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

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one way anova calculator

One-way ANOVA calculator with table and Welch ANOVA

In the "ANOVA" tab, each group is a box to paste the values (or three fields: mean, standard deviation and n). Add up to 8 groups. The calculator builds the ANOVA table, computes F and the p-value, measures the effect with η² and ω² and checks whether the group variances are similar.

Between groups against within groups

One-way ANOVA tests H0: μ₁ = μ₂ = … = μₖ by comparing two estimates of variance. The mean square between groups, MS between = SS between / (k − 1), measures how far the group means are from the grand mean; the mean square within, MS within = SS within / (N − k), measures variation inside each group. The statistic F = MS between / MS within follows an F with k − 1 and N − k degrees of freedom if H0 is true. It is always right-tailed: a large F says the means differ more than chance would explain.
In the R PlantGrowth data (a control and two treatments, 10 plants each), SS between = 3.766, SS within = 10.492, F = 4.846 with 2 and 27 df and p = 0.0159. The 5% critical value is 3.354, so H0 falls. The η² = SS between / SS total = 0.264 says 26% of the weight variance is explained by the groups (a large effect, above 0.14), and the less biased ω² gives 0.204. In a four-group example with 8 observations each (means 23.5, 26.9, 29.3 and 25.1), F = 4.902 with 3 and 28 df, p = 0.0073 and η² = 0.344.
ANOVA assumes independent groups, approximately normal residuals and similar variances. With raw data the calculator runs the Levene (Brown-Forsythe) test; with summaries it warns if the largest standard deviation exceeds twice the smallest. In those cases look at Welch ANOVA, which does not assume equal variances: on the same PlantGrowth data it gives F = 5.181, df = 2 and 17.13 and p = 0.0174. A significant F does not say which groups differ; for that use a post hoc test, such as Tukey, not several t tests without correction. With only two groups, F = t² of the pooled t.

Checked examples

PlantGrowth, raw data

Input: controle, tratamento 1, tratamento 2 (10 cada)
Expected output: F = 4,846, gl = 2 e 27, p = 0,0159

η² = 0.264; critical value 3.354.

PlantGrowth, Welch ANOVA

Input: mesmos dados / same data
Expected output: F = 5,181, gl = 2 e 17,13, p = 0,0174

Same conclusion.

Four groups of 8 (summaries)

Input: médias 23,5 26,9 29,3 25,1; s 3,1 3,4 2,9 3,3
Expected output: F = 4,902, gl = 3 e 28, p = 0,0073

η² = 0.344; critical value 2.947.

Two groups, ANOVA and t

Input: sono de Student / Student sleep data
Expected output: F = t² = 3,4625

p = 0.0792, same as the pooled t.

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

How many groups can I compare?

From 2 to 8 groups, each with at least 2 observations (or n ≥ 2 in the summaries). For more groups or for planning, use the summaries of each and check the table.

Can I use ANOVA with different group sizes?

You can, but robustness to unequal variances drops when the smaller groups have the larger variances. With very unequal sizes, prefer Welch ANOVA.

Why does the result only show F and not the pairs of groups?

Because F is an omnibus test. Comparing pairs requires correction for multiple comparisons (Tukey, Bonferroni or Holm), which is not in this calculator; each group data and means are in the table for you to take to that step.

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

Between groups against within groups

One-way ANOVA tests H0: μ₁ = μ₂ = … = μₖ by comparing two estimates of variance. The mean square between groups, MS between = SS between / (k − 1), measures how far the group means are from the grand mean; the mean square within, MS within = SS within / (N − k), measures variation inside each group. The statistic F = MS between / MS within follows an F with k − 1 and N − k degrees of freedom if H0 is true. It is always right-tailed: a large F says the means differ more than chance would explain.

In the R PlantGrowth data (a control and two treatments, 10 plants each), SS between = 3.766, SS within = 10.492, F = 4.846 with 2 and 27 df and p = 0.0159. The 5% critical value is 3.354, so H0 falls. The η² = SS between / SS total = 0.264 says 26% of the weight variance is explained by the groups (a large effect, above 0.14), and the less biased ω² gives 0.204. In a four-group example with 8 observations each (means 23.5, 26.9, 29.3 and 25.1), F = 4.902 with 3 and 28 df, p = 0.0073 and η² = 0.344.

ANOVA assumes independent groups, approximately normal residuals and similar variances. With raw data the calculator runs the Levene (Brown-Forsythe) test; with summaries it warns if the largest standard deviation exceeds twice the smallest. In those cases look at Welch ANOVA, which does not assume equal variances: on the same PlantGrowth data it gives F = 5.181, df = 2 and 17.13 and p = 0.0174. A significant F does not say which groups differ; for that use a post hoc test, such as Tukey, not several t tests without correction. With only two groups, F = t² of the pooled t.

Checked examples

PlantGrowth, raw data

Input: controle, tratamento 1, tratamento 2 (10 cada)
Expected output: F = 4,846, gl = 2 e 27, p = 0,0159

η² = 0.264; critical value 3.354.

PlantGrowth, Welch ANOVA

Input: mesmos dados / same data
Expected output: F = 5,181, gl = 2 e 17,13, p = 0,0174

Same conclusion.

Four groups of 8 (summaries)

Input: médias 23,5 26,9 29,3 25,1; s 3,1 3,4 2,9 3,3
Expected output: F = 4,902, gl = 3 e 28, p = 0,0073

η² = 0.344; critical value 2.947.

Two groups, ANOVA and t

Input: sono de Student / Student sleep data
Expected output: F = t² = 3,4625

p = 0.0792, same as the pooled t.

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

How many groups can I compare?

From 2 to 8 groups, each with at least 2 observations (or n ≥ 2 in the summaries). For more groups or for planning, use the summaries of each and check the table.

Can I use ANOVA with different group sizes?

You can, but robustness to unequal variances drops when the smaller groups have the larger variances. With very unequal sizes, prefer Welch ANOVA.

Why does the result only show F and not the pairs of groups?

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