QUICK PEEKQUICK STATS · TWO GROUPS

Welch's t-test calculator, with Cohen's d and an APA sentence.

Q: Why Welch's t-test and not Student's?

Welch's test doesn't assume the two groups have equal variances, and when they do, it gives almost the same answer as Student's test. That's why many statisticians recommend it as the default for two independent groups.

Q: How are the Welch degrees of freedom calculated?

With the Welch–Satterthwaite formula: df = (s1²/n1 + s2²/n2)² / [(s1²/n1)²/(n1 − 1) + (s2²/n2)²/(n2 − 1)]. The result is usually not a whole number; the calculator reports it to two decimal places, as in t(20.87).

Q: How is the p value computed?

Exactly, from the t distribution's cumulative distribution function with the fractional degrees of freedom, via the regularized incomplete beta function. It isn't looked up in a table. The p value is two-sided, and α = .05 decides whether the sentence says the groups differed significantly.

Q: How is Cohen's d calculated here?

d = (mean of group A − mean of group B) divided by the pooled standard deviation, √[((n1 − 1)s1² + (n2 − 1)s2²) / (n1 + n2 − 2)]. No small-sample (Hedges' g) correction is applied. The label uses Cohen's rough benchmarks on |d|: below 0.2 negligible, below 0.5 small, below 0.8 medium, otherwise large.

Q: Can I use it for paired data or a one-sided test?

No. This calculator is for two independent groups and gives a two-sided p value. Before-and-after measurements on the same people need a paired t-test, which this peek doesn't run.

Q: Is my data uploaded?

No. The numbers stay in your browser while you work. If you press Share result, the two groups of numbers and their names are written into the link itself, so anyone who opens it, and our server, which draws the page and its preview image, sees them. If you leave your email, we only record the group sizes and whether the result was significant.

Paste two independent groups of numbers to compare their means with Welch's t-test, which doesn't assume equal variances. You get t, the Welch–Satterthwaite degrees of freedom, an exact two-sided p value, Cohen's d, a 95% confidence interval for the difference and an APA-style sentence to adapt.

Compare two groups All quick peeks

NO ACCOUNT · RUNS IN YOUR BROWSER · ABOUT 30 SECONDS

Fig 0CONTROL VS TREATMENT

Two samples, one question: is the difference bigger than the noise?

FIG 1THE PEEK

Paste the numbers, read the test.

Two made-up groups are pre-filled. Rename them and paste your own values; everything updates as you type.

Fig 1WELCH t · COHEN'S d · APA

GROUP A · NUMBERS12 VALUES

GROUP B · NUMBERS12 VALUES

SEPARATE WITH SPACES, COMMAS OR NEW LINES · USE A POINT FOR DECIMALS · SAMPLE NUMBERS ARE MADE UP

THE LINK CONTAINS YOUR INPUTS AND REOPENS THIS RESULT

−4.61

WELCH t

20.87

< .001

p · TWO-SIDED

−1.88

COHEN'S d · LARGE

−0.89

DIFF · 95% CI −1.29 TO −0.49

DESCRIPTIVE STATISTICS

	CONTROL	TREATMENT
n	12	12
Mean	6.29	7.18
SD	0.53	0.42
Median	6.25	7.15
Min	5.50	6.60
Max	7.10	7.90

Cohen's d = −1.88 is a large effect by Cohen's rough benchmarks (0.2 small, 0.5 medium, 0.8 large). What counts as a meaningful difference depends on your field.

BOX + STRIP PLOT◇ MEAN · ▬ MEDIAN

BOX = QUARTILES · WHISKERS = MIN TO MAX · DOTS = EACH VALUE

APA-STYLE SENTENCE · ADAPT IT TO YOUR STUDY

A Welch's t-test showed that Control (M = 6.29, SD = 0.53) had a significantly lower mean than Treatment (M = 7.18, SD = 0.42), t(20.87) = −4.61, p < .001, d = −1.88.

α = .05 · TWO-SIDED · d USES THE POOLED SD · CHECK YOUR COURSE'S REPORTING RULES

RUNS ENTIRELY IN YOUR BROWSER — NOTHING IS UPLOADED.

FIG 2HOW TO USE

How to use the Welch's t-test calculator

01 →
Paste two groups
Numbers separated by spaces, commas or new lines, at least two per group. Name the groups so the sentence reads naturally.
02 →
Read the test
t, df, the two-sided p value, Cohen's d and the 95% CI of the difference update as you type, with descriptives and a box plot.
03
Adapt the sentence, then share
Copy the APA-style sentence into your results section and edit it to fit your study, or share a link that reopens this exact analysis.

MethodWHAT THE CODE DOES

How it works: the method

Means and sample standard deviations (dividing by n − 1) are computed for each group. Then:

t  = (M₁ − M₂) / SE,   SE = √(s₁²/n₁ + s₂²/n₂)
df = (s₁²/n₁ + s₂²/n₂)²
     / [ (s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1) ]
p  = I_{df/(df+t²)}(df/2, 1/2)   (two-sided)
95% CI = (M₁ − M₂) ± t₀.₉₇₅,df × SE
d  = (M₁ − M₂) / s_pooled
s_pooled = √[ ((n₁−1)s₁² + (n₂−1)s₂²) / (n₁+n₂−2) ]

The p value comes from the regularized incomplete beta function, so fractional degrees of freedom are handled exactly. Medians and quartiles for the table and box plot use linear interpolation between order statistics (R type 7, Excel QUARTILE.INC).

Example12 VS 12 · MADE-UP DATA

Worked example

The pre-filled groups (made-up numbers) give Control n = 12, M = 6.29, SD = 0.53 and Treatment n = 12, M = 7.18, SD = 0.42.

difference = 6.29 − 7.18 = −0.89
SE = 0.194,  t = −4.61,  df = 20.87
p < .001 (p ≈ .00015)
95% CI −1.29 to −0.49
s_pooled = 0.474,  d = −1.88 (large)

The sentence it writes: A Welch's t-test showed that Control (M = 6.29, SD = 0.53) had a significantly lower mean than Treatment (M = 7.18, SD = 0.42), t(20.87) = −4.61, p < .001, d = −1.88.

FIG 3THE FULL WORKSPACE

What the full workspace adds.

Fig 3IN MOREMIND

✓Describe the analysis you need and get R or Python code you can read, run and keep with your data
✓Go beyond two groups: paired tests, ANOVA, non-parametric tests and regression, in code you control
✓Charts and figures for your results, numbered and captioned
✓Verified references for the methods you cite
✓You interpret the results; MoreMind doesn't write your discussion

You still write the argument.

MoreMind handles the mechanics around your work (sources, data, figures and formatting) so your time goes into the thinking. Every result is something you check and decide on.

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FIG 4QUESTIONS

Welch's t-test: common questions

Why Welch's t-test and not Student's?

Welch's test doesn't assume the two groups have equal variances, and when they do, it gives almost the same answer as Student's test. That's why many statisticians recommend it as the default for two independent groups.

How are the Welch degrees of freedom calculated?

With the Welch–Satterthwaite formula: df = (s1²/n1 + s2²/n2)² / [(s1²/n1)²/(n1 − 1) + (s2²/n2)²/(n2 − 1)]. The result is usually not a whole number; the calculator reports it to two decimal places, as in t(20.87).

How is the p value computed?

Exactly, from the t distribution's cumulative distribution function with the fractional degrees of freedom, via the regularized incomplete beta function. It isn't looked up in a table. The p value is two-sided, and α = .05 decides whether the sentence says the groups differed significantly.

How is Cohen's d calculated here?

d = (mean of group A − mean of group B) divided by the pooled standard deviation, √[((n1 − 1)s1² + (n2 − 1)s2²) / (n1 + n2 − 2)]. No small-sample (Hedges' g) correction is applied. The label uses Cohen's rough benchmarks on |d|: below 0.2 negligible, below 0.5 small, below 0.8 medium, otherwise large.

Can I use it for paired data or a one-sided test?

No. This calculator is for two independent groups and gives a two-sided p value. Before-and-after measurements on the same people need a paired t-test, which this peek doesn't run.

Is my data uploaded?

No. The numbers stay in your browser while you work. If you press Share result, the two groups of numbers and their names are written into the link itself, so anyone who opens it, and our server, which draws the page and its preview image, sees them. If you leave your email, we only record the group sizes and whether the result was significant.

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