Density Curves

Reading

The book refers to them as “continuous probability functions”.
Think of them like “smooth histograms”.
Used to describe large (infinite) populations.
Key property:

Area under the curve and between \(v_1\), \(v_2\) is equal to the percent of data values that are between \(v_1\) and \(v_2\).

Total area under curve equals 1 (100%).
To really compute things we would need to know the equation of this curve, and do some Calculus.
Distributions have parameters that specify their exact shape.
Some examples of distributions that are easy to work with:

Uniform

Straight line from \(a\) to \(b\). Represents the idea that all numbers between \(a\) and \(b\) are equally likely. For example most computers are equipped with “random number generators” that produce uniformly random numbers between \(0\) and \(1\).

Normal

Has well known formula and tables to use. We will see this in next section. Plays a dominant role in statistics because of the “Central Limit Theorem” we will discuss later.

Graph is a straight line at height \(\frac{1}{b-a}\), extending from \(x=a\) and \(x=b\).
Can compute areas as they are just rectangles.

Shape

Symmetric

Mean

\(\frac{a+b}{2}\)

Median

\(\frac{a+b}{2}\)

Std Dev

\(\sqrt{\frac{(b-a)^2}{12}}\)

Quartiles

One fourth and three fourths of the way from \(a\) to \(b\).

IQR

\(\frac{b-a}{2}\)

Example: Consider the uniform distribution from \(a=1\) to \(b=3\).