Module 04 · Probability distributions
The normal distribution.
One curve. Two parameters. Explore how the mean and standard deviation shape a distribution, then find the probability to the left of any value.
Follow the area under the curve
Normal distributionThe shaded region is the area to the left of the cutoff x. The display shows the curve within four standard deviations of the mean.
With mean 0 and standard deviation 1, the probability of a value at or below 0 is 50 percent.
The idea
The mean μ locates the center. The standard deviation σ controls the spread. Increasing σ makes the curve wider and lowers its peak; the total area remains 1.
A z-score expresses the cutoff in standard deviations: z = (x − μ) / σ. Probability is an area under the curve, not the height of the curve.
TRY THIS Move one standard deviation
Place the cutoff at μ + σ. About 84.13% of the distribution is now to its left. Change μ and σ while keeping that same standardized cutoff. What stays the same?
Notation, interpretation & reference
Here, σ > 0 and the variance is σ². When μ = 0 and σ = 1, the distribution is the standard normal. P(X ≤ x) is the cumulative probability, including the tail outside the visible chart. Probabilities are computed numerically.
This explorer assumes a normal model; a bell-shaped illustration alone does not establish that real data are normal.
Reference: NIST / SEMATECH e-Handbook: Normal distribution ↗