# One-way ANOVA calculator

> One-way ANOVA calculator: the F statistic, p-value, sums of squares, mean squares and η² for two or more groups, with the full ANOVA table.

النسخة التفاعلية: https://www.calcopenly.com/ar/statistics/one-way-anova-calculator
الموضوع: حاسبات الإحصاء والاحتمالات

One-way analysis of variance (ANOVA) tests whether two or more independent groups share the same mean. It splits the total variation into a between-group sum of squares and a within-group sum of squares, divides each by its degrees of freedom (k − 1 and N − k) to get mean squares, and takes their ratio as the F statistic. The p-value is the area of the F distribution beyond that F.

Typical uses are comparing crop yields under three fertilisers, exam scores across teaching methods or response times across product versions. The default data, the NIST e-Handbook example with three groups of five, give F = 9.59 on 2 and 12 degrees of freedom and p = 0.0032, so at α = 0.05 the three means are not all equal.

The test assumes independent observations, roughly normal data in each group and similar group variances. A significant F says that at least one mean differs, not which one.

## المدخلات

- **Groups**: One group per line; separate values with spaces or commas. Groups may have different sizes.
- **Significance level α**

## النتائج

- p-value — النتيجة الرئيسية
- Decision
- F statistic
- df between groups
- df within groups
- Sum of squares between
- Sum of squares within
- Mean square between
- Mean square within
- η² (share of variance explained)
- Critical F

## الصيغة

$$
F = \frac{SS_B/(k-1)}{SS_W/(N-k)},\quad SS_B = \sum_i n_i(\bar x_i - \bar x)^2,\quad SS_W = \sum_i\sum_j (x_{ij} - \bar x_i)^2
$$

## أمثلة محلولة

### NIST e-Handbook example, 3 groups of 5 (defaults)

- Groups: 6.9 5.4 5.8 4.6 4.0 / 8.3 6.8 7.8 9.2 6.5 / 8.0 10.5 8.1 6.9 9.3
- Significance level α: 0.05
- **Sum of squares between: 27.897**
- **Sum of squares within: 17.452**
- **F statistic: 9.59**
- **df between groups: 2**
- **df within groups: 12**
- **p-value: 0.003248**
- مصدر التحقق: ⁨NIST/SEMATECH e-Handbook §7.4.3.3 ANOVA table (SS 27.897 / 17.452, F = 9.59); p = (12/(12 + 2F))⁶ by A&S 26.6.4 in Python⁩

### Unequal group sizes

- Groups: 23 25 21 22 / 28 30 27 26 29 / 24 26 25
- Significance level α: 0.05
- **F statistic: 13.5**
- **df between groups: 2**
- **df within groups: 9**
- **p-value: 0.001953**
- **η² (share of variance explained): 0.75**
- مصدر التحقق: ⁨Sums of squares with Python fractions (SSB 62.25, SSW 20.75); A&S 26.6.4 for d₁ = 2 gives p = (9/(9 + 2F))^4.5 = 4^−4.5 = 1/512⁩

### Two groups equals the pooled t-test

- Groups: 12 15 11 14 13 16 / 17 14 18 16 19 15
- Significance level α: 0.05
- **F statistic: 7.714286**
- **df within groups: 10**
- **p-value: 0.019536**
- مصدر التحقق: ⁨F = t² for two groups; pooled t from Python fractions, two-sided p from the A&S 26.7.4 closed form with ν = 10⁩

### Edge case: identical group means

- Groups: 1 2 3 / 2 1 3 / 3 2 1
- Significance level α: 0.05
- **F statistic: 0**
- **p-value: 1**
- **Decision: Fail to reject H₀**
- مصدر التحقق: ⁨Every group mean is 2, so SS between = 0 and F = 0⁩

## الأسئلة

### What does the p-value in ANOVA mean?

It is the probability of an F statistic at least as large as the one observed if every group mean were equal. With the default data p = 0.0032: equal means would produce F ≥ 9.59 in about 3 samples out of 1,000. It is not the probability that the null hypothesis is true, and a small p-value does not say how large the differences are; η² measures that.

### How do you interpret the F statistic?

F is the between-group mean square divided by the within-group mean square. Equal population means give F values near 1; larger values point to real differences. How large is large enough depends on the degrees of freedom: with 2 and 12 df the 5% critical value is 3.885, so F = 9.59 is significant at α = 0.05 and F = 3 would not be.

### What are the assumptions of one-way ANOVA?

Independent observations, a roughly normal distribution in each group, and equal population variances. The F test tolerates moderate non-normality when groups are of similar size. Moore and McCabe's rule of thumb accepts the equal-variance assumption if the largest group standard deviation is less than twice the smallest; otherwise use Welch's ANOVA, and for clearly non-normal data the Kruskal–Wallis test.

### How do you find which groups differ after ANOVA?

Run a post-hoc test. Tukey's honestly significant difference (HSD) compares every pair while holding the family-wise error rate at α; the Bonferroni method tests each of the m pairs at α/m, which is 0.05/3 ≈ 0.0167 for three groups. Separate t-tests at 0.05 on every pair push the chance of at least one false positive well above 5%, towards 1 − 0.95³ ≈ 14% for three comparisons.

### What is a good eta squared value?

η² is the between-group sum of squares divided by the total sum of squares: the share of variation explained by group membership. Cohen (1988) proposed 0.01, 0.06 and 0.14 as small, medium and large effects. The default data give 27.897/45.349 = 0.615, a very large effect. η² overstates the population effect in small samples; ω² corrects for that bias.

### ما مدى دقة «⁨One-way ANOVA calculator⁩»؟

تعتمد الدقة على مدخلاتك وافتراضات الطريقة. يستخدم الحساب العشري 50 رقمًا معنويًا، لكن التقديرات والأساليب العددية وبيانات المصدر قد تكون أقل دقة؛ تقريب القيم المعروضة لا يزيل هذه الحدود. أمثلة محلولة جرى التحقق منها بمصادر مستقلة: 4. مثلًا، يجري التحقق من «⁨NIST e-Handbook example, 3 groups of 5 (defaults)⁩» بالرجوع إلى ⁨NIST/SEMATECH e-Handbook §7.4.3.3 ANOVA table (SS 27.897 / 17.452, F = 9.59); p = (12/(12 + 2F))⁶ by A&S 26.6.4 in Python⁩.

### ما مصدر هذه الطريقة؟

NIST/SEMATECH e-Handbook of Statistical Methods, §7.4.3 Are the means equal? (one-way ANOVA and worked example); Abramowitz & Stegun, Handbook of Mathematical Functions, §26.6 (F distribution).

## المصادر

- [NIST/SEMATECH e-Handbook of Statistical Methods, §7.4.3 Are the means equal? (one-way ANOVA and worked example)](https://www.itl.nist.gov/div898/handbook/prc/section4/prc43.htm)
- Abramowitz & Stegun, Handbook of Mathematical Functions, §26.6 (F distribution)