In my posts about statistics, I've tried to pay some attention to the assumptions of different statistical tests. One of the key assumptions of many tests is that data are normally distributed. I should add that this is a key assumption for many of what we call 'parametric' tests.
Remember that in statistics lingo, parameter is the term we use to describe values that apply to populations, whereas statistics are values created with samples. When we try to generalize back to the population, we want our sample data to follow a similar distribution as the population - this distribution is often normal but not always. In any case, anytime we make/have assumptions about the distribution of data, we use parametric tests that include these assumptions. The t-test is considered a parametric test, because it includes assumptions about the sample (and hence, the population) distribution.
But if your data are not normally distributed, there are still many tests you can use, specifically ones that are known as distribution-free or 'non-parametric' tests. During April A to Z, I talked about Frank Wilcoxon. Wilcoxon contributed two tests that are analogues to the t-test, but have no assumptions about distribution.
To be considered a parametric test, it isn't necessary to have an assumption that data are normally distributed, because there are many types of distributions data can follow; an assumption of normality is a sufficient but not necessary condition. What is necessary to be a parametric test is to have some assumption of what the data should look like. If test assumptions make no mention about data distribution, it would be considered a non-parametric test. One well-known non-parametric test is the chi-square, which I'll blog about in the near future.