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Centerpoint

Definition

A centerpoint is an experimental run with quantitative factors set at the middle of their selected low and high settings. In coded units, these factor values are usually zero. Centerpoint runs can help assess repeatability, monitor stability and check whether a simple model based on factorial corner settings misses curvature near the center of the design region.

What it means

A two-level factorial experiment examines combinations of low and high factor settings. Adding the middle setting provides information about what happens between those corners. For example, the midpoint of temperatures 40 and 60 degrees is 50 degrees on that scale. The meaning depends on the experimental scale: a midpoint after a logarithmic transformation is different from the arithmetic midpoint in original units. A purely categorical factor, such as supplier identity, does not have a numerical midpoint merely because its levels are coded with numbers.

How it works

Define meaningful quantitative ranges and their coding, then add one or more runs at the center. Independently repeated center runs can estimate variation under those conditions; repeated readings of a single run may estimate only measurement repeatability. Compare the center response with the average predicted by the factorial portion under the chosen model and design. A discrepancy can indicate curvature or another source of model inadequacy. Preserve run order to check drift, and plan randomization, blocking or interspersed control runs according to the study's objectives. Additional design points are generally needed to estimate separate quadratic effects for multiple factors.

Three practical examples

• Process development: add middle temperature and pressure settings to a two-level experiment to check whether a straight-line response approximation is adequate.
• Simulation analysis: evaluate a model at the center of a parameter region and compare it with predictions inferred from boundary combinations.
• Experimental monitoring: repeat a reference condition during a run sequence to detect possible changes in the process or measurement system.

Think of it this way

Think of checking the middle of a bridge after measuring its ends. The endpoints alone cannot show whether the middle bends upward or downward. One middle measurement can reveal a discrepancy, but it cannot fully describe every possible bend across a wider structure.

Worked example

Let two factors have coded settings x and z, each at -1 or +1 at the factorial corners. Consider the hypothetical response y = 10 + 2x + 3z + 4x squared + z squared. The four corner responses are 10 at (-1,-1), 16 at (-1,+1), 14 at (+1,-1) and 20 at (+1,+1). Their mean is 15. At the centerpoint (0,0), the response is 10, so the corner mean exceeds the center by five. This is the sum of the two quadratic coefficients, 4 + 1. A model with quadratic coefficients 2 and 3 would give the same corner-to-center contrast, so these runs cannot identify the separate curvatures. If independent center responses were 9, 10 and 11, their sample mean would be 10 and sample variance would be one. That provides local repeatability information, not a universal variance estimate without assumptions. Finally, coefficients +4 and -4 would cancel in this contrast: the absence of a center difference would not prove that the surface has no curvature.

Illustrative case study

Fictional case: Atlas Coatings tests low and high settings for two process factors. Added centerpoint runs produce responses inconsistent with the fitted linear approximation, so engineers avoid extrapolating a straight-line optimization recommendation. They investigate time drift and measurement consistency, then augment the design with suitable axial points to estimate quadratic terms. The team keeps confirmation runs separate from fitting data. This fictional case illustrates sequential experimental design and reports no actual improvement in coating quality.

Common misconceptions and mistakes

A centerpoint is not automatically an optimum, a control treatment or the center of the observed response distribution. Repeated center runs do not replace replication across the whole design region. A significant curvature contrast does not identify which factor bends or where the best setting lies. A nonsignificant contrast does not establish linearity, particularly with limited power or canceling quadratic effects. The center of numeric codes must correspond to a meaningful feasible setting. If experimental conditions change systematically with run order, a center-versus-corner difference may reflect timing as well as the intended factor settings.

Business use

Businesses can use centerpoints to reduce the risk of making decisions from an overly simple response model. Plan them before data collection, document their purpose and preserve the distinction between independent process runs and repeated measurements. Check that intermediate settings are feasible and that the chosen scale matches the science or engineering. In AI-system experiments, a midpoint is meaningful for some numeric controls but not for model names or prompt categories. Use the results to decide whether a richer design is warranted, not to declare an optimum from one promising response.

People also ask

Can a centerpoint detect all curvature?

No. It provides a particular comparison, and different curved patterns can produce the same or a canceling contrast.

How many centerpoint runs are needed?

That depends on variability, the study's objectives, resources and the precision needed; one universal count is not sufficient for every design.

Is repeating a measurement at the center the same as repeating the process run?

No. They can measure different sources of variation.

Check your understanding

Q: Why can the worked example identify a total curvature contrast of five but not separate coefficients four and one?

Answer: A: At every factorial corner both squared factors equal one, and at the center both equal zero. Their contributions therefore change together in the available observations.

Related terms

• Factorial design
• Replication (statistics)
• coded factor
• curvature
• pure error
• replication
• axial point
• central composite design and response surface

Learn next

calculate corner and center responses for several quadratic surfaces, including one with canceling curvature. Then add points along one factor axis at a time and examine how those observations separate the individual squared-factor effects.

More in Statistics and probability

• Censoring - Censoring occurs when an observation provides only partial information about a value, such as knowing that an event time exceeds a recorded follow-up time.
• Census - A census is a statistical collection that seeks information from every unit in a defined population, rather than selecting only a sample of those units.
• Centering - Centering transforms a variable by subtracting a reference value from each observation, most commonly the variable's mean.
• Central composite design - A central composite design (CCD) is an experimental design for fitting a second-order response surface.
• CERNO test - The CERNO test, short for Coincident Extreme Ranks in Numerical Observations, is a rank-based enrichment test.
• Chain linking - Chain linking constructs an index or volume series by connecting changes measured between successive periods.

Sources

Sources: NIST Engineering Statistics Handbook, adding centerpoints: https://itl.nist.gov/div898/handbook/pri/section3/pri337.htm ; NIST, central composite designs and axial points: https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm ; NIST, response surface design properties: https://www.itl.nist.gov/div898/handbook/pri/section3/pri336.htm .

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