Z-Scores and the Normal Distribution

Percentiles

The pth percentile is the value with p% of the data at or below it.

Z-scores

z = (x − μ) / σ

The number of standard deviations a value is from the mean. It allows comparison across different scales.

Example. A score of 85 on a test with mean 70 and SD 10 has z = 1.5. A score of 30 on a test with mean 24 and SD 3 has z = 2, so it is relatively better.

Normal distributions

Symmetric, bell-shaped, described by N(μ, σ).

The empirical rule (68-95-99.7)

In a normal distribution, about:

• 68% of values lie within 1 σ of the mean,
• 95% within 2 σ,
• 99.7% within 3 σ.

Example. Heights N(170, 6): about 95% are between 158 and 182 cm; about 16% are above 176 cm.

Normal calculations

1. Draw and shade the region.
2. Convert to z, or use technology (normalcdf, invNorm).
3. Answer in context.

Example. For N(500, 100), P(X > 650) = P(Z > 1.5) ≈ 0.0668.

Finding a value: the 90th percentile of N(500, 100) is 500 + 1.28(100) ≈ 628.

Assessing normality

A normal probability plot that is roughly straight suggests the data are approximately normal.

Quick summary

• z = (x − μ)/σ compares values on different scales.
• 68-95-99.7 rule for normal distributions.
• Use z-tables or technology for exact areas and percentiles.