2Research methods
Distributions and sampling
Inferential statistics work because sample means behave predictably even when the underlying data do not. The central limit theorem is why a t-test holds up on skewed data in a large enough sample, provided its other assumptions hold.
Practice
10 questions
Level
Foundation
Reading time
2 min
The normal distribution
A normal distribution is symmetrical, unimodal and defined entirely by its mean and standard deviation. Many tests assume normality, but of the sampling distribution rather than 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 hold up in large samples even when the data themselves are skewed (Fagerland, 2012), provided their other assumptions hold.
How large is large enough depends on the skew. The theorem sets no threshold, and with strong skew or outliers the often-quoted 30 per group may not be enough.
- The theorem is about the sampling distribution of the mean, not the data.
- It deals with skew only. A large sample does not make related observations independent.
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 often confused (Altman and Bland, 2005).
Worked example:Four times the sample
- Scores on a questionnaire have a standard deviation of 12.
- With 36 people, the standard error of the mean isSE = 12 ÷ √36 = 12 ÷ 6 = 2
- With four times as many, 144 people:SE = 12 ÷ √144 = 12 ÷ 12 = 1
- Four times the sample halves the standard error. The standard deviation stays around 12 however many are measured, because it describes the scores, not the estimate.
Source: Altman and Bland (2005), checked .
Common mistakes
- 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 only makes the wrong answer more precise.
Sources
- Altman and Bland (2005), Standard deviations and standard errors. BMJ, 331(7521), 903. Checked .
- Fagerland (2012), t-tests, non-parametric tests, and large studies—a paradox of statistical practice? BMC Medical Research Methodology, 12, 78. Checked .
- Field (2024), Discovering Statistics Using IBM SPSS Statistics, sixth edition. SAGE. Checked .