# 2/15/2023

## Mathematical Models for Data

A mathematical model for data, commonly referred to as a density curve or p.d.f., provides a way to describe the entire distribution with a single expression.

A density curve is a function f(x) that satisfies:

1. f(x)>=0 for all x
2. The total area under the curve f(x) is one.

• The density curve is an idealized description of the distribution of the data.
• The mean, or balancing point, of the density curve is denoted by the Greek letter mu.
• The median of the density curve is the value that has area .5 to the right and .5 to the left.
• The standard deviation of the density curve measures the variability from the idealized mean, and is denoted by the Greek letter sigma.

## Normal Distributions

• Normal Distributions are symmetric, single-peaked, bell-shaped density curves.
• All normal distributions have the same overall shape.
• The exact density curve for a particular normal distribution is given by specifying mu and sigma.

## The Empirical Rule  In any normal distribution:

1. 68% of the observations fall within 1 standard deviation of the mean.
2. 95% of the observations fall within 2 standard deviations of the mean.
3. 99.7% of the observations fall within 3 standard deviations of the mean.

## Calculations for Normal Distributions

• If X~N(mu, sigma) then the standardized variable Z=(X-mu)/sigma~N(0,1).
• You may always use RStudio to do calculations for normal distributions. The commands pnorm(x, mean, sd), xpnorm(x, mean, sd), qnorm(x, mean, sd), and xqnorm(x, mean, sd) will be very helpful.

# Examples

• Find the area under the standard normal curve to the left of -1.4.
• Find the area under the N(0,1) curve between .76 and 1.4.
• Find the value, z, of the N(0,1) distribution which has area .25 to the right of z.
• Suppose X~N(275, 43).  Find P(X>200).
• Suppose X~N(275, 43).  Find P(200<X<375).
• Suppose verbal SAT scores follow the N(430, 100) distribution.  How high must a student score in order to place in the top 5%?