1.10 The Normal Distribution

What is the Normal Distribution?

The Normal distribution is a symmetric, bell-shaped curve where mean = median = mode, centered at μ with spread σ. It describes many real-world variables and is fully defined by its mean and standard deviation. Areas under the curve represent proportions of data, and total area always equals 1.

The Empirical Rule (68-95-99.7)

For approximately Normal data: about 68% falls within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD. This rule gives a quick way to estimate proportions without a calculator, and helps check whether data are roughly Normal by comparing actual percentages to these benchmarks.

Standardizing with Z-scores

A z-score = (x − μ)/σ tells how many standard deviations a value is from the mean. Standardizing lets you compare values from different Normal distributions on the same scale and use the standard Normal table or calculator functions (like normalcdf) to find proportions or percentiles.

Key terms

Normal Distribution

A symmetric, bell-shaped distribution defined by mean (μ) and standard deviation (σ), where mean = median = mode.

Empirical Rule

In a Normal distribution, about 68% of data is within 1 SD, 95% within 2 SD, and 99.7% within 3 SD of the mean.

Z-score

z = (x − μ)/σ; measures how many standard deviations a value is from the mean.

Standard Normal Distribution

A Normal distribution with mean = 0 and standard deviation = 1, used with z-scores to find proportions.

Why standardize data?

To compare values from different Normal distributions on a common scale using z-scores.

Try an AP-style question

Scores are N(70, 5). What is the z-score of 80?

1. 10
2. 0.5
3. 1
4. 2