Definition · Plain-language
Normal distribution
A normal distribution is a symmetric, bell-shaped continuous probability distribution defined by its mean μ and standard deviation σ, with the mean, median and mode all at the centre.
The step most authors miss
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The shape and its two parameters
A normal distribution — also called a Gaussian distribution or bell curve — is a continuous distribution that is perfectly symmetric about its centre. It is described completely by two numbers: the mean μ, which locates the peak, and the standard deviation σ, which controls how wide or narrow the bell is. A larger σ spreads the curve out and flattens it; a smaller σ makes it tall and narrow. Because the curve is symmetric and unimodal, its mean, median and mode all fall at the same central point.
The empirical (68-95-99.7) rule
For any normal distribution, the proportion of values within a given number of standard deviations of the mean is fixed. About 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three — the empirical or "68-95-99.7" rule. This regularity makes the normal distribution exceptionally useful: knowing only μ and σ, you can describe where most data lie and how unusual any single value is. Converting a value to a z-score expresses how many standard deviations it sits from the mean.
Why it appears so often
The normal distribution is central to statistics largely because of the central limit theorem, which states that the means of sufficiently large samples tend toward a normal distribution regardless of the shape of the underlying population. This is why many inferential tests — t-tests, ANOVA, regression — rest on an assumption of normality, at least of the sampling distribution. Real-world data are rarely exactly normal, so analysts check the assumption rather than assume it, using plots or formal tests before relying on normal-based methods.
Key facts
At a glance
- Definition: symmetric, bell-shaped continuous probability distribution
- Parameters: mean μ (centre) and standard deviation σ (spread)
- Centre: mean = median = mode at the peak
- Empirical rule: ~68% within 1σ, ~95% within 2σ, ~99.7% within 3σ
- Also called: Gaussian distribution or bell curve
- Why it matters: underpins many tests via the central limit theorem
Common misconceptions
What people often get wrong
Often heard: Any bell-shaped or symmetric curve is a normal distribution.
Actually: Not necessarily. The normal distribution has a specific mathematical form. Other symmetric, bell-like distributions (such as the t-distribution) exist with heavier tails, so symmetry alone does not make a curve normal.
Often heard: Real-world data must be exactly normally distributed to use statistics.
Actually: No. Few datasets are perfectly normal. Many methods rely on the sampling distribution being approximately normal — which the central limit theorem often supplies for large samples — rather than the raw data being normal.
Often heard: The mean and median are different in a normal distribution.
Actually: They are identical. Because a normal distribution is perfectly symmetric and unimodal, its mean, median and mode all coincide at the central peak. A gap between mean and median signals skew, not normality.
Going deeper
