test-scores

Correlation tells us the strength of the relationship between two variables.
Regression goes one step further: it gives us an equation to predict one variable from another.

The Regression Equation

The regression line predicts Y from X.

Symbolic formula:
$$\hat{Y} = a + bX$$

Formula in words:
$$\text{Predicted Y} = \text{intercept} + (\text{slope} \times X)$$

Where:

• $$\hat{Y}$$ = predicted value of Y
• $$a$$ = intercept (value of Y when X = 0)
• $$b$$ = slope (change in Y for each 1-unit change in X)

Slope and Intercept

The slope is calculated as:

$$b = \frac{\sum (X - \bar{X})(Y - \bar{Y})}{\sum (X - \bar{X})^2}$$

The intercept is:

$$a = \bar{Y} - b\bar{X}$$

Example

Study hours (X) and test scores (Y):

• X = [2, 4, 6]
• Y = [50, 60, 80]
• $$\bar{X} = 4, \quad \bar{Y} = 63.3$$

Step 1: Slope

• Numerator = Σ(X – X̄)(Y – Ȳ) = 60
• Denominator = Σ(X – X̄)² = 8
• $$b = \tfrac{60}{8} = 7.5$$

Step 2: Intercept

• $$a = 63.3 - (7.5)(4) = 33.3$$

Regression equation:
$$\hat{Y} = 33.3 + 7.5X$$

Interpretation: each extra study hour adds about 7.5 points to the predicted test score.

Coefficient of Determination

The square of the correlation, $$r^2$$, shows the proportion of variance explained by regression.

Here: $$r^2 = 0.98$$, so 98% of score variation is explained by study hours.

Definition

• Regression: predicts one variable from another using a line
• Slope (b): how much Y changes per unit change in X
• Intercept (a): expected value of Y when X = 0
• r²: proportion of variance explained by regression

Visuals

Figure 10.1 — Scatterplot with regression line (Y predicted from X).

Figure 10.2 — Illustration of slope (rise/run) and intercept.

Why This Matters

Regression is a predictive tool.
It connects statistical description to practical forecasting: how much outcome (Y) changes with predictor (X).
It is the basis for more advanced models used in science, business, and data analysis.

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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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