Calculator Inputs
Example Data Table
This sample shows a three-factor circumscribed plan with a rotatable alpha.
| Item | Value | Interpretation |
|---|---|---|
| Factors | 3 | Temperature, Time, Catalyst |
| CCD Type | Circumscribed CCD | Star points extend beyond the cube |
| Alpha | 1.682 | Rotatable value for three factors |
| Factorial Points | 8 | 2^3 cube points |
| Axial Points | 6 | Two axial runs per factor |
| Center Points | 6 | Improves error estimation and curvature checking |
| Total Runs | 20 | 8 + 6 + 6 |
| Model Terms | 10 | Intercept, linear, interaction, and quadratic effects |
Formula Used
1) Factorial points
F = 2k
k is the number of factors.
2) Axial points
A = 2k
Each factor gets one positive and one negative star point.
3) Total runs
N = (F × rf) + (A × ra) + nc
rf is factorial replicates, ra is axial replicates, and nc is center points.
4) Rotatable alpha
α = F1/4 = (2k)1/4
This preserves equal prediction variance at equal distances from the center.
5) Actual value conversion
Actual = Center + (Coded × (High - Low) / 2)
Used for circumscribed and face-centered designs.
For inscribed designs: Actual = Center + ((Coded / α) × (High - Low) / 2)
This rescales the cube inward while keeping star points at the chosen limits.
6) Second-order model term count
p = 1 + k + [k(k - 1)/2] + k
The model contains one intercept, k linear terms, interaction terms, and k quadratic terms.
How to Use This Calculator
- Enter the number of experimental factors you want to study.
- Choose the CCD style: circumscribed, face-centered, or inscribed.
- Select rotatable alpha or provide your own custom alpha.
- Set the number of center points and any replicate counts.
- Add factor definitions as comma-separated lines using low, center, and high settings.
- Optionally randomize the run order for execution planning.
- Press the generate button to build the summary and design matrix.
- Use the export buttons to download a CSV file or create a PDF copy.
Frequently Asked Questions
1) What does a central composite design do?
It builds a second-order experimental plan for response surface analysis. You can estimate linear, interaction, and quadratic effects without testing every possible factor combination.
2) When should I choose a face-centered design?
Choose face-centered when you cannot go beyond the tested low and high settings. Its star points stay on the cube faces because alpha equals one.
3) What is the benefit of rotatable alpha?
Rotatable alpha makes prediction variance depend mainly on distance from the center. That helps keep precision balanced in every direction around the design space.
4) Why are center points important?
Center points help estimate pure error, detect curvature, and stabilize the design. More center runs usually improve reliability when process noise is noticeable.
5) What is the difference between circumscribed and inscribed CCD?
Circumscribed designs place star points outside the cube. Inscribed designs shrink the factorial cube inward so the star points sit at your allowed operating limits.
6) Can I use actual factor units here?
Yes. Enter each factor with low, center, and high values. The tool then converts coded settings into actual values for every generated run.
7) Does the calculator estimate the response model coefficients?
No. This page plans the experimental design and exports the matrix. After running experiments, you can fit the second-order regression with your measured response data.
8) Why would I randomize the run order?
Randomization reduces bias from time trends, operator drift, temperature shifts, and other hidden process changes that could distort your fitted model.