Normal Distribution

Shape and parameters

A normal distribution is bell-shaped, symmetric, unimodal, and continuous. It is determined by

and

We write $X\sim N(\mu,\sigma^2)$ when the second parameter is variance.

Worked example: interpret a z-score

The percentile is an area, not the z-score itself.

Empirical rule

For a normal distribution, approximately:

• $68\%$ of values lie within $1\sigma$ of the mean;
• $95\%$ lie within $2\sigma$;
• $99.7\%$ lie within $3\sigma$.

This $68$–$95$–$99.7$ rule provides fast estimates and reasonableness checks.

Finding interval probabilities

Convert interval endpoints to z-scores, then use a standard-normal table or technology. Sketch and shade the requested region first.

For a right-tail probability, subtract cumulative left area from $1$. For an interval, subtract the lower cumulative area from the upper one.

Finding a value from a percentile

If a z-score or percentile is known, reverse the standardization:

This can determine cutoffs, warranty thresholds, or score boundaries. Keep the correct tail direction and interpret whether the cutoff is above or below the mean.

Common mistakes

Using variance in the z-score denominator. Divide by standard deviation.

Interpreting a z-score as a percentage. Convert it to an area.

Forgetting the right-tail subtraction. Most tables/calculators report left cumulative area.

Assuming every bell-ish sample is exactly normal. Assess fit and context.

Ignoring continuity correction in a binomial approximation. Discrete and continuous boundaries differ.