE}t=SExˉ1​−xˉ2​​
    ↓
    t=20.14−30.480.81901=−12.621t = \dfrac{20.14 - 30.48}{0.81901} = -12.621t=0.8190120.14−30.48​=−12.621
  7. 7p-value and critical value
    p and tc (df=326)p\ \text{and}\ t_c\ (df = 326)p and tc​ (df=326)
    ↓
    p=5.273×10−30, tc=−1.967 and 1.967p = 5.273\times 10^{-30},\ t_c = -1.967\ \text{and}\ 1.967p=5.273×10−30, tc​=−1.967 and 1.967
  8. 8Confidence interval
    (xˉ1−xˉ2)±tα/2 SE(\bar x_1 - \bar x_2) \pm t_{\alpha/2}\,SE(xˉ1​−xˉ2​)±tα/2​SE
    ↓
    −10.34±1.967×0.81901=[−11.95, −8.725]-10.34 \pm 1.967\times 0.81901 = [-11.95,\ -8.725]−10.34±1.967×0.81901=[−11.95, −8.725]
  9. 9Cohen's d
    d=xˉ1−xˉ2spd = \dfrac{\bar x_1 - \bar x_2}{s_p}d=sp​xˉ1​−xˉ2​​
    ↓
    d=−10.346.3426=−1.63d = \dfrac{-10.34}{6.3426} = -1.63d=6.3426−10.34​=−1.63

    large effect

  10. 10Decision
    p<αp < \alphap<α
    ↓
    reject H0\text{reject } H_0reject H0​
  1. Tools
  2. Statistics
  3. T-Test Calculator
  4. Two-Sample T-Test Calculator

About Two-Sample T-Test Calculator

This two-sample t-test calculator compares the means of two independent groups with the Student test, which pools the variances, and shows Welch alongside for comparison. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

Enter the mean, standard deviation and size of each group, or paste the raw data. The result gives the pooled variance, the standard error, the t statistic with n1 + n2 minus 2 degrees of freedom, the p-value, the critical value and Cohen’s d.

The page opens on the NIST car-mileage example, which uses this exact test. A recommendation line tells you whether Welch would be the safer choice for your variances and group sizes.

How to use Two-Sample T-Test Calculator

  1. 1Choose the test: one-sample, paired or two-sample.
  2. 2Choose raw data or summary statistics. For raw data, paste each list; for paired data, keep the two lists in the same order and the same length.
  3. 3For a one-sample test type the hypothesised mean. For two samples pick Welch or Student.
  4. 4Pick the alternative hypothesis (two-tailed, right or left) and the significance level.
  5. 5Read the t statistic, p-value, interval and Cohen's d, then open the working to see every formula.

Features

  • One-sample, paired and independent two-sample t-tests.
  • Welch (unequal variances) and Student (pooled) two-sample tests, compared side by side with a recommendation.
  • Raw data or summary statistics (mean, standard deviation, n).
  • Cohen's d with a size label, plus the standard error and a confidence interval.
  • Left, right or two-tailed tests, with the critical value for your significance level.
  • A bell curve with the rejection region shaded and the t statistic marked.
  • Step-by-step KaTeX working and a plain-English conclusion.

Student two-sample t-test

The two sample variances are pooled into one estimate, weighted by their degrees of freedom.

s_p² = ((n1 − 1) s1² + (n2 − 1) s2²) / (n1 + n2 − 2)
t = (x̄1 − x̄2) / √(s_p² (1/n1 + 1/n2)),   df = n1 + n2 − 2

Worked examples

  • US cars: n 249, mean 20.14458, SD 6.41470. Japanese cars: n 79, mean 30.48101, SD 6.10771 → gives t = -12.6206, df = 326, critical values ±1.9673

    NIST/SEMATECH e-Handbook, section 1.3.5.3: T = -12.62059, df 326, critical value 1.9673.

Privacy: how it works

  • The test is calculated by code in your own browser tab. Your data is not sent to Ulkaa or to any other server.
  • There is no account and nothing is stored. Closing the tab clears what you entered.

Tips

  • Use the paired test when the same people or items are measured twice. Using the two-sample test on paired data throws away the pairing and usually loses power.
  • Decide one-tailed or two-tailed before you see the data. A two-tailed test is the safe default.
  • Report the effect size and the interval with the p-value. A tiny p-value with a negligible Cohen's d is statistically significant but may not matter in practice.
  • If your lists have different lengths, that is fine for the two-sample test, but the paired test needs equal lengths.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Your data are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Your data are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

What does the Student t-test assume?

Independent observations, roughly normal data in each group and equal population variances. If the variances may differ, use Welch instead.

How are the degrees of freedom found?

They are n1 plus n2 minus 2. For the NIST example, 249 plus 79 minus 2 is 326.

More T-Test Calculator pages

  • T-Test CalculatorRun a one-sample, paired or two-sample t-test from raw data or summary statistics, with Cohen's d, the p-value and the working.
  • One-Sample T-Test CalculatorCompare the mean of one sample with a value you expect and see the t statistic, p-value and working.
  • Paired T-Test CalculatorPaste two matched lists and get the paired t statistic, p-value and working.
  • Welch's T-Test CalculatorCompare two means without assuming equal variances, with Welch-Satterthwaite degrees of freedom.
  • Cohen's d CalculatorGet Cohen's d for two groups from summary statistics, with the matching t-test and a size label.

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