Please put the following data

in your calculator. Thanks!Volume

(cm3)Pressure

(atmospheres)

6 2.9589

8 2.4073

10 1.9905

12 1.7249

14 1.5288

16 1.349

18 1.2223

20 1.1201

Transforming to

Achieve

Linearity12.2!

Gapminder.orgMission: “Fighting devastating ignorance with

fact-based worldviews everyone can

understand” Gapminder was founded in Stockholm on February 25, 2005, by Ola Rosling,

Anna Rosling Rönnlund and Hans Rosling. In 2006, Hans Rosling held his first TED talk on “The best statistics you’ve never seen”. It became one of the most seen TED talks ever, thanks to its unique combination of knowledge-testing, animating bubble charts and storytelling about global development.

The animated bubble graph is a software called Trendalyzer, which Gapminder had developed to make global public data understandable. In 2007 Google acquired Trendalyzer and the team of developers moved to Google’s headquarters in California. Over three years the team improved the user experience for search and exploration of global public data. In 2010 Anna and Ola decided to leave Google and return to Gapminder to develop free teaching material. In order to prioritize what content to include in the teaching material, they started measuring public knowledge (or rather the lack of knowledge) with the Ignorance project. They soon realized that spreading facts was not enough since the problem was bigger. People had an incorrect dramatic worldview.

The Factfulness project was born to help people get a fact-based worldview. Since the world is better than most people think, a fact-based worldview actually reduces stress and anxiety. Using Factfulness you can dismantle the misconceptions that shape the overdramatic worldview.

How would you describe the relationship

between wealth and life expectancy?

As the years go on, in general, what is

happening?

Why do we need a regression

line?

Regression lines are used to predict linear

data, by using an equation to model the

relationship between two variables.

Other models that could fit data:

Power models

Exponential Models

Trigonometric Models

But if data’s not linear, how can we

apply a regression line?

Straightening Curved Data

To straighten curved patterns on a

scatterplot, quantitative variables can be

transformed.

This is when the scale of measurement is

changed using a function such as a square

root, reciprocal, or logarithm to one or both

quantitative variables.

Example 1:

Boyle’s Law (p785, #32)If you have taken a chemistry or physics class, then you are probably familiar with Boyle’s law: for gas in a confined space kept at a constant temperature, pressure times volume is a constant (in symbols, PV = k). Students collected the following data on pressure and volume using a syringe and a pressure probe.

Make a scatterplot of this data.

Volume (cm3)

Pressure (atmospheres)

6 2.9589

8 2.4073

10 1.9905

12 1.7249

14 1.5288

16 1.349

18 1.2223

20 1.1201

Transforming Method #1: using powers/roots

Using PV = k, algebraically isolate…

P V

Transformation 1: ( 1/V, P )

Find reciprocal of every

explanatory variable, keep response the same

Transformation 2: ( V, 1/ P )

Find reciprocal of every response

variable, keep explanatory the same

Using your calculators, keep L1, L2 as is, and then…

L3: type in as 1/L1

Your scatterplot will be for

(L3, L2)

Label the axes below for the

scatterplot, make a rough

sketch of the scatterplot and

comment what you see.

• L4: type in as 1/L2

• Your scatterplot will be for (L1, L4)

• Label the axes below for the

scatterplot, make a rough sketch

of the scatterplot and comment

what you see.

Continued…Do a residual plot for each to determine if a line is the best

model for the transformed data.

Using your calculators, determine the equation for the least-

squares regression line for these

transformations…remember your units are no longer the

same! Define any variables used. L3: type in as 1/L1

Your scatterplot will be for

(L3, L2)

Label the axes below for the

scatterplot, make a rough

sketch of the scatterplot and

comment what you see.

• L4: type in as 1/L2

• Your scatterplot will be for (L1,

L4)

• Label the axes below for the

scatterplot, make a rough

sketch of the scatterplot and

comment what you see.

Use the model to predict the pressure in the syringe when

the volume is 17 cm3.

1.303 atmospheres 1.287 atmospheres

We perform specific transformations on

specific types of models that two variables

follow. The most popular are …

Power Models: (note, x is the

base)

Exponential Models: (note, x is

the exponent)

Many biological models are

described by using power

models

Abundance of species follows

a power model, with x = body

weights

With x as the explanatory,

the same relationship is

seen with pulse rate, life

length, number of eggs a

bird lays, etc.

Populations grow

exponentially (uninterrupted)

Compounded interest

Power vs. Exponential Model?

But what if we are unsure of the type of relationship between our

variables?

Transforming Method #2:

To transform either of these models, the best option is to take

logarithms.

If a power model describes the

relationship between two

variables, transform the

relationship by taking the

logarithm of both sides.

A scatterplot of logs of both

variables should produce a linear

pattern.

If an exponential model

describes the relationship

between two variable,

transform by taking the

logarithm of response variable

ONLY.

A scatterplot of the

explanatory variable and the

logarithm of the response

variable should produce a

linear pattern.

Back to Boyle

g. So…do you think when we transform the

equation for Boyle’s low, is it a Power model

or Exponential?

h. Give the equation of the least-squares

regression line, defining any variables you

use.

Which to choose?

The type of model that best represents a set

of nonlinear data can be deducted if we

transform it as if it was a power model and

then as an exponential: whichever

transformation creates a linear pattern

indicates which type of model it is.

Which to choose?If a power model best describes a linear relationship, plot log x and log y.

If an exponential model best describes a linear relationship, then plot x to log y.

For both, ask yourself:

is there a line in the scatterplot?

What does the residual plot of the explanatory to resididual supposed to look like?

Which to choose?

If both (log x, log y) and (x, log y) look

linear…

Fit least-squares regression line to both sets

of transformed data

Compare residual plots: look for most

random scatter

Use s (which is smaller?) and r2 (which is

bigger?)

The Best Stats You've Ever

Seen: TED talk, 2007