Linear transformations

When the values are replaced by other using an equation of the form

new value   =   a  +  b  ×  old value

we say that there has been a linear transformation of the original values. The original and transformed data can be displayed together with dual axes.

Centre and spread

The centre and spread of the data are different, but the shape of the distribution otherwise remains unchanged. The mean and standard deviation are related:

new mean   =   a  +  b  ×  old mean

new sd   =  |b|  ×  old sd

Note that if the scale factor, b, is negative, we must change its sign since the standard deviation must always be positive.

Most other measures of centre (e.g. the median) and spread (e.g. the interquartile range) are similarly related.