Center point runs are experimental runs performed at the midpoint of all factor ranges (the center of the design space) in a DOE. They serve multiple critical statistical purposes and are an essential component of well-designed factorial and response surface experiments.

What Center Points Are

In a $2^k$ factorial design where factors vary between low (−1) and high (+1) levels, center points are run at the zero level (0) for all factors simultaneously.

• Factor A: (Low + High) / 2
• Factor B: (Low + High) / 2
• etc.

Typically 3–5 center point replicates are added to a factorial design.

Why Center Points Are Important

• Curvature Detection: The most important role. If the average response at the center point differs significantly from the average of the factorial points, this indicates curvature (nonlinear response) — meaning a linear model is inadequate and a response surface design (RSM) may be needed.
• If center point average ≈ factorial average → linear model is adequate.
• If center point average ≠ factorial average → curvature exists → consider RSM.

• Pure Error Estimation: Because center points are replicated (run multiple times at the same conditions), they provide a direct estimate of experimental error (pure error) — independent of any model.
• The variation among center point replicates reflects the inherent noise of the process.
• This pure error estimate is used to test the significance of main effects and interactions.

• Process Insight: The center point runs are at the nominal operating condition — they directly show process performance at the current baseline settings.

Center Points in Practice

• A typical $2^4$ factorial design (16 runs) might add 4 center points = 20 total runs.
• Center points should be randomized with the factorial points — interspersed throughout the run order, not grouped.
• Good: runs 1,2,3(CP),4,5,6(CP),7,8(CP),... etc.
• This also provides monitoring of time-related drift during the experiment.

Semiconductor Example

• Etch DOE with factors: Power (200–400W), Pressure (20–50 mTorr), Gas Flow (50–100 sccm).
• Center point: Power=300W, Pressure=35 mTorr, Flow=75 sccm — run 3–4 times.
• If the etch rate at center points differs significantly from the average of the 8 factorial corner points, the etch rate has a nonlinear dependence on one or more factors.

Center points are the cheapest and most informative addition to any factorial DOE — they provide curvature detection, error estimation, and baseline verification for just a few extra runs.