Hypothesis Tests
Welch's T-Test Calculator
Welch's t test compares two means without assuming equal spread, and it spends a Satterthwaite degrees of freedom instead of pooling.
Worksheet
Press Calculate to see the result.
Welch's t test compares two means without assuming equal spread, and it spends a Satterthwaite degrees of freedom instead of pooling.
What this calculator answers
Welch's t test is the two-mean comparison that refuses to pretend the groups share a variance. Each sample keeps its own variance, the standard error is the square root of s1 squared over n1 plus s2 squared over n2, and the degrees of freedom are the Satterthwaite fraction rather than n1 plus n2 minus 2. That fraction gets smaller when one variance dominates, which keeps the critical t honest. The loaded example is built so a pooled test and Welch do not tell the same decision story. Lists are preferred. The summary boxes are the fallback when both lists are empty.
The headline is the Welch t. The page also shows the Satterthwaite df, the unpooled standard error, the p-value, the critical t, and the decision. Nothing is pooled. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Welch t statistic uses this relationship.
t = (mean1 - mean2) / sqrt(s1^2/n1 + s2^2/n2). df = (u1+u2)^2 / (u1^2/(n1-1) + u2^2/(n2-1)), where u = s^2/n.
- u1 and u2. Each sample variance divided by its sample size.
- se. Square root of u1 plus u2. Not a pooled standard error.
- df. Satterthwaite's approximation, often a fraction.
- t. Difference of means divided by the Welch standard error.
The variance of a difference of independent means is the sum of the two variances of the means, with no shared-variance step. The distribution of that ratio is approximated by a t curve whose df matches the first two moments of the mixture, which is the Satterthwaite quotient. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The loaded lists are 8, 18, 9, 17 and a tight group of twelve scores at 5 and 6.
The lists were used. Group 1 has 4 values and group 2 has 12. Means are 13 and 5.4167. Sample variances are 27.3333 and 0.2652. They are not pooled. u1 equals s1 squared over n1, which is 6.8333. u2 equals s2 squared over n2, which is 0.0221.
The Welch standard error is the square root of u1 plus u2, which is 2.6183. Welch t is 2.8963 and the Satterthwaite df is 3.0194. The two-sided p-value is 0.0622 and the critical t at alpha 0.05 is 3.1709.
At the 0.05 level, the result is outside the rejection region. The data are still compatible with the null hypothesis. A pooled test on the same numbers would report t equal to 5.3332 on 14 degrees of freedom, with p-value 0.0001. That pooled figure is shown only as a contrast. It is not the result of this page.
| Welch t | 2.8963 |
|---|---|
| P-value | 0.0622 |
| Critical t | 3.1709 |
| Satterthwaite df | 3.0194 |
| Decision | At the 0.05 level, the result is outside the rejection region. The data are still compatible with the null hypothesis. |
A second situation
The same comparison typed as summaries, with the lists cleared
Both lists were empty, so the summaries were used for n1 equal to 4 and n2 equal to 12. Means are 13 and 5.4167. Sample variances are 27.333 and 0.2651. They are not pooled. u1 equals s1 squared over n1, which is 6.8333. u2 equals s2 squared over n2, which is 0.0221.
The Welch standard error is the square root of u1 plus u2, which is 2.6183. Welch t is 2.8963 and the Satterthwaite df is 3.0194. The two-sided p-value is 0.0622 and the critical t at alpha 0.05 is 3.1709.
At the 0.05 level, the result is outside the rejection region. The data are still compatible with the null hypothesis. A pooled test on the same numbers would report t equal to 5.3332 on 14 degrees of freedom, with p-value 0.0001. That pooled figure is shown only as a contrast. It is not the result of this page.
| Welch t | 2.8963 |
|---|---|
| P-value | 0.0622 |
| Critical t | 3.1709 |
| Satterthwaite df | 3.0194 |
| Decision | At the 0.05 level, the result is outside the rejection region. The data are still compatible with the null hypothesis. |
The summary path uses the same unpooled formula once the lists are cleared. A pooled test on a lopsided pair of spreads can cross alpha while Welch stays outside the rejection region, which is why the procedures are not aliases. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
Group 1 was replaced by a tight cluster at 12, 14, 13, and 15. Group 2 stayed put. The headline result went from 2.8963 to 12.2033.
The nudged inputs
The lists were used. Group 1 has 4 values and group 2 has 12. Means are 13.5 and 5.4167. Sample variances are 1.6667 and 0.2652. They are not pooled. u1 equals s1 squared over n1, which is 0.4167. u2 equals s2 squared over n2, which is 0.0221.
The Welch standard error is the square root of u1 plus u2, which is 0.6624. Welch t is 12.2033 and the Satterthwaite df is 3.3241. The two-sided p-value is 0.0007 and the critical t at alpha 0.05 is 3.0139.
At the 0.05 level, the result is in the rejection region. The data are unlikely if the null hypothesis is the whole story. A pooled test on the same numbers would report t equal to 18.6185 on 14 degrees of freedom, with p-value 2.831e−11. That pooled figure is shown only as a contrast. It is not the result of this page.
| Welch t | 12.2033 |
|---|---|
| P-value | 0.0007 |
| Critical t | 3.0139 |
| Satterthwaite df | 3.3241 |
| Decision | At the 0.05 level, the result is in the rejection region. The data are unlikely if the null hypothesis is the whole story. |
How to read the result
Read the fractional df as part of the result, not as a rounding error. The p-value is a t tail on that fractional df. The decision uses your alpha. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Quote the Welch t, the fractional df, and both sample sizes. Rounding the df to an integer before you look up the p-value changes the test. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- The groups are independent.
- Each group is close enough to normal for its mean to be well behaved.
- The sample variances are the n minus 1 versions.
- You are unwilling to assume the population variances are equal.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Pooling first and then calling the result Welch.
- Replacing the fractional df with n1 plus n2 minus 2.
- Using this test on paired rows. Pairing is a different standard error.
- Typing variances into the sd boxes. The boxes want standard deviations when you use summaries.
Where a different calculator is the honest choice
The two-sample t-test calculator is the pooled procedure. The degrees of freedom calculator can show the Satterthwaite fraction alone. The t-test chooser does not run Welch. It tells you when Welch is the safer two-group page. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Two Sample T-Test Calculator, Degrees of Freedom Calculator, T-Test Calculator.
Check yourself
Group 1 is 8, 18, 9, 17 and group 2 is twelve values clustered just above 5. What is the Welch t?
The Welch t statistic is 2.8963 on 3.0194 degrees of freedom.
Using the result in a sentence
A precise instrument compared with a noisy one, or a large calm group compared with a small wild group, is where the unpooled standard error earns its keep. Similar spreads can stay on the pooled page. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
Welch is safer than pooling when spreads differ, and it is still a normal-means test. A handful of points in the noisy group can dominate the df, and the test does not pick a winner for a decision that is not about means. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Welch's T-Test Calculator commit to?
This page uses Welch t statistic: t = (mean1 - mean2) / sqrt(s1^2/n1 + s2^2/n2). df = (u1+u2)^2 / (u1^2/(n1-1) + u2^2/(n2-1)), where u = s^2/n.. The variance of a difference of independent means is the sum of the two variances of the means, with no shared-variance step. The distribution of that ratio is approximated by a t curve whose df matches the first two moments of the mixture, which is the Satterthwaite quotient. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Group 1 data, Group 2 data, Mean 1, s1, n1, Mean 2, and related inputs, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the welch t is 2.8963. Read the fractional df as part of the result, not as a rounding error. The p-value is a t tail on that fractional df. The decision uses your alpha. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Pooling first and then calling the result Welch. Welch is safer than pooling when spreads differ, and it is still a normal-means test. A handful of points in the noisy group can dominate the df, and the test does not pick a winner for a decision that is not about means. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
The two-sample t-test calculator is the pooled procedure. The degrees of freedom calculator can show the Satterthwaite fraction alone. The t-test chooser does not run Welch. It tells you when Welch is the safer two-group page. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The Welch t statistic is 2.8963 on 3.0194 degrees of freedom. Quote the Welch t, the fractional df, and both sample sizes. Rounding the df to an integer before you look up the p-value changes the test. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.