Normal Distribution

Definition

The normal distribution is a continuous probability distribution characterised by its symmetric bell-shaped curve. It is fully described by two parameters: the mean (μ) and the variance (σ²). The Central Limit Theorem guarantees that the sampling distribution of the mean converges to the normal distribution as sample size grows, regardless of the underlying population distribution.

Why It Matters

The normal distribution is central to statistics because many natural phenomena approximate it, and because parametric tests such as the t-test, ANOVA, and linear regression assume normality of residuals or sampling distributions. Understanding when and why the normal distribution applies helps researchers choose appropriate methods and justify their analytical decisions to reviewers and regulators.

Example

The heights of adult men in a homogeneous population often follow a normal distribution with a mean around 175 cm and a standard deviation of 7 cm. This means approximately 68% of men fall between 168 and 182 cm, and 95% fall between 161 and 189 cm. A researcher testing a new growth supplement can use parametric methods confidently if the height residuals in her sample are normally distributed.

Related Terms

• Standard Deviation
• Variance
• Mean
• Skewness
• Kurtosis
• Central Limit Theorem (not a separate page, but worth mentioning)

Software Notes

• SPSS: Analyze > Descriptive Statistics > Explore includes normality tests (Shapiro-Wilk, Kolmogorov-Smirnov) and Q-Q plots under the Plots option.
• R: shapiro.test(x) for normality. qqnorm(x); qqline(x) for visual inspection. hist(x, prob = TRUE); curve(dnorm(x, mean(x), sd(x)), add = TRUE) overlays the normal curve.
• Stata: histogram x, normal overlays a normal curve. sktest x performs skewness and kurtosis tests for normality. swilk x runs the Shapiro-Wilk test.