Statistical Methods for Psychology

fall within a set called linear transformations,in which we multiply each Xby some

constant (possibly 1) and add a constant (possibly 0):

where aand bare our constants. (Transformations that use exponents, logarithms, trigono-

metric functions, etc., are classed as nonlinear transformations.) An example of a linear

transformation is the formula for converting degrees Celsius to degrees Fahrenheit:

As long as we content ourselves with linear transformations, a set of simple rules de-

fines the mean and variance of the observations on the new scale in terms of their means

and variances on the old one:

1. Adding (or subtracting) a constant to (or from) a set of data adds (or subtracts) that same
   constant to (or from) the mean:
   For


2. Multiplying (or dividing) a set of data by a constant multiplies (or divides) the mean by
   the same constant:
   For
   For


3. Adding or subtracting a constant to (or from) a set of scores leaves the variance and
   standard deviation unchanged:
   For


4. Multiplying (or dividing) a set of scores by a constant multiplies (or divides) the vari-
   ance by the square of the constant and the standard deviation by the constant:
   For and
   For and

The following example illustrates these rules. In each case, the constant used is 3.

Addition of a constant:

Old New

Data s Data s

4, 8, 12 8 16 4 7, 11, 15 11 16 4

Multiplication by a constant:

Old New

Data s Data s

4, 8, 12 8 16 4 12, 24, 36 24 144 12

Reflection as a Transformation

A very common and useful transformation concerns reversing the order of a scale. For exam-

ple, assume that we asked subjects to indicate on a 5-point scale the degree to which they agree

X s^2 X s^2

X s^2 X s^2

Xnew=Xold>b: s^2 new=s^2 old>b^2 snew=sold>b.

Xnew=bXold: s^2 new=b^2 s^2 old snew=bsold.

Xnew=Xold 6 a: s^2 new=sold^2.

Xnew=Xold>b: Xnew=Xold>b.

Xnew=bXold: Xnew=bXold.

Xnew=Xold 6 a: Xnew=Xold 6 a.

F= 9 >5(C) 1 32.

Xnew=bXold 1 a

Section 2.12 The Effect of Linear Transformations on Data 53

linear
transformations