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Density Curves: You know you're earning baccalaureate credit when you start throwing around terminology like "density curves."









Density Curve Basics:

1. The area under ANY density curve is 1. This essentially means 100%.
2. The fraction of observations between two values is the area under the curve.

A VERY Special Density Curve: The Normal Distributions.

A normal distribution is often colloquially referred to as a "bell curve."

Normal distributions model certain kinds of data very well.













Examples of Normal Distributions: Heights, Weights, Manufactured Items (sizes, weights), IQ test scores, precipitation records.

There are many examples found in biology and other natural systems.

In other words, normal distributions are EVERYWHERE!













Example: Iowa Test Scores.













Example: Iowa Test Scores.















Research Question: is lexical density normally distributed?















Normal Distribution Basics

Every normal distribution is determined by two numbers: the mean $\mu$, and the standard deviation $\sigma$.

$\mu$ tells us where the center of our distribution is.

$\sigma$ tells us how wide, or "spread out," the distribution is.













A normal distribution with mean $\mu$ and standard deviation $\sigma$ is denoted as $N(\mu,\sigma)$.











Example: Iowa Test Scores, AGAIN.

Question: Approximately, what is $\mu$? Approximately, what is $\sigma$?













Example: Iowa Test Scores, AGAIN.
The mean of the Iowa test score data is 6.84, and the standard deviation is 1.55.
The normal distribution $N(6.84,1.55)$ models this data set very well.









Normal Distribution Basics: the 68-95-99.7 Rule
For ANY normal distribution:
About 68% of observations lie within 1 standard deviation of the mean.
About 95% of observations lie within 2 standard deviations of the mean.
About 99.7% of observations lie within 3 standard deviations of the mean.













Example: Iowa Test Scores, AGAIN. Recall: The normal distribution $N(6.84,1.55)$ describes the Iowa test score data set very well.











Example: Iowa Test Scores, AGAIN. Question:What percentage of students scored 9 or below?

To answer this question, we need to find the area under the curve corresponding to 9 or less.







Example: Iowa Test Scores, AGAIN.

The normal distribution $N(6.84,1.55)$ describes the Iowa test score data set very well.

What percentage of students scored 9 or less?









To find the area under a standard normal curve $N(\mu, \sigma)$ for values BELOW an observation $x$:

1. Compute the value $z=\frac{x-\mu}{\sigma}$.
2. Use Table A in the course text, or use this table.

We say that $z$ is the standardized value of $x$. This is also called a $z$-score. It is the number of standard deviations above or below the mean.

Example: Find the proportion of students who scored ABOVE $x=9$ on the Iowa Vocabulary Test.













Example: Find the proportion of students who scored above $x=4$, but below $x=9$, on the Iowa Vocabulary Test.











Example: Find the score on the Iowa Vocabulary Test which represents the 90th percentile.