Certaint tests, such as the t-test, require a normal distribution. Yet I have never quite understood what exactly has to be normally distributed.
Let's use age as an example. Age is rarely normally distributed in the population:
Yet, in a specific sample age might very well appear normally distributed. Here is an example from a recent study:
age.in.months <- c(156, 127, 160, 183, 137, 158, 132, 131, 126, 199, 138, 141, 126, 154, 130, 148, 163, 158, 139, 101, 176, 155, 115, 177, 150, 123, 124, 151, 177, 121, 170, 137, 162, 136, 161, 146, 141, 152, 154, 170, 137, 137, 160, 142, 126, 150, 129, 144, 175, 134, 117, 158, 157, 157, 127, 175, 128, 130, 137, 148, 130, 194, 151, 133, 184, 132, 125, 160, 175, 137, 115, 191, 171, 147, 162, 136, 134, 178, 165, 173, 133, 117, 138, 145, 141, 154)
Shapiro-Wilk normality test W = 0.97908, p-value = 0.1769 Shapiro-Francia normality test W = 0.98057, p-value = 0.1913 Anderson-Darling normality test A = 0.67431, p-value = 0.07562 Cramer-von Mises normality test W = 0.11377, p-value = 0.07141
In this sample, age appears to be borderline normally distributed.
With most data samples, we do not know how that trait is distributed in the population and have only the sample distribution to go by. But with some traits, such as age, we either know or have a very strong indication as to the distribution of that trait in the population.
For these traits, where we know the distribution in the population, which distribution – the one in the population or the one in the sample – has to be normally distributed for us to use tests that require normality?
As I see it, the (almost) normal distribution of age in the example sample is merely an artifact of the sampling. We wanted participants to be within a certain age range, and the limits we have set may have caused a drop in participant numbers close to the edges of that range.
If I understand it correctly, the Wikipedia article on parametric statistics states that it is the population in which the trait must be normally distributed: "Parametric statistics is a branch of statistics which assumes that sample data comes from a population that follows a probability distribution based on a fixed set of parameters. ... we assume all ... test scores are random samples from a normal distribution ..."
And in the slides from his lecture on inferential statistics, my professor states that: "To answer the question, whether the means in both samples are the same, we have to go beyond the current samples and look at the situation in the population that we want to study. To infer the situation in the population from the situation in the sample data, we assume that the characteristic that we want to study is normally distributed in the population. This modelling is not theoretical but must be established empirically. We therefore check whether the sampled data can be considered as coming from a normal distribution." (my translation) This, to me, would imply that if we know that the trait is not normally distributed in the population, we must not use parametric tests, no matter the sample distribution.