Continuous Distributions, the Normal Distribution, and Excel Calculations

Practice vocabulary flashcards covering continuous density functions, normal distributions, standardization, Excel normal distribution functions, and empirical rules.

Density Function

A function, usually denoted by $f(x)$, that specifies the probability distribution of a continuous random variable $X$. The higher $f(x)$ is, the more likely $x$ is; it is nonnegative for all possible values of $X$, and the total area between its graph and the horizontal axis equals $1$.

Normal Distribution

A continuous distribution with a bell-shaped appearance whose possible values range over the entire number line from minus infinity to plus infinity, although only a relatively small range has much chance of occurring.

Normal Density Function

The mathematical function representing a normal distribution, defined as $f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-(x-\mu)^2 / (2\sigma^2)}$, where $\mu$ is the mean and $\sigma$ is the standard deviation.

Standard Normal Distribution

A normal distribution with a mean of $0$ and a standard deviation of $1$, denoted by $N(0,1)$ and also referred to as the $Z$ distribution.

Standardizing

The process of subtracting a variable's mean and dividing the difference by its standard deviation using the formula $Z = \frac{X - \mu}{\sigma}$, allowing variables with different means and/or standard deviations to be measured on a single scale.

Z-Value

The number of standard deviations an original value falls to the right (if positive) or to the left (if negative) of the mean.

Inverse Calculations

Percentile calculations for normal distributions that take a probability as an input and return a corresponding value.

NORM.DIST

The Excel function used to compute normal probability calculations for any general normal distribution.

NORM.S.DIST

The Excel function used to compute normal probability calculations that applies strictly to standardized $N(0,1)$ calculations.

NORM.INV

The Excel inverse percentile function that takes a probability and returns a value for any general normal distribution.

NORMS.INV

The Excel inverse percentile function that takes a probability and returns a value specifically for the standard normal distribution.

Empirical Rules

Rules applying to data distributions that are at least approximately symmetric and bell-shaped, stating that about $68\%$ of the data fall within one standard deviation of the mean, about $95\%$ fall within two standard deviations, and almost all fall within three standard deviations.