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📘 Lesson 8 of 8 · Probability

🔔 Normal Distribution

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.

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01 Key Concepts

Shape of the Normal Curve

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

Mean 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.

The 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.

Standardizing 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.

Standard Normal Distribution

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

The normal curve: 68-95-99.7 within 1-2-3 SD

02 Key Formulas

03 Solved Examples

Example 1 Test scores are normally distributed with mean 70 and standard deviation 10. Find the z-score for a score of 85.
  1. z = (x - mean) / standard deviation = (85 - 70) / 10.
Answer: z = 1.5
Example 2 Using the Empirical Rule, what percentage of test scores (mean 70, standard deviation 10) fall between 60 and 80?
  1. 60 is 1 standard deviation below the mean (70-10=60). 80 is 1 standard deviation above (70+10=80).
  2. The Empirical Rule states about 68% of values fall within 1 standard deviation of the mean.
Answer: Approximately 68%
Example 3 A value has a z-score of -2 in a distribution with mean 50 and standard deviation 5. Find the original value x.
  1. Rearrange the z-score formula: x = mean + z*standard deviation.
  2. x = 50 + (-2)*5 = 50 - 10.
Answer: x = 40

04 Practice Questions

1Find the z-score for x=90 if mean=80, standard deviation=5.
2
2Find the z-score for x=75 if mean=80, standard deviation=5.
-1
3Using the Empirical Rule, what % of values fall within 2 standard deviations of the mean?
About 95%
4If mean=100, standard deviation=15, and z=1, find x.
115
5What are the mean and standard deviation of the standard normal distribution?
Mean=0, standard deviation=1

📄 Normal Distribution — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet