The Normal Distribution

Continuous random variables

A continuous random variable can take any value within a specified domain, such as a height or a time. You cannot list its values one by one, as you could for the discrete variables in 2.8 Introduction to Random Variables and Probability Distributions. Instead, every interval of values has a probability, and that probability is the area under a curve above the interval. The total area under the curve is 1.

A single exact value has no width, so it has no area. For a continuous variable, $P(X<72)$ and $P(X\le 72)$ are the same number.

The normal distribution

Many continuous variables are well modeled by a normal distribution: a continuous, unimodal, bell-shaped, symmetric curve. A normal curve can model a distribution of data, like the heights of the women in a survey, or a random variable, like the height of one woman chosen at random.

Two parameters identify a normal distribution: the mean $\mu$, which sits at the center, and the standard deviation $\sigma$, which sets the spread. The smaller the standard deviation, the taller the curve and the more tightly it is concentrated around the mean. The larger the standard deviation, the shorter and wider it is. The area stays 1 either way, so a curve that spreads out has to come down.

The standard normal distribution is the normal distribution with $\mu =0$ and $\sigma =1$. A z-score from 1.9 Comparisons of the Distributions for One Quantitative Variable moves any normal value onto it: $z=\frac{x-\mu }{\sigma }$.

The empirical rule

For any normal distribution, approximately 68% of values lie within 1 standard deviation of the mean, approximately 95% lie within 2, and approximately 99.7% lie within 3. This is the empirical rule, or the 68-95-99.7 rule. It gives quick estimates with no calculator.

Areas from technology or a table

For an interval the empirical rule does not cover, use technology or a table. On a calculator, normalcdf(lower, upper, $\mu$, $\sigma$) gives the area between two values; use a very large number, or infinity, for an open end. By hand, convert to a z-score and read Table A, which gives the area to the left of z under the standard normal curve.

Working backward: from an area to a cutoff

Sometimes you know the area and need the boundary: what battery life marks the top 10%? invNorm(area, $\mu$, $\sigma$) returns the value with that much area to its left. The top 10% has 90% of the area to its left, so the cutoff is invNorm(0.90, 22, 2.5) $\approx 25.20$ hours. By hand, find $z\approx 1.28$ in the body of Table A and convert back: $x=\mu +z\sigma =22+1.28(2.5)\approx 25.2$.

Every question of this kind matches one of four region shapes. Set up the area statement first, then the calculator command follows from it.

Working backward to the mean or standard deviation

The z-score link also runs the other way, when a parameter is the unknown. A company wants only 5% of its phones to die before 18 hours, and its process gives a standard deviation of 2.5 hours. The 5th percentile sits at $z=$ invNorm(0.05) $\approx -1.645$, so 18 hours must be 1.645 standard deviations below the mean:

When the mean is known and the standard deviation is not, solve $z=\frac{x-\mu }{\sigma }$ for $\sigma$ in the same way.

Comparing relative positions

Percentiles let you compare values from different normal distributions. Jordan throws the discus 40 meters in a league where throws are approximately normal with mean 34 and standard deviation 4.5 meters. Priya long-jumps 5.6 meters in a league where jumps are approximately normal with mean 5.0 and standard deviation 0.4 meters.

Jordan's throw has $z=\frac{40-34}{4.5}\approx 1.33$ and beats about 90.9% of throws. Priya's jump has $z=\frac{5.6-5.0}{0.4}=1.50$ and beats about 93.3% of jumps. Relative to her league, Priya's performance is the more impressive one, even though 5.6 and 40 cannot be compared directly.