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Welch's T-Test Calculator

Compare two means without assuming equal variances, with Welch-Satterthwaite degrees of freedom.

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

Welch's two-sample t-test

t = 2.323, p = 0.026
Degrees of freedom
34.805
Critical values
-2.031 and 2.031
Difference of means
5.2
Standard error
2.2385
95% interval
0.6547 to 9.745
Cohen's d
0.634 (medium)

p = 0.026 < α = 0.05, so reject H0: μ1 = μ2. There is evidence that the two population means differ. The sample means differ by 5.2 (95% interval from 0.6547 to 9.745). Cohen's d = 0.634, a medium effect.

Student or Welch?

The variances differ by a factor of 5.37 and the group sizes are 12 and 25. Use Welch, which does not assume equal variances.

Methodtdfp-value
Welch2.32334.8050.026
Student1.8067350.079

Rejection region and t statistic

Rejection region and t statisticRejection region and t statistic. The two tails is shaded. The t is 2.323.-4-3-2-101234t (df = 34.81)t = 2.323

The shaded area is the rejection region (total area 0.05): values below -2.031 or above 2.031. 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​
    ↓
    83.2, 4.1, 12 and 78, 9.5, 2583.2,\ 4.1,\ 12\ \text{and}\ 78,\ 9.5,\ 2583.2, 4.1, 12 and 78, 9.5, 25
  3. 3Standard error
    SE=s12n1+s22n2SE = \sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}SE=n1​s12​​+n2​s22​​​
    ↓
    SE=16.8112+90.2525=2.2385SE = \sqrt{\dfrac{16.81}{12}+\dfrac{90.25}{25}} = 2.2385SE=1216.81​+2590.25​​=2.2385
  4. 4Welch-Satterthwaite df
    df=(s12/n1+s22/n2)2(s12/n1)2n1−1+(s22/n2)2n2−1df = \dfrac{(s_1^2/n_1 + s_2^2/n_2)^2}{\frac{(s_1^2/n_1)^2}{n_1-1}+\frac{(s_2^2/n_2)^2}{n_2-1}}df=n1​−1(s12​/n1​)2​+n2​−1(s22​/n2​)2​(s12​/n1​+s22​/n2​)2​
    ↓
    df=34.805df = 34.805df=34.805
  5. 5t statistic
    t=xˉ1−xˉ2SEt = \dfrac{\bar x_1 - \bar x_2}{SE}t=SExˉ1​−xˉ2​​
    ↓
    t=83.2−782.2385=2.323t = \dfrac{83.2 - 78}{2.2385} = 2.323t=2.238583.2−78​=2.323
  6. 6p-value and critical value
    p and tc (df=34.81)p\ \text{and}\ t_c\ (df = 34.81)p and tc​ (df=34.81)
    ↓
    p=0.02615, tc=−2.031 and 2.031p = 0.02615,\ t_c = -2.031\ \text{and}\ 2.031p=0.02615, tc​=−2.031 and 2.031
  7. 7Confidence interval
    (xˉ1−xˉ2)±tα/2 SE(\bar x_1 - \bar x_2) \pm t_{\alpha/2}\,SE(xˉ1​−xˉ2​)±tα/2​SE
    ↓
    5.2±2.031×2.2385=[0.6547, 9.745]5.2 \pm 2.031\times 2.2385 = [0.6547,\ 9.745]5.2±2.031×2.2385=[0.6547, 9.745]
  8. 8Cohen's d
    d=xˉ1−xˉ2spd = \dfrac{\bar x_1 - \bar x_2}{s_p}d=sp​xˉ1​−xˉ2​​
    ↓
    d=5.28.1957=0.6345d = \dfrac{5.2}{8.1957} = 0.6345d=8.19575.2​=0.6345

    medium effect

  9. 9Decision
    p<αp < \alphap<α
    ↓
    reject H0\text{reject } H_0reject H0​
  1. Tools
  2. Statistics
  3. T-Test Calculator
  4. Welch's T-Test Calculator

About Welch's T-Test Calculator

This Welch's t-test calculator compares the means of two independent groups without assuming that they have equal variances, and shows the Student test next to it. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

Welch’s test uses each group’s own variance in the standard error and adjusts the degrees of freedom with the Welch-Satterthwaite formula, which is usually not a whole number. That is why it stays accurate when the groups have different spreads or very different sizes.

The page opens on two groups with unequal spread and size. The table under the answer shows both tests, so you can see how much the choice changes the p-value.

How to use Welch's 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.

Welch's t-test

The standard error adds the variance of each mean, and the degrees of freedom come from the Welch-Satterthwaite approximation.

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

Worked examples

  • Group 1: mean 83.2, SD 4.1, n 12. Group 2: mean 78, SD 9.5, n 25 → gives Welch t = 2.323, df = 34.81, p = 0.026. Student t = 1.807, df = 35, p = 0.079

    When the smaller group also has the smaller spread, Student and Welch can give clearly different answers.

  • Sleep data, drug 1 and drug 2, raw values → gives t = -1.861, df = 17.78, p = 0.079

    R: t.test(extra ~ group, data = sleep) gives t = -1.8608, df = 17.776, p = 0.07939.

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.

Why are the degrees of freedom not a whole number?

Welch’s formula estimates an effective degrees of freedom between the smaller group’s n minus 1 and n1 + n2 minus 2. Software uses the fractional value, and so does this calculator.

Should I always use Welch?

It is a sound default. It gives nearly the same result as Student when the variances are equal and the correct one when they are not, so there is little to lose.

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.
  • Two-Sample T-Test CalculatorCompare the means of two independent groups with the classic pooled Student t-test.
  • 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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