Is all data normally distributed?

No. Income and home prices, for example, are heavily skewed.

What does Z = 1.96 mean?

It's the cutoff leaving 5% in the two tails combined, used for 95% confidence intervals.

Normal Distribution

➗ Mathematics & Statistics By CampusDown Wiki Editorial Team Last updated October 26, 2026

The normal distribution is a symmetric, bell-shaped probability distribution centered on the mean. Heights, test scores, and measurement errors often follow it.

Contents

1. What are its features and the 68-95-99.7 rule?
2. How do you use the standard normal and z-scores?
3. FAQ
4. References

What are its features and the 68-95-99.7 rule?

The curve peaks at the mean μ and falls off symmetrically on both sides; the standard deviation σ sets how spread out it is. About 68% of values fall within μ±1σ, 95% within μ±2σ, and 99.7% within μ±3σ. For a test with a mean of 70 and SD of 10, about 95% of students score between 50 and 90.

How do you use the standard normal and z-scores?

Any normal distribution can be converted with Z = (X − μ)/σ into the standard normal (mean 0, SD 1). Then a z-table gives cumulative probabilities. On that test, a score of 85 is Z = 1.5, cumulative probability ≈ 0.933—top 6.7% or so. Thanks to the central limit theorem, sample means tend toward normal, which is why it underpins so much of statistical testing.

FAQ

Is all data normally distributed?: No. Income and home prices, for example, are heavily skewed.
What does Z = 1.96 mean?: It's the cutoff leaving 5% in the two tails combined, used for 95% confidence intervals.

Sources reviewed: 2026-10-25

References

Gauss, C. F. (1809). Theoria motus corporum coelestium.
Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.

Reports and materials that use this theory (Korean)

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This article was last updated on October 26, 2026. It is based on widely recognized original works and textbooks; when citing it in a paper, please check the original sources listed in the references.