Example Data Table
| Design |
Levels |
Term |
Effect input |
n per cell |
Alpha |
Use case |
| 2 × 3 |
A=2, B=3, C=1 |
A × B |
f = 0.25 |
12 |
0.05 |
Detector calibration trial |
| 3 × 3 |
A=3, B=3, C=1 |
Main A |
η² = 0.06 |
10 |
0.05 |
Material stress comparison |
| 2 × 2 × 2 |
A=2, B=2, C=2 |
A × B × C |
r = 0.20 |
18 |
0.01 |
Thermal response experiment |
Formula Used
This calculator uses a balanced fixed effect factorial ANOVA power approximation. It treats the selected effect as a noncentral F test.
Total cells: cells = a × b × c
Total sample size: N = cells × n per cell
Error degrees of freedom: df₂ = N − cells
Noncentrality: λ = f² × N
Power: Power = 1 − Fcdf(Fcritical, df₁, df₂, λ)
Effect conversions: f = √(η² / (1 − η²)) and f = r / √(1 − r²)
How to Use This Calculator
Enter the number of levels for each factor. Use C = 1 for a two factor design. Add the planned replications per cell. Choose alpha and target power. Select the effect type. Enter Cohen f, partial eta squared, or r. Pick the tested ANOVA term. Press calculate. Review power, degrees of freedom, lambda, and needed sample size.
Factorial ANOVA Power in Physics Studies
Why Power Matters
Power planning protects a physics experiment from weak design. A factorial ANOVA can test several factors at once. It can test voltage, material, temperature, angle, pressure, or timing. It can also test interactions. Interactions are often the real discovery. They show that one factor changes the effect of another factor.
Balanced Design Planning
This calculator assumes a balanced layout. Each cell has the same number of observations. Balanced designs are easier to inspect. They also keep degrees of freedom clear. The tool multiplies all factor levels. It then multiplies the cells by replications per cell. This gives the total sample size.
Effect Size Choices
The calculator accepts Cohen f, partial eta squared, or correlation r. Cohen f is common for ANOVA power. Partial eta squared is often found in reports. Correlation r is useful when a pilot effect is stored as an association. The calculator converts eta squared and r into f. That keeps the final power step consistent.
Noncentral F Method
The tested term defines numerator degrees of freedom. A main effect uses its levels minus one. A two way interaction multiplies two reduced level counts. A three way interaction multiplies all three. The error degrees of freedom come from total observations minus design cells. The method forms a noncentrality value. It then compares a noncentral F distribution with the critical F value.
Interpreting Results
A power near 0.80 is a common planning target. Higher power needs more observations. Smaller alpha also needs more observations. Smaller effects are harder to detect. Interactions usually need larger samples than main effects. This is because interaction degrees of freedom can grow quickly.
Practical Physics Use
Use the result before collecting lab data. Compare several designs. Try different levels, alpha values, and effect sizes. Export the output for notes, proposals, or lab records. The result is an estimate. Real experiments may include missing data, unequal variance, instrument drift, and measurement noise. Review assumptions before final sampling.
FAQs
What does this calculator estimate?
It estimates statistical power for a balanced factorial ANOVA. It also reports sample size, degrees of freedom, critical F, and noncentrality.
Can I use it for two factors only?
Yes. Set Factor C levels to 1. The calculator will treat the design as a two factor factorial ANOVA.
What is Cohen f?
Cohen f is an ANOVA effect size. Larger f values indicate stronger group differences or stronger interaction effects.
Can I enter partial eta squared?
Yes. Select partial eta squared as the effect input type. The calculator converts it into Cohen f before estimating power.
Can I enter correlation r?
Yes. Select correlation r. This helps when a pilot result or prior study reports an association instead of Cohen f.
Why do interactions need more samples?
Interactions often have more degrees of freedom. Their effects may also be smaller. Both points can reduce power.
Is this calculator for unbalanced designs?
No. It assumes equal replications in every cell. Use dedicated statistical software for strongly unbalanced layouts.
Is 80 percent power always enough?
No. It is only a common planning target. Critical physics experiments may need higher power and stricter error control.