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I'm not sure I understand this concept. In $2^k$ designs, the independent variables have only two values and are coded as either being -1 (low value) or +1 (high value).

We can add a center point to the model (where the independent variables equal 0) but what on earth does this mean?

Is this some sort of data manipulation technique, so that we're adding fake data points to the model with plausible values?

Or are be actually adding new observations with intermediate factor levels to the model? If so, how is this still a $2^k$-model?

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Yes, the center point is an additional observations added to the experiment, generally it is midway between the high and low values. This is done in order to estimate the curvature (non-linearity) in the system.

For example, take a simple 2x2 system Variable A: will have a value of say 1 and 10, and Variable B: will have a value of -2 and 2. The classic experiment is

A    B
1    -2
1     2
10   -2
10    2
5.5   0  This is the added center point

This is still a 2^2 design since we adding just one additional measurement and are not full testing each variable at three different values (or nine tests).

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