68‑95‑99.7 explorer
Shade each band, click segments for exact percentages, and use the z-score calculator.
Open explorerLearn the Concepts
1 · The 68‑95‑99.7 rule
A normal distribution is symmetric and bell-shaped, centered on its mean (μ) with spread set by the standard deviation (σ). The empirical rule says:
- 68% of values fall within 1σ of the mean,
- 95% within 2σ,
- 99.7% within 3σ.
The interactive explorer below lets you shade each band and see the percentages — it's the fastest way to build intuition for this module.
Learn the Concepts
2 · Z-scores
A z-score tells you how many standard deviations a value sits from the mean:
z = (x − μ) ÷ σ
A z-score of 0 is exactly average; z = +2 is two standard deviations above the mean (high); z = −1 is one below. Z-scores let you compare values from different normal distributions on a common scale.
Worked Examples
See it done, step by step
Example 1 — Test scores are normal with μ = 80, σ = 5. What percent scored 75 to 85?
75 = μ − σ and 85 = μ + σ, so this is the 1σ band.
That's 68% of students.
Example 2 — Men's heights are normal with μ = 70 in, σ = 3 in. What percent are 64 to 76 inches?
64 = μ − 2σ and 76 = μ + 2σ, the 2σ band.
That's 95% of men.
Example 3 — IQ scores have μ = 100, σ = 15. Find a z-score of 130.
Use z = (x − μ) ÷ σ.
z = (130 − 100) ÷ 15 = 30 ÷ 15 = 2 — two standard deviations above average.
Watch & Review
Resources for this module
Using the 68‑95‑99.7 rule
A short walkthrough of empirical-rule problems.
Watch videoLecture slides
The Module 10 slides on the normal distribution.
Open slidesCheck Yourself
Quick self-check
1. About what percent of a normal distribution lies within 1 standard deviation of the mean?
About 68%.
2. A value has μ = 50, σ = 10, x = 35. Find its z-score.
z = (35 − 50) ÷ 10 = −1.5 (one and a half standard deviations below the mean).
3. Scores are normal with μ = 500, σ = 100. Between what scores do about 95% fall?
μ ± 2σ = 500 ± 200, so 300 to 700.
Ready to turn it in?
When the empirical rule and z-scores click, complete the Module 10 homework and quiz in Canvas.
Submit in Canvas