Distributions and sampling

The normal distribution

A normal distribution is symmetrical, unimodal, and defined entirely by its mean and standard deviation. Many statistical tests assume normality — but crucially, they assume it of the sampling distribution, not of the raw data.

The central limit theorem

The distribution of sample means approaches normal as sample size increases, whatever the shape of the population distribution. This is why parametric tests are reasonably robust with samples of roughly 30 or more per group even when the data themselves are skewed.

• The theorem is about the sampling distribution of the mean, not the data.
• It is why large samples rescue parametric tests, and why small ones do not.

Standard error

The standard error of the mean is the standard deviation divided by the square root of n. It estimates how much a sample mean would vary across repeated samples, which is what makes it the denominator of most test statistics.

Because n is under a square root, precision improves slowly: quadrupling the sample halves the standard error.

• SE = SD / √n
• SD describes the spread of scores. SE describes the precision of an estimate. They are routinely confused.

Where marks get lost

• Saying the data must be normally distributed, when the assumption concerns the sampling distribution.
• Reporting SE when SD was meant, which makes variability look artificially small.
• Assuming a large sample fixes a biased sampling method. It does not — it produces a precise wrong answer.