What is alpha (α) in a central composite design?

Alpha (α) is the distance of each axial point (also called star point) from the center in a central composite design.

A value less than one puts the axial points in the cube; a value equal to one puts them on the faces of the cube; and a value greater than one puts them outside the cube.

Alpha, along with the number of center points, determines whether a design can be orthogonally blocked and is rotatable. Orthogonally blocked designs let model terms and block effects be estimated independently and minimize the variation in the regression coefficients. Rotatable designs provide the preferred property of constant prediction variance at all points that are equidistant from the design center, thus improving the quality of the prediction.

Minitab's default designs achieve rotatability and orthogonal blocking when both properties can be achieved at the same time. When the design includes blocks and you cannot achieve both properties at the same time, the default designs use α such that the design includes orthogonal blocks. When there are no blocks, the default designs use α such that the design is rotatable.