### Introduction

In earlier Workbooks we have looked at a number of significance tests, such as the
$t$
-test, the
$F$
-test and the
${\chi}^{2}$
test. All of these depend on the assumption that the data are drawn from normal distributions. Although the normal distribution is very common, and this is what gave it its name, there are clearly cases when the data are not drawn from normal distributions and there are other cases when we might simply be unwilling to make that assumption. It is possible to make tests for cases where the data are drawn from some other specified distribution but sometimes we are unable or unwilling to say what kind of distribution it is. In such cases we can use tests which are designed to do without an assumption of a specific distribution. Sometimes these tests are called
**
distribution-free
**
tests, which seems like a very sensible name, but usually they are called
**
non-parametric
**
tests because they do not refer to the parameters which distinguish members of a particular family of distributions. For example, a
$t$
-test is used to consider questions concerning the statistic
$\mu ,$
the mean of a normal distribution, which distinguishes one normal distribution from another. In a non-parametric test we do not have a parametric formula for the form of the underlying probability distribution.

#### Prerequisites

- be familiar with the general ideas and terms of significance tests
- be familiar with $t$ -tests
- understand and be able to apply the binomial distribution

#### Learning Outcomes

- explain what is meant by a nonparametric test and decide when such a test should be used
- use a sign test
- use and interpret the results of a Wilcoxon signed rank test

#### Contents

1 Non-parametric tests2 The sign test

3 The sign test for paired data

3.1 Ties

3.2 Method 1

3.3 Method 2

3.4 General comments about the sign test

4 The Wilcoxon signed-rank test

4.1 Note

4.2 General comments about the Wilcoxon signed-rank test