Normal Distribution
The normal distribution, also called the Gaussian distribution or bell curve, is a continuous probability distribution that is symmetric about its mean. It is the most important distribution in statistics because of the Central Limit Theorem, which states that the sum of many independent random variables tends toward a normal distribution regardless of the underlying distribution. Approximately 68% of data falls within one standard deviation, 95% within two, and 99.7% within three.
Formula
f(x) = (1 / (σ√(2π))) · e^(-(x - μ)² / (2σ²))
Mean (Expected Value)
μ
Variance
σ²
Parameters
The center of the distribution, representing the expected value. Can be any real number.
Controls the spread of the distribution. Must be positive (σ > 0). A larger σ produces a wider, flatter curve.
Key Properties
- •Symmetric about the mean (μ), so mean = median = mode
- •The Empirical Rule: 68% of data within μ ± 1σ, 95% within μ ± 2σ, 99.7% within μ ± 3σ
- •The standard normal distribution has μ = 0 and σ = 1, denoted Z ~ N(0, 1)
- •Skewness = 0 and excess kurtosis = 0, making it the reference for other distributions
- •The sum of independent normal random variables is also normally distributed
- •Completely determined by its first two moments (mean and variance)
Example
The heights of adult women in a population are normally distributed with a mean of 64.5 inches and a standard deviation of 2.5 inches. What is the probability that a randomly selected woman is taller than 69.5 inches?
Step 1: Calculate the z-score: z = (x - μ) / σ = (69.5 - 64.5) / 2.5 = 5 / 2.5 = 2.0. Step 2: Look up P(Z > 2.0) in the standard normal table. P(Z > 2.0) = 1 - P(Z ≤ 2.0) = 1 - 0.9772 = 0.0228.
Result: P(X > 69.5) = 0.0228, or about 2.28%
Approximately 2.28% of adult women in this population are taller than 69.5 inches. This makes sense because 69.5 inches is exactly 2 standard deviations above the mean, and we know about 2.5% of data lies beyond 2σ in one tail.
When to Use
- ✓When modeling measurements that cluster symmetrically around a central value (heights, weights, test scores, measurement errors)
- ✓When applying the Central Limit Theorem to approximate the distribution of sample means from large samples
- ✓When constructing confidence intervals or performing hypothesis tests for population means
- ✓When data has been verified to be approximately normal via a histogram, Q-Q plot, or normality test
Common Mistakes
- ✗Assuming data is normally distributed without checking. Many real datasets are skewed, heavy-tailed, or multimodal.
- ✗Confusing the standard deviation (σ) with the variance (σ²) when plugging values into the PDF formula or calculating z-scores.
- ✗Forgetting to convert to z-scores before using the standard normal table. The table only applies to Z ~ N(0, 1).
- ✗Using the normal approximation for small sample sizes when the population is not normal. The CLT requires sufficiently large n (often n ≥ 30 as a guideline).
Need Help with Distribution Problems?
Snap a photo of any distribution problem for instant step-by-step solutions.
Download StatsIQFAQs
Common questions about Normal Distribution
A normal distribution can have any mean (μ) and any positive standard deviation (σ), written as X ~ N(μ, σ²). The standard normal distribution is a special case with μ = 0 and σ = 1, written as Z ~ N(0, 1). Any normal random variable X can be converted to a standard normal variable using the z-score formula: Z = (X - μ) / σ. This standardization allows you to use a single z-table for any normal distribution problem.
You can use the normal approximation to the binomial when both np ≥ 10 and n(1 - p) ≥ 10, where n is the number of trials and p is the probability of success. The approximation uses μ = np and σ = √(np(1 - p)). For better accuracy, apply the continuity correction: to find P(X ≤ k), compute P(Y ≤ k + 0.5) where Y is the normal approximation.
The normal distribution is central to statistics for three reasons. First, the Central Limit Theorem guarantees that sample means from any population approach a normal distribution as sample size increases, enabling inference about population parameters. Second, many natural phenomena (heights, errors, IQ scores) are approximately normal due to being the sum of many small independent effects. Third, many statistical methods (t-tests, ANOVA, regression) assume normality of residuals, making it foundational for inferential statistics.