Correlation measures the strength and direction of the relationship between two variables.
It tells us whether high values of one variable go with high (or low) values of another.
Pearson’s r
The most common measure is Pearson’s correlation coefficient, $$r$$.
It ranges from –1 to +1.
- $$r = +1$$ → perfect positive correlation (as X increases, Y increases).
- $$r = –1$$ → perfect negative correlation (as X increases, Y decreases).
- $$r = 0$$ → no linear relationship.
Symbolic formula:
$$r = \frac{\sum (X - \bar{X})(Y - \bar{Y})}{\sqrt{\sum (X - \bar{X})^2 , \sum (Y - \bar{Y})^2}}$$
Formula in words:
$$r = \frac{\text{sum of the cross-products of deviations from the mean}}{\text{square root of (sum of squared deviations in X × sum of squared deviations in Y)}}$$
Example
Suppose study hours (X) and test scores (Y) are:
- X = [2, 4, 6]
- Y = [50, 60, 80]
Means:
- $$\bar{X} = 4$$
- $$\bar{Y} = 63.3$$
Deviations:
- (2–4)(50–63.3) = (–2)(–13.3) = 26.6
- (4–4)(60–63.3) = (0)(–3.3) = 0
- (6–4)(80–63.3) = (2)(16.7) = 33.4
Sum cross-products = 60
Sum squares X = (–2)² + 0² + 2² = 8
Sum squares Y = (–13.3)² + (–3.3)² + 16.7² ≈ 466.7
So:
$$r = \frac{60}{\sqrt{8 \times 466.7}} = \frac{60}{\sqrt{3733}} = \frac{60}{61.1} = 0.98$$
A very strong positive correlation.
Coefficient of Determination
The square of correlation is $$r^2$$.
It represents the proportion of variance in Y explained by X.
Example above:
$$r^2 = (0.98)^2 = 0.96$$
So about 96% of the variation in scores is explained by study hours.
Definition
- Correlation: degree of linear relationship between two variables.
- Pearson’s r: ranges from –1 to +1.
- Coefficient of determination (r²): proportion of explained variance.
Visual Placeholders
Figure 9.1 — Scatterplot with positive correlation (points rising, line upward).
Figure 9.2 — Scatterplots showing r ≈ +1, r ≈ 0, r ≈ –1.
Why This Matters
Correlation is the first step in studying relationships.
It helps identify whether variables move together, setting the stage for regression analysis.
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