The Normal Distribution Explained

What Is a Normal Distribution?

A normal distribution (also called the Gaussian distribution or bell curve) is a symmetric, bell-shaped probability distribution. Many natural phenomena roughly follow it: human height, measurement errors, blood pressure in healthy populations.

• ●Perfectly symmetric around the mean
• ●Mean = Median = Mode
• ●Defined entirely by two parameters: mean (μ) and standard deviation (σ)
• ●68% of data falls within 1 SD of mean; 95% within 2 SD; 99.7% within 3 SD (the "68-95-99.7 rule")

Why It Matters for Statistical Tests

Most common parametric tests (t-tests, ANOVA, linear regression, Pearson correlation) assume that the residuals or the outcome variable are approximately normally distributed. Violating this assumption can lead to invalid p-values.

The Central Limit Theorem

The Central Limit Theorem (CLT) is one of the most important results in statistics. It states: if you draw many large random samples from any population and calculate the sample mean each time, the distribution of those sample means will be approximately normal — regardless of the original population's shape.

This is why t-tests and ANOVA work even when the underlying data is slightly skewed: the test statistics rely on sample means, not raw data. The rule of thumb is that this approximation is good enough for n ≥ 30 per group.

How to Check for Normality

Never assume normality — always verify it. Use multiple approaches:

1. 1.Visual check: Histogram (should look bell-shaped) and Q-Q plot (points should fall near the diagonal line)
2. 2.Shapiro-Wilk test: Most reliable for small to medium samples (n < 2000). A p-value < 0.05 suggests significant non-normality.
3. 3.Kolmogorov-Smirnov test: Common in larger samples, but less powerful than Shapiro-Wilk.
4. 4.Check skewness and kurtosis: Values of skewness between -1 and +1 and kurtosis between -2 and +2 are generally acceptable.

What to Do When Data Is Not Normal

When your data is substantially non-normal and your sample is small, you have several options:

• ●Data transformation: log transformation is effective for right-skewed data (e.g., lab values, income). Log-transform, run your test, then back-transform results.
• ●Nonparametric tests: Use Mann-Whitney U instead of independent t-test; Wilcoxon signed-rank instead of paired t-test; Kruskal-Wallis instead of one-way ANOVA.
• ●Collect more data: If n > 50 per group, parametric tests are often robust enough.
• ●Report both: When in doubt, run parametric and nonparametric tests — if conclusions agree, you're safe.