Unit 13
Statistics and Probability
Class 12 Mathematics
Correlation and Regression
Class 12 Mathematics – Correlation and Regression Notes PDF
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Contents / Quick Navigation
NEB / CDC Focus
This unit covers correlation and its nature, Karl Pearson’s correlation coefficient and interpretation, and regression with the regression lines of y on x and x on y.
1. Correlation
Correlation measures the degree and direction of association between two quantitative variables.
Examples include height and weight, advertising and sales, temperature and electricity use, or marks in two subjects. Correlation describes association; by itself it does not prove causation.
A strong correlation can occur because of common causes, coincidence, selection effects, or indirect relationships. Do not automatically interpret correlation as cause and effect.
2. Nature and Types of Correlation
| Type | Description |
|---|---|
| Positive | As x increases, y tends to increase. |
| Negative | As x increases, y tends to decrease. |
| Zero / no linear correlation | No clear linear tendency. |
| Perfect positive | r=+1 |
| Perfect negative | r=−1 |
3. Scatter Diagram
A scatter diagram plots every paired observation (x,y). Before calculating a coefficient, it is useful for checking direction, form, outliers and whether a linear summary is sensible.
- Points rising left-to-right suggest positive correlation.
- Points falling left-to-right suggest negative correlation.
- Closer clustering around a straight line suggests stronger linear correlation.
4. Karl Pearson’s Correlation Coefficient
Karl Pearson’s coefficient measures the strength and direction of linear correlation between two variables.
4.1 Deviation-from-Mean Form
Let x=X−X̄ and y=Y−Ȳ. Then
4.2 Direct Computational Form
4.3 Calculation Table
| X | Y | X² | Y² | XY |
|---|---|---|---|---|
| … | … | … | … | … |
| Compute ΣX, ΣY, ΣX², ΣY² and ΣXY. | ||||
5. Interpretation of the Correlation Coefficient
| r | Linear association |
|---|---|
| r=+1 | Perfect positive |
| 0<r<1 | Positive, strength increases as r approaches +1 |
| r=0 | No linear correlation |
| −1<r<0 | Negative, strength increases as r approaches −1 |
| r=−1 | Perfect negative |
The magnitude of r measures linear association. A value near zero can still occur when variables have a strong nonlinear relationship.
6. Regression
Regression studies the average relationship between variables and provides an equation for estimating one variable from the other.
There are two regression lines:
- Regression of y on x: used to estimate y from a known x.
- Regression of x on y: used to estimate x from a known y.
The line of y on x and the line of x on y are generally not the same equation and are used for different prediction directions.
7. Regression Lines
7.1 Regression Line of y on x
where
7.2 Regression Line of x on y
where
Both regression lines pass through (x̄,ȳ).
8. Regression Coefficients
b_yx and b_xy are the regression coefficients.
Useful properties:
- Both regression coefficients have the same sign as r.
- The product satisfies b_yx b_xy = r².
- Therefore r=±√(b_yx b_xy), with sign matching the regression coefficients.
9. Correlation and Regression Together
| Correlation | Regression |
|---|---|
| Measures degree/direction of linear association. | Provides estimating equations. |
| Symmetric in x and y. | Distinguishes y on x from x on y. |
| Coefficient r is unit-free and lies from −1 to +1. | Regression coefficients depend on scale and direction. |
| Does not by itself give a prediction equation. | Used to estimate one variable from the other. |
10. Worked Examples
For paired values (1,2),(2,4),(3,6), the points lie exactly on Y=2X, so r=+1.
For (1,6),(2,4),(3,2), Y=8−2X exactly, so r=−1.
Suppose x̄=10, ȳ=20, r=0.8, σ_x=2, σ_y=5.
b_yx=0.8(5/2)=2.
Thus y−20=2(x−10), or y=2x.
Using the same values,
b_xy=0.8(2/5)=0.32.
So x−10=0.32(y−20).
If b_yx=0.75 and b_xy=0.48, both positive:
r=+√(0.75×0.48)=√0.36=0.6.
If regression line of y on x is y=3+2x, then for x=5, estimated y=13.
11. Common Mistakes
Association alone does not establish cause and effect.
A correctly calculated Pearson coefficient must satisfy −1≤r≤1.
Multiply corresponding paired observations only.
Use y on x to estimate y from x; x on y to estimate x from y.
The signs of both regression coefficients match the sign of r.
r=0 means no linear correlation, not necessarily no relationship at all.
12. Formula Sheet
13. Important Exam Questions
Short
- Define correlation and regression.
- Describe positive, negative and zero correlation.
- State the range and interpretation of Pearson’s r.
- Distinguish correlation from regression.
- State equations of both regression lines.
Long / Numerical
- Find Karl Pearson’s coefficient from paired data and interpret it.
- Construct and interpret a scatter diagram.
- Find regression line of y on x.
- Find regression line of x on y.
- Find r from two regression coefficients.
- Use a regression line to estimate an unknown value.
14. One-Minute Revision
- Correlation measures direction and strength of linear association.
- −1≤r≤1.
- Positive r means variables tend to move together; negative r means opposite directions.
- r=0 means no linear correlation.
- Pearson’s r is unit-free.
- Regression provides prediction equations.
- Use y on x when x is given and y is to be estimated.
- Use x on y when y is given and x is to be estimated.
- Both regression lines pass through (x̄,ȳ).
- b_yx b_xy=r².
- Correlation does not prove causation.
15. Diagram Practice
- Positive, negative and zero scatter plots.
- Strong positive scatter plot.
- Pearson calculation flow.
- Correlation scale from −1 to +1.
- Two regression lines through the mean point.
- Relationship between r and regression coefficients.
- Regression-line selection diagram.
Also Visit
Original Nepal eNotes page: Correlation and Regression Notes.
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