The Normal Distribution

You have undoubtedly seen a bell curve. It appears in discussions of exam scores, measurement error, biological traits, and rather more dubious claims about human behavior.

The curve is not automatically the right model merely because it is familiar. But when a normal distribution is appropriate, its mathematical structure makes many calculations elegant.

Where a bell curve comes from

A normally distributed random variable $X$ has two parameters: its mean $\mu$ and standard deviation $\sigma>0$.

Its probability density function is

$$f(x)=\frac{1}{\sigma\sqrt{2\pi}} \exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right].$$

The mean determines the center. The standard deviation determines how wide or narrow the curve is.

The height of a density curve is not itself a probability. Probability corresponds to area under the curve. The entire area equals 1.

For a continuous random variable, the probability of any single exact value is zero. We calculate the probability of an interval instead.

Standardization

If $X$ is normal with mean $\mu$ and standard deviation $\sigma$, the standardized variable is

$$Z=\frac{X-\mu}{\sigma}.$$

$Z$ has a standard normal distribution: mean 0 and standard deviation 1. A $z$-score of 2 means an observation is two standard deviations above the mean.

For example, if exam scores are approximately normal with mean 70 and standard deviation 10, a score of 85 has

$$z=\frac{85-70}{10}=1.5.$$

We can use the standard normal distribution to interpret the associated proportion of observations.

The 68–95–99.7 rule

For an exactly normal distribution, about 68% of the probability lies within one standard deviation of the mean, about 95% within two, and about 99.7% within three.

These are approximations. The commonly used interval $-1.96<Z<1.96$ captures approximately 95% of the standard normal probability.

That does not mean every dataset contains exactly 95% of its values within 1.96 sample standard deviations. The result is a statement about an idealized probability model.

Explore the probability area

Move the slider in the laboratory to change the symmetric interval $[-z,z]$. The shaded area and its approximate probability update together.

This is a model probability, not automatically a confidence interval for an unknown parameter. Later, we will examine how sampling distributions help construct confidence intervals and what their coverage really means.

Check your understanding

1. If $\mu=100$ and $\sigma=15$, what is the $z$-score of 130?
2. What does the total area under a probability density equal?
3. Is a tall density curve at a particular point evidence that the point itself has positive probability?

Suggested responses

1. 2. 2. Exactly 1. 3. No; for a continuous distribution the probability at a single exact point is zero. Probability is associated with intervals.

Translate an applied interval to a probability statement

For $X\sim N(100,15^2)$, the chance of a value between 85 and 130 is

$$P(85<X<130)=P(-1<Z<2)=\Phi(2)-\Phi(-1)\approx0.81859.$$

The standard deviation is 15, while the second distribution parameter as written is the variance $225$. Books and software use different conventions; define which one is supplied. Standardization subtracts the mean and divides by the standard deviation, producing a dimensionless variable.

An upper 5% cutoff is $100+15z_{.95}\approx124.6728$, with $z_{.95}\approx1.644854$. A central 95% interval uses $z_{.975}\approx1.959964$ instead because 5% is split between both tails. Confusing a one-tail percentile with a two-tail interval critical value changes the answer.

The empirical 68–95–99.7 pattern is a normal-model approximation. It is not a property of every distribution with a mean and standard deviation. A normal model also assigns positive probability to every real interval; if a variable is bounded or strongly skewed, examine whether its tail predictions make sense in the application.

Additional practice

1. Standardize 70 for a normal variable with mean 50 and standard deviation 10.
2. Find its approximate upper-tail probability using $\Phi(2)\approx0.97725$.
3. Explain the difference between a 95th percentile of individual values and a 95% confidence interval for an unknown mean.

Worked additional solutions

1. $z=(70-50)/10=2$.
2. $1-0.97725=0.02275$.
3. The percentile describes individual variability under a distribution. A confidence interval describes uncertainty from an estimation procedure and depends on sample size and study design.