One-Way ANOVA Calculator
Statistics, Inference
Intro: Runs a one-way ANOVA to compare k group means, returning the full ANOVA table and a clear interpretation.
Worked example
- Three methods (A, B, C) are tested with exam scores: A: 78, 85, 80 B: 72, 70, 68 C: 90, 88, 92 Test if at least one mean differs at α=0.05.
- We use a one-way ANOVA with $k=3$ groups (A, B, C). The null hypothesis is $H_0: \mu_A=\mu_B=\mu_C$, and the alternative is that at least one mean is different.
- Compute group means: A: $(78+85+80)/3 = 81$ B: $(72+70+68)/3 = 70$ C: $(90+88+92)/3 = 90$.
- Compute the overall mean: total of all 9 scores is $78+85+80+72+70+68+90+88+92 = 723$, so $\bar{x}_\text{overall}=723/9=80.333\dots$.
- Between-group sum of squares: $\text{SS}_B = \sum n_j(\bar{x}_j-\bar{x}_\text{overall})^2$ with $n_j=3$ each.
- Compute deviations: A: $(81-80.33)^2 \approx 0.444$ B: $(70-80.33)^2 \approx 106.78$ C: $(90-80.33)^2 \approx 93.444$.
- Now $\text{SS}_B = 3(0.444+106.78+93.444) \approx 3 \times 200.668 \approx 602.004$.
- Within-group sum of squares $\text{SS}_W$ is the sum of squared deviations within each group. A: $(78-81)^2+(85-81)^2+(80-81)^2 = 9+16+1=26$. B: $(72-70)^2+(70-70)^2+(68-70)^2 = 4+0+4=8$. C: $(90-90)^2+(88-90)^2+(92-90)^2 = 0+4+4=8$. So $\text{SS}_W=26+8+8=42$.
- Degrees of freedom: between $df_B = k-1 = 2$; within $df_W = N-k = 9-3 = 6$.
- Mean squares: $\text{MS}_B = \text{SS}_B/df_B \approx 602.004/2 \approx 301.002$, and $\text{MS}_W = 42/6 = 7$.
- Compute F-statistic: $F = \text{MS}_B/\text{MS}_W \approx 301.002/7 \approx 43.0$.
- Using an F-distribution with (2,6) degrees of freedom, the p-value for F ≈ 43 is extremely small (p < 0.0001).
- Since $p < 0.05$, we reject $H_0$ and conclude that at least one teaching method has a different mean score.
- Answer: $\boxed{F \approx 43.0,\; df=(2,6),\; p<0.0001 \Rightarrow \text{reject }H_0.}$
FAQs
Does this tool perform post-hoc tests?
The core calculator focuses on ANOVA. Post-hoc comparisons (like Tukey HSD) can be added as a next feature.
Why choose MathGPT?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.