# Mathematics and Applied Statistics Lesson of the Day – Contrasts

A contrast is a linear combination of a set of variables such that the sum of the coefficients is equal to zero.  Notationally, consider a set of variables

$\mu_1, \mu_2, ..., \mu_n$.

Then the linear combination

$c_1 \mu_1 + c_2 \mu_2 + ... + c_n \mu_n$

is a contrast if

$c_1 + c_2 + ... + c_n = 0$.

There is a reason for why I chose to use $\mu$ as the symbol for the variables in the above notation – in statistics, contrasts provide a very useful framework for comparing multiple population means in hypothesis testing.  In a later Statistics Lesson of the Day, I will illustrate some examples of contrasts, especially in the context of experimental design.

### 3 Responses to Mathematics and Applied Statistics Lesson of the Day – Contrasts

1. msanregret says:

I remember learning about contrasts in my multivariate statistics class. Unfortunately, I’ve forgotten how to use them exactly… I’ll stay tuned for the experimental design lessons, I look forward to learning more!

• Hi Mitchell,

Note that I have already written many Statistics Lessons of the Day on experimental design, though this is my first one about contrasts.

Hope you find the other lessons to be useful.

Eric