ANOVA Calculator

This free ANOVA calculator runs a one-way ANOVA on raw group columns (paste straight from Excel or Sheets) or on summary stats (n, mean, SD). You get the full ANOVA table with SS, df, MS, the F statistic, p-value and F critical value, effect sizes η² (eta squared), ω² and Cohen’s f, Tukey–Kramer HSD pairwise comparisons with confidence intervals, a means chart, a shaded F curve, step-by-step working and the exact TI-84 ANOVA( keystrokes. A separate tab turns any F and degrees of freedom into a p-value or critical value.

Contents

How to Use the ANOVA Calculator

1. Pick the input. One-way ANOVA (data) to paste group columns, From summary stats if you already have n, mean and SD for each group, or P-value / F critical if you already have the F statistic.
2. Enter the groups. Paste one column per group from a spreadsheet (optional header = group names), or type the summary lists. Tap an example to load a known dataset.
3. Choose α. Type any significance level or tap 0.10, 0.05, 0.025, 0.01 or 0.001.
4. Read the answer. Results update as you type: F, p-value, ANOVA table, η², Tukey HSD pairs, the decision and TI-84 steps. Copy share link saves your exact inputs in the URL.

One-Way ANOVA Formula and ANOVA Table

One-way ANOVA splits the total variation in the data into variation between group means and variation within groups:

SST = SSB + SSW    F = MSB / MSW

Source	Sum of squares	df	Mean square
Between (Factor)	SSB = Σ ni(x̄i − x̄)²	dfB = k − 1	MSB = SSB / dfB
Within (Error)	SSW = Σ (ni − 1)si²	dfW = N − k	MSW = SSW / dfW
Total	SST = Σ (xij − x̄)²	dfT = N − 1

H₀ says all group means are equal (μ₁ = μ₂ = … = μ_k). The p-value is the right-tail area P(F_{dfB, dfW} ≥ your F). ANOVA is always right-tailed, because any real mean differences make F larger.

One-Way ANOVA Worked Example

The default example is the classic PlantGrowth-style dataset: three treatments (ctrl, trt, trt2) with 10 observations each.

Group	n	Mean	SD
ctrl	10	5.032	0.583
trt	10	4.661	0.794
trt2	10	5.526	0.443

1. Grand mean: x̄ = (sum of all 30 values) / 30 = 5.073.
2. SSB: 10[(5.032−5.073)² + (4.661−5.073)² + (5.526−5.073)²] = 3.766. dfB = 2, MSB = 1.883.
3. SSW: pooled within SS = 10.492. dfW = 27, MSW = 0.389.
4. F: 1.883 / 0.389 = 4.846. p = P(F_2,27 ≥ 4.846) = 0.0159.
5. Decision: 0.0159 ≤ 0.05 (and 4.846 > F_0.05(2,27) = 3.354), so reject H₀: at least one mean differs.

η² = 3.766 / 14.258 = 0.264 (large). Tukey HSD then shows which pairs differ: trt vs trt2 is significant; the other two pairs are not. Tap PlantGrowth example in the calculator to see every cell.

Effect Size: Eta Squared, Omega Squared and Cohen’s f

A significant F says “somewhere there is a difference,” not how big it is. This calculator reports three common effect sizes:

• η² (eta squared) = SSB / SST — the proportion of total variance explained by the factor. Rough guide: 0.01 small, 0.06 medium, 0.14 large.
• ω² (omega squared) = (SSB − dfB·MSW) / (SST + MSW) — a less biased estimate of the population effect; usually a bit smaller than η².
• Cohen’s f = √(η² / (1 − η²)) — the effect-size metric used in power analysis (0.10 / 0.25 / 0.40 ≈ small / medium / large).

Tukey HSD Post-Hoc Test

When ANOVA rejects H₀ you still need to know which means differ. The Tukey–Kramer HSD (Honestly Significant Difference) compares every pair while keeping the family-wise Type I error at α, and it allows unequal sample sizes.

HSD = q_crit · √(MSW / 2 · (1/n_i + 1/n_j))

q_crit is the studentized-range critical value for k groups and dfW. A pair is significant when |x̄_i − x̄_j| ≥ HSD (equivalently, when the Tukey confidence interval excludes 0). In the PlantGrowth example, q_crit ≈ 3.506 and only trt vs trt2 clears the HSD bar.

F Critical Value and P-Value from F

Already have F? Open the P-value / F critical tab, enter F, dfB and dfW, and read the right-tail p-value plus a row of critical values. On a TI-84, p-value = Fcdf(F, 1E99, dfB, dfW). For example, F = 3.354 with df (2, 27) gives p ≈ 0.0500.

F Critical Value Table (α = 0.05)

Reject H₀ in a one-way ANOVA when your F is greater than or equal to the critical value for your dfB (columns), dfW (rows) and α = 0.05.

dfW \ dfB	1	2	3	4	5	6	8	10	12
5	6.608	5.786	5.409	5.192	5.050	4.950	4.818	4.735	4.678
6	5.987	5.143	4.757	4.534	4.387	4.284	4.147	4.060	4.000
7	5.591	4.737	4.347	4.120	3.972	3.866	3.726	3.637	3.575
8	5.318	4.459	4.066	3.838	3.687	3.581	3.438	3.347	3.284
9	5.117	4.256	3.863	3.633	3.482	3.374	3.230	3.137	3.073
10	4.965	4.103	3.708	3.478	3.326	3.217	3.072	2.978	2.913
12	4.747	3.885	3.490	3.259	3.106	2.996	2.849	2.753	2.687
15	4.543	3.682	3.287	3.056	2.901	2.790	2.641	2.544	2.475
20	4.351	3.493	3.098	2.866	2.711	2.599	2.447	2.348	2.278
24	4.260	3.403	3.009	2.776	2.621	2.508	2.355	2.255	2.183
30	4.171	3.316	2.922	2.690	2.534	2.421	2.266	2.165	2.092
40	4.085	3.232	2.839	2.606	2.449	2.336	2.180	2.077	2.003
60	4.001	3.150	2.758	2.525	2.368	2.254	2.097	1.993	1.917
120	3.920	3.072	2.680	2.447	2.290	2.175	2.016	1.910	1.834

Need a different α or exact df? Use the P-value / F critical tab above — it computes F_α(dfB, dfW) for any α.

How to Do ANOVA on a TI-84

1. Press STAT 1:Edit. Put each group in its own list (L1, L2, L3, …).
2. Press STAT, go to TESTS, choose H:ANOVA(.
3. Type ANOVA(L1,L2,L3) with every list that holds a group, then press ENTER.
4. The screen shows F, P, Factor df/SS/MS, Error df/SS/MS and Sxp (√MSW).

For a p-value from a known F only, use 2nd VARS Fcdf(lower, 1E99, df1, df2). Practice on our TI-84 Plus CE calculator online. The TI-84 does not run Tukey HSD; use this calculator for the post-hoc table.

Assumptions of One-Way ANOVA

• Independent observations within and between groups (no repeated measures — that needs a repeated-measures ANOVA).
• Approximately normal residuals within each group. ANOVA is fairly robust for moderate samples.
• Equal variances across groups (homogeneity). With very unequal variances, consider Welch’s ANOVA; this calculator’s classic F-test matches the TI-84 ANOVA( command.
• At least 2 observations per group and preferably 3+ groups (two groups reduce to a pooled two-sample t-test).

Why This ANOVA Calculator Is Different

• Paste group columns from a spreadsheet with optional headers, or enter summary n / mean / SD when that is all you have.
• Full ANOVA table plus η², ω² and Cohen’s f — not just F and p.
• Tukey–Kramer HSD for every pair, with q statistic, adjusted p-value, HSD threshold and confidence interval (works with unequal n).
• Shaded F curve, group-means chart, plain-English decision and TI-84 ANOVA( keystrokes with the exact screen output.
• Separate p-value / F-critical tool for homework that already has the statistic.
• Checked against SciPy and statsmodels (F, p, η² and Tukey p-values). Share links, dark mode and a phone-friendly layout. See all tools on the math and statistics calculators page, or compare two means with the hypothesis test calculator.

ANOVA Calculator FAQ

Ti84CalcPro is not affiliated with or endorsed by Texas Instruments. TI-84 Plus CE is a trademark of Texas Instruments.