How to Calculate Normal Distribution
Learn how the normal distribution works and how to calculate probabilities using it. This guide covers the probability density function, the empirical rule, standardization, and using z-tables to find area under the curve.
What Is the Normal Distribution?
The normal distribution (also called the Gaussian distribution) is a symmetric, bell-shaped probability distribution fully described by two parameters: its mean μ (which determines the center) and its standard deviation σ (which controls the spread). It is arguably the most important distribution in statistics because many natural phenomena approximate it and because of the Central Limit Theorem, which states that the sampling distribution of the mean approaches normality as sample size grows, regardless of the underlying population distribution.
The Probability Density Function
The probability density function (PDF) of a normal distribution is: f(x) = (1 / (σ√(2π))) × e^[−(x−μ)² / (2σ²)]. This equation gives the height of the bell curve at any value x. The area under the entire curve equals 1 (it is a valid probability distribution). The PDF itself does not give the probability at a single point; for continuous distributions, probabilities are computed as areas under the curve between two values.
The Standard Normal Distribution
The standard normal distribution is the special case where μ = 0 and σ = 1, denoted N(0,1). Any normal random variable X ~ N(μ, σ²) can be standardized to Z ~ N(0,1) using: Z = (X − μ) / σ. Once standardized, you can use the standard normal table (z-table) to look up cumulative probabilities P(Z ≤ z), which represent the area under the curve to the left of z.
The Empirical Rule (68-95-99.7 Rule)
For any normal distribution: approximately 68.27% of values fall within μ ± 1σ, approximately 95.45% fall within μ ± 2σ, and approximately 99.73% fall within μ ± 3σ. This rule provides a fast way to estimate probabilities without a table. For example, if heights are N(170, 8²) cm, then roughly 95% of people have heights between 170 − 16 = 154 cm and 170 + 16 = 186 cm.
Finding Probabilities Using the Z-Table
To find P(a < X < b) for X ~ N(μ, σ²): Step 1: Standardize both bounds: z_a = (a − μ)/σ and z_b = (b − μ)/σ. Step 2: Look up the cumulative probabilities Φ(z_b) and Φ(z_a) from the z-table, where Φ(z) = P(Z ≤ z). Step 3: P(a < X < b) = Φ(z_b) − Φ(z_a). For example, if X ~ N(50, 10²) and you want P(40 < X < 65): z_a = −1, z_b = 1.5. P = Φ(1.5) − Φ(−1) = 0.9332 − 0.1587 = 0.7745.
Inverse Normal — Finding a Value from a Probability
Sometimes you need the value x corresponding to a given cumulative probability p (e.g., the 90th percentile). Step 1: Find z such that Φ(z) = p using the inverse z-table or the formula z ≈ the quantile function. Step 2: Back-transform: x = μ + z × σ. For N(50, 10²) at the 90th percentile, z ≈ 1.282 (since Φ(1.282) ≈ 0.90), so x = 50 + 1.282 × 10 = 62.82.
Testing for Normality
Before assuming your data follow a normal distribution, test this assumption. Common visual checks include the histogram (look for a symmetric bell shape) and the Q-Q plot (points should lie along a straight diagonal line if data are normal). Formal tests include the Shapiro-Wilk test (preferred for small to medium samples), the Kolmogorov-Smirnov test, and the Anderson-Darling test. For very large samples, even small departures from normality will be statistically significant; judgment about practical significance is important.
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