AP Statistics 2.7-2.8: The Normal Distribution

How can a general normal probability be expressed using the standard normal CDF $\Phi$?

If $X\sim N(\mu,\sigma^2)$, then $P(X\le x)=\Phi\left(\frac{x-\mu}{\sigma}\right)$. Standardization converts the original value to a standard normal value.

Normal distribution

A continuous probability distribution characterized by a symmetric, bell-shaped curve. It is specified by its mean $\mu$ and standard deviation $\sigma$, and is written $X \sim N(\mu,\sigma^2)$.

What do the parameters $\mu$ and $\sigma^2$ represent in $X \sim N(\mu,\sigma^2)$?

$\mu$ is the mean, which determines the center of the distribution, and $\sigma^2$ is the variance. The standard deviation is $\sigma$, the positive square root of the variance.

Standard normal distribution

The normal distribution with mean $0$ and standard deviation $1$, written $Z \sim N(0,1)$. Its variable is commonly called a $z$-score.

What is the probability density function for a normal random variable?

For $X \sim N(\mu,\sigma^2)$, the density is $f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$. The total area under the curve is $1$.

What are the mean, median, and mode of a normal distribution?

All three are equal to $\mu$, the center of the symmetric distribution.

What is the shape of a normal distribution with respect to skewness?

A normal distribution is symmetric about its mean and has skewness $0$. Values equally far above and below the mean have equal density.

What does the area under a normal curve represent?

Area represents probability. For a continuous random variable, the area between two values is the probability that the variable falls in that interval.

How are left-tail and right-tail probabilities related for a continuous normal variable?

$P(X>x)=1-P(X<x)$. For continuous distributions, $P(X<x)=P(X\le x)$ and $P(X>x)=P(X\ge x)$ because the probability of any single exact value is $0$.

How do you standardize a value $x$ from a normal distribution?

Compute the $z$-score using $z=\frac{x-\mu}{\sigma}$. It measures how many standard deviations $x$ lies above or below the mean.

What does a positive or negative $z$-score indicate?

A positive $z$-score means the value is above the mean, while a negative $z$-score means it is below the mean. The magnitude gives the distance from the mean in standard deviations.

How do you find the probability that a normal variable lies between two values?

For $X\sim N(\mu,\sigma^2)$, use $P(a<X<b)=P(X<b)-P(X<a)$, or equivalently $\Phi\left(\frac{b-\mu}{\sigma}\right)-\Phi\left(\frac{a-\mu}{\sigma}\right)$.

How do you find a normal probability to the left of a cutoff value?

Standardize the cutoff and find the cumulative area: $P(X<x)=\Phi\left(\frac{x-\mu}{\sigma}\right)$. A calculator can evaluate this with a normal CDF function.

How do you find a normal probability to the right of a cutoff value?

First find the left-tail probability, then use the complement: $P(X>x)=1-\Phi\left(\frac{x-\mu}{\sigma}\right)$.

What calculator function finds the probability between two bounds for a normal distribution?

Use a normal CDF function, commonly written $\text{normalcdf}(\text{lower},\text{upper},\mu,\sigma)$. Very large positive or negative bounds can approximate infinite tails.

What does a percentile mean in a normal distribution?

The $p$th percentile is the value $k$ such that $P(X\le k)=p$. Thus, $p$ of the observations are at or below $k$, and $1-p$ are above it.

How do you find a cutoff value corresponding to a specified percentile?

Use the inverse normal function with the area to the left: $k=\mu+\sigma z_p$, where $\Phi(z_p)=p$. On a calculator this is commonly $\text{invNorm}(p,\mu,\sigma)$.

How do you find the cutoff above which a specified proportion of a normal population lies?

If $P(X\ge k)=q$, then the area to the left is $1-q$. Find $k$ with $k=\text{invNorm}(1-q,\mu,\sigma)$.

What is the first quartile $Q_1$ of a normal distribution?

$Q_1$ is the 25th percentile, so it satisfies $P(X\le Q_1)=0.25$. It can be calculated with $Q_1=\text{invNorm}(0.25,\mu,\sigma)$.

What is the third quartile $Q_3$ of a normal distribution?

$Q_3$ is the 75th percentile, so it satisfies $P(X\le Q_3)=0.75$. It can be calculated with $Q_3=\text{invNorm}(0.75,\mu,\sigma)$.

Interquartile range for a normal distribution

The interquartile range is $IQR=Q_3-Q_1$, the width of the middle $50\%$ of the distribution.

How do you find the interval containing the middle $p$ proportion of a symmetric normal distribution?

The two tails together contain $1-p$, so each tail contains $\frac{1-p}{2}$. Find the lower and upper cutoffs at cumulative areas $\frac{1-p}{2}$ and $1-\frac{1-p}{2}$.

How can a standard normal variable be transformed into a general normal variable?

If $Z\sim N(0,1)$, then $X=\mu+\sigma Z$ has distribution $N(\mu,\sigma^2)$. The factor $\sigma$ stretches the distribution and $\mu$ shifts its center.

Why does increasing $\sigma$ change a normal curve?

A larger standard deviation spreads values farther from the mean, producing a wider and lower curve. A smaller standard deviation concentrates values near the mean, producing a narrower and taller curve.

What is the relationship between the standard normal density $\phi(z)$ and its cumulative distribution function $\Phi(z)$?

$\phi(z)$ gives the height or density at $z$, while $\Phi(z)$ gives the area to the left of $z$: $\Phi(z)=P(Z\le z)$. Probabilities require areas, not individual density heights.

What are the inflection points of the standard normal density?

The standard normal curve changes concavity at $z=-1$ and $z=1$. These points are one standard deviation below and above the mean.

What assumption is made when a real-world variable is modeled with a normal distribution?

The variable is assumed to have an approximately symmetric, continuous distribution with most observations near the mean and progressively fewer observations farther away. The model should be checked for reasonable fit rather than applied automatically.

Why are normal distributions often used to model measurement errors or combined effects?

The central limit theorem says that averages or sums of many independent contributions tend toward a normal distribution under broad conditions. Consequently, aggregate measurement errors and similar quantities are often approximately normal.

What does the central limit theorem imply about averages of independent observations?

As the number of independent observations increases, the sampling distribution of their average approaches a normal distribution when the underlying variable has a finite mean and variance.

A variable has $\mu=63$ and $\sigma=5$. What is the $z$-score for $x=65$?

$z=\frac{65-63}{5}=0.40$. The value is $0.40$ standard deviations above the mean.

For $X\sim N(63,5^2)$, how would you calculate $P(X>65)$?

Standardize first: $z=0.40$. Then calculate $P(X>65)=P(Z>0.40)=1-\Phi(0.40)$, which is approximately $0.3446$.

For $X\sim N(63,5^2)$, how would you find the 90th percentile?

Use $k=\text{invNorm}(0.90,63,5)$, or $k=63+5z_{0.90}$. The result is approximately $69.4$.

For $X\sim N(2,0.5^2)$, how would you find $P(1.8<X<2.75)$?

Use the interval area: $\text{normalcdf}(1.8,2.75,2,0.5)$. The probability is approximately $0.5886$.

For $X\sim N(2,0.5^2)$, what value marks the maximum of the bottom quartile?

The maximum of the bottom quartile is the 25th percentile: $k=\text{invNorm}(0.25,2,0.5)\approx1.66$. About $25\%$ of observations are at or below this value.

How would you calculate the interquartile range for $X\sim N(36.9,13.9^2)$?

Calculate $Q_1=\text{invNorm}(0.25,36.9,13.9)$ and $Q_3=\text{invNorm}(0.75,36.9,13.9)$, then subtract: $IQR=Q_3-Q_1\approx18.8$.

If $40\%$ of a normal population is at least $k$, what cumulative area should be used to find $k$?

The area to the right is $0.40$, so the area to the left is $1-0.40=0.60$. Therefore, use the 60th percentile: $k=\text{invNorm}(0.60,\mu,\sigma)$.

If a normal distribution has mean $\mu$ and standard deviation $\sigma$, what is the 90th percentile in terms of a standard-normal critical value?

The 90th percentile is $\mu+\sigma z_{0.90}$, where $z_{0.90}$ is the standard-normal value with $90\%$ of the area to its left. Since $z_{0.90}\approx1.282$, the cutoff is approximately $\mu+1.282\sigma$.

What is the complement rule used for normal-distribution probabilities?

For any event $A$, $P(A^c)=1-P(A)$. In particular, a right-tail probability can be found by subtracting the corresponding left-tail probability from $1$.

Why can $<$ and $\le$ be used interchangeably for normal probabilities?

A normal variable is continuous, so $P(X=x)=0$ for every exact value $x$. Therefore, $P(X<x)=P(X\le x)$, and the analogous equality holds for right-tail events.