Statistics 101 Chapter 4 Section 1 Transforming Relationships

Statistics 101 Chapter 4 Section 1

Transforming Relationships • Linear transformations do not fit all sets of observations • We can use – Logarithm – Exponential

Monotonic Functions • Monotonic function f(t) – moves in one direction as its argument t increases • Monotonic increasing function: preserves the order of data. That is, if a>b, then f(a)>f(b). • Monotonic decreasing function: reverses the order of data. That is, if a>b, then f(a) < f(b).

Examples of increasing monotonic functions

Examples of monotonic decreasing

Ladder of power transformations

Concavity of power functions • Transformations tp for power p greater than 1 are concave up. • Transformations tp for power p less than 1 are concave down.

Life expectancy and gross domestic product for 115 nations.

Exponential Growth • Linear growth increases by a fixed amount each time period. • Exponential growth increases by a fixed percentage of the previous total. • Exponential growth model » Y = a • bx

Exponential examples

Logarithm transformation • Logbx = y iff by = x

Examples

Prediction in exponential growth model • If a variable grows exponentially, its logarithm grows linearly • Technology toolbox p 210

Lets try a technology toolbox to practice this stuff • P 210 will guide us through a transformation.

Power law models • Y = a • xp • Power law models become linear when we apply the logarithm transformation to both variables. • Prediction in power laws – Example 4. 9 (walk thru)