Probability · Quick Reference

Normal Distribution Cheat Sheet

Probability · Lesson 8/8

In one line: the normal distribution is the classic 'bell curve' — a continuous distribution that describes many natural phenomena, from heights to test scores, symmetric around its mean.

Key Ideas

1Shape of the Normal Curve. Symmetric, bell-shaped, with most values clustered near the mean and fewer values farther away in either direction.

2Mean and Standard Deviation. The mean (written 'mu') sets the curve's center; the standard deviation (written 'sigma') controls its spread — a larger sigma means a wider, flatter curve.

3The Empirical Rule (68-95-99.7 Rule). About 68% of values fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.

4Standardizing with Z-Scores. A z-score converts any normal value to a standard scale: z = (x - mean) / standard deviation, telling you how many standard deviations x is from the mean.

5Standard Normal Distribution. The special case where mean = 0 and standard deviation = 1; z-scores let any normal distribution be compared using this standard version.

Worked Examples

Test scores are normally distributed with mean 70 and standard deviation 10. Find the z-score for a score of 85.

z = 1.5

Using the Empirical Rule, what percentage of test scores (mean 70, standard deviation 10) fall between 60 and 80?

Approximately 68%

A value has a z-score of -2 in a distribution with mean 50 and standard deviation 5. Find the original value x.

x = 40

Formulas

z = (x - mean) / standard deviation

Practice Yourself

Find the z-score for x=90 if mean=80, standard deviation=5.

Find the z-score for x=75 if mean=80, standard deviation=5.

-1

Using the Empirical Rule, what % of values fall within 2 standard deviations of the mean?

About 95%