MYP Extended Mathematics Topic Correlation Revision Notes

MYP Extended Mathematics Topic Correlation Revision Notes | RevisionDojo

Measuring Linear Relationships With Correlation

Two variables form bivariate data when each individual provides a pair of values, written as $(x_1,y_1),(x_2,y_2),\dots,(x_n,y_n)$.
A scatter diagram (scatter plot) is usually the first representation because it lets you see whether an increase in $x$ tends to be associated with an increase or decrease in $y$.

Definition

Correlation

A measure of the association between two variables, describing whether they tend to increase together, decrease together, or show no consistent pattern.

Note

Correlation is useful because it supports generalization: if a relationship is strong and stable, you can often make cautious predictions using a line of best fit.

Pearson's Correlation Coefficient Standardizes Covariance

From the previous article you should know that covariance which measures whether $x$ and $y$ tend to deviate from their means in the same direction, and its common "population-style" formula is $$s_{xy}=\frac{\sum (x-\bar{x})(y-\bar{y})}{n}$$
To make the strength of the relationship meaningful, we divide by the spread (standard deviation) of each variable.

Definition

Pearson’s product-moment correlation coefficient (PMCC)

A standardized measure of linear correlation given by
$$r=\frac{\frac{1}{n}\sum xy-\bar{x}\bar{y}}{\sigma_x\sigma_y},$$
where $\sigma_x$ and $\sigma_y$ are the standard deviations of $x$ and $y$.

Because we divide by $\sigma_x\sigma_y$, the coefficient $r$ has two key properties:

It is unit-free.
It always lies in the interval $-1\le r\le 1$.

Hint

Interpretation (linear patterns only):

$r=1$: points lie exactly on an upward-sloping straight line (perfect positive linear correlation).
$r=-1$: points lie exactly on a downward-sloping straight line (perfect negative linear correlation).
$r=0$: no linear relationship (there could still be a curved relationship).

Nice try, unfortunately this paywall isn't as easy to bypass as you think. Want to help devleop the site? Join the team at https://revisiondojo.com/join-us. exercitation voluptate cillum ullamco excepteur sint officia do tempor Lorem irure minim Lorem elit id voluptate reprehenderit voluptate laboris in nostrud qui non Lorem nostrud laborum culpa sit occaecat reprehenderit

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Duis aute irure dolor in reprehenderit

Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit aut fugit, sed quia consequuntur magni dolores eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci velit.

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Unlock the rest of this chapter with a free account

Measuring Linear Relationships With Correlation

Two variables form bivariate data when each individual provides a pair of values, written as $(x_1,y_1),(x_2,y_2),\dots,(x_n,y_n)$.
A scatter diagram (scatter plot) is usually the first representation because it lets you see whether an increase in $x$ tends to be associated with an increase or decrease in $y$.

Definition

Correlation

A measure of the association between two variables, describing whether they tend to increase together, decrease together, or show no consistent pattern.

Note

Correlation is useful because it supports generalization: if a relationship is strong and stable, you can often make cautious predictions using a line of best fit.

Pearson's Correlation Coefficient Standardizes Covariance

From the previous article you should know that covariance which measures whether $x$ and $y$ tend to deviate from their means in the same direction, and its common "population-style" formula is $$s_{xy}=\frac{\sum (x-\bar{x})(y-\bar{y})}{n}$$
To make the strength of the relationship meaningful, we divide by the spread (standard deviation) of each variable.

Definition

Pearson’s product-moment correlation coefficient (PMCC)

A standardized measure of linear correlation given by
$$r=\frac{\frac{1}{n}\sum xy-\bar{x}\bar{y}}{\sigma_x\sigma_y},$$
where $\sigma_x$ and $\sigma_y$ are the standard deviations of $x$ and $y$.

Because we divide by $\sigma_x\sigma_y$, the coefficient $r$ has two key properties:

It is unit-free.
It always lies in the interval $-1\le r\le 1$.

Hint

Interpretation (linear patterns only):

$r=1$: points lie exactly on an upward-sloping straight line (perfect positive linear correlation).
$r=-1$: points lie exactly on a downward-sloping straight line (perfect negative linear correlation).
$r=0$: no linear relationship (there could still be a curved relationship).

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

MYP Extended Mathematics Correlation Notes

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

Measuring Linear Relationships With Correlation

Pearson's Correlation Coefficient Standardizes Covariance

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Measuring Linear Relationships With Correlation

Pearson's Correlation Coefficient Standardizes Covariance

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account