A bit esoteric but you may meet it in reports, particularly those using multi-level modelling (MLM).

#### Details #

It’s just the process of replacing the values of a variable with their values after subtracting their mean to “centre” them.

To give a very simple example, suppose you had a tiny dataset of ages: 67, 74, 93 then their mean is 78 and their centred values are 67-78, 74-78 and 93-79: -11, -4 and +15.

##### Why? #

The point of doing this is that the internal algorithms, the way that the software computes MLM (and some other sophisticated statistical analyses) works by convergent approximations and can struggle, stopping without finding an answer, or worse, giving incorrect analyses when handling input variables that differ markedly in their numeric ranges. For example these might be ages (under 110?!), family income which might be in thousands (of whatever currency!) and perhaps number of children (0 to 15?). Software may give a “failed to converge” warning or an error in which case centring the predictor variables often solves the problem.

##### Equations! #

In equation terms this is:

$$x^{‘}=(x – mean_{x})$$

Or more precisely:

$$x^{‘}=x – \frac{\sum_{i=1}^{i=n}x_{i}}{n}$$

Where that \(\sum_{i=1}^{i=n}x_{i}\) bit means “the sum of the *n* values of *x*” so \(\frac{\sum_{i=1}^{i=n}x_{i}}{n}\) is the formula for the arithmetic mean of x.

#### Try also #

Mean (arithmetic mean, average)

Standardising/normalising

Transforming

#### Chapters #

Not mentioned in the OMbook.

#### Online resources #

Not yet.

#### Dates #

First created 1.iv.24.