Understanding the Normal Distribution

The normal (Gaussian) distribution is the familiar bell-shaped curve that describes many natural measurements — heights, blood pressure, measurement errors, test scores. It is fully defined by two parameters: the mean (μ), which locates the center, and the standard deviation (σ), which controls the spread. The standard normal distribution is the special case with μ = 0 and σ = 1.

Z-Scores

A Z-score converts any value to standard-normal units: z = (x − μ) / σ. It tells you how many standard deviations a value sits above or below the mean. If exam scores have μ = 75 and σ = 8, a score of 91 has z = (91 − 75) / 8 = 2.0 — two standard deviations above average, better than about 97.7% of test takers.

The Empirical (68-95-99.7) Rule

• About 68.27% of values fall within 1σ of the mean.
• About 95.45% fall within 2σ.
• About 99.73% fall within 3σ.

You can verify these with the Between tab: set μ = 0, σ = 1, and compute P(−1 ≤ X ≤ 1), P(−2 ≤ X ≤ 2), and P(−3 ≤ X ≤ 3).

What Each Mode Does

• P(X ≤ x): the cumulative distribution function (CDF) — the probability a value falls at or below x. Example: with μ = 100, σ = 15 (IQ scale), P(X ≤ 130) ≈ 0.9772, so an IQ of 130 is at the 97.7th percentile.
• Inverse: the reverse question — given a probability, find the cutoff value x. Useful for finding percentile thresholds, e.g. the 90th percentile of a μ = 100, σ = 15 distribution is about 119.2.
• Between: P(x₁ ≤ X ≤ x₂), the area under the curve between two values.
• Z-Score: works directly on the standard normal, giving left-tail, right-tail, and two-tail probabilities for any z.

Common Reference Z-Values

These critical values appear constantly in statistics: z = 1.645 leaves 5% in one tail (90% two-sided confidence), z = 1.96 leaves 2.5% in one tail (95% confidence), and z = 2.576 corresponds to 99% confidence. Hypothesis tests, confidence intervals, and quality-control limits are all built from these numbers.