Correlation, Regression, and Prediction

Manual: General · Subject: Statistics

Model linear relationships and use them for explanation and prediction with caution.

Modeling Relationships Between Variables

Correlation and regression

Correlation describes the strength and direction of linear association between two numerical variables. Simple linear regression models the average change in a response variable $y$ as the explanatory variable $x$ changes.

Core formulas

Correlation

r

A standardized measure of linear association with

-1\le r\le 1

Regression line

\hat{y}=a+bx

Slope

b

Predicted change in

\hat{y}

for a one-unit increase in

x

Residual

y-\hat{y}

, the vertical prediction error.

Interpreting regression output

Quantity	Meaning	Caution
Slope	Rate of change in predicted response	Not necessarily causal
Intercept	Predicted response when $x=0$	May be outside data range
$r^2$	Proportion of variation explained by the linear model	Does not prove the model is correct

⚠️

Extrapolation risk

A regression line is most reliable within the range of observed $x$ values. Predictions far outside that range can be misleading.

Using regression responsibly

1

Step 1: Make a scatter plot and check whether the relationship looks roughly linear.
2

Step 2: Fit a line and inspect residuals for unusual patterns.
3

Step 3: Use the model for interpolation when the data support it.
4

Step 4: Avoid causal claims unless the study design justifies them.

What does a correlation of $r=0$ mean?

There is no possible relationship of any kind There is no linear association The slope must be positive The variables are identical

Why is the residual important in regression?

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Correlation, Regression, and Prediction

Modeling Relationships Between Variables

Correlation and regression

Using regression responsibly