Normal Distribution

In a normal model, the mean sets the centre and the standard deviation sets the horizontal scale.

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.

Area represents probability

The total area under a probability density curve is $1$. Probability over an interval is the area under the curve above that interval.

Because the model is continuous,

for any exact point. Therefore $P(X<a)=P(X\le a)$ under the ideal continuous model.

Standardizing with z-scores

A z-score tells how many standard deviations a value lies from the mean:

Positive $z$ is above the mean, negative $z$ below, and $z=0$ at the mean. Standardization converts any normal variable to the standard normal distribution $N(0,1)$.

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.

Effects of changing parameters

Changing $\mu$ shifts the curve horizontally without changing its shape. Increasing $\sigma$ spreads the same total area across a wider, lower curve; decreasing $\sigma$ makes it narrower and taller.

Area remains $1$ in every case.

Assessing normality

Check a histogram or density plot for approximate symmetry, one central peak, and absence of extreme outliers. A normal probability plot can provide a more formal visual check.

Do not use a normal model solely because a variable is numerical. Many distributions are skewed, bounded, multimodal, or heavy-tailed.

Normal approximation to binomial

When $np$ and $n(1-p)$ are sufficiently large, a binomial count may be approximated normally with

A continuity correction such as $x+0.5$ or $x-0.5$ aligns discrete bars with continuous area.

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.

Quick self-check

• Are $\mu$ and $\sigma$ identified in matching units?
• Does the graph/model appear reasonably normal?
• Is the z-score sign and magnitude sensible?
• Has the requested area been shaded before calculation?
• Is a cumulative area, tail, or interval being reported?
• If reversing a percentile, did I use $x=\mu+z\sigma$ and interpret the cutoff?

Related topics

• Standard Deviation
• Z-Scores
• Binomial Distribution
• Confidence Intervals