# Central limit theorem and the normality assumption

Due to central limit theory, the assumption of normality implied in many statistical tests and estimators is not a problem.

The normal distribution is the basis of much statistical theory. Hypothesis tests and interval estimators based on the normal distribution are often more powerful than their non-parametric equivalents. When the distribution assumption can be met they are preferred as the increased power means a smaller sample size can be used to detect the same difference.

However, violation of the assumption is often not a problem, due to the central limit theorem. The central limit theorem states that the sample means of moderately large samples are often well-approximated by a normal distribution even if the data are not normally distributed. For many samples, the test statistic often approaches a normal distribution for non-skewed data when the sample size is as small as 30, and for moderately skewed data when the sample size is larger than 100. The downside in such situations is a reduction in statistical power, and there may be more powerful non-parametric tests.

Sometimes a transformation such as a logarithm can remove the skewness and allow you to use powerful tests based on the normality assumption.

**Available in Analyse-it Editions**

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- What is Analyse-it?
- Administrator's Guide
- User's Guide
- Statistical Reference Guide
- Distribution
- Continuous distributions
- Univariate descriptive statistics
- Calculating univariate descriptive statistics
- Univariate plot
- Creating a univariate plot
- Frequency distribution
- Normality
- Normal distribution
- Normal probability (Q-Q) plot
- Creating a normal probability plot
- Normality hypothesis test
- Tests for normality
- Testing the normality of a distribution
- Central limit theorem and the normality assumption
- Inferences about distribution parameters
- Discrete distributions
- Study design
- Compare groups
- Compare pairs
- Contingency tables
- Correlation and association
- Principal component analysis (PCA)
- Factor analysis (FA)
- Item reliability
- Fit model
- Method comparison
- Measurement systems analysis (MSA)
- Reference interval
- Diagnostic performance
- Control charts
- Process capability
- Pareto analysis
- Study Designs
- Bibliography

Version 5.40

Published 29-Jul-2019