Linear Regression and Correlation

Linear Regression and Correlation

Linear Regression and Correlation - Complete Interactive Lesson

📈 Linear Regression and Correlation

📈 Linear Regression and Correlation

📈 Linear Regression and Correlation

📈 Linear Regression and Correlation

📈 Linear Regression and Correlation

Linear Regression and Correlation - Complete Interactive Lesson

📈 Linear Regression and Correlation

📈 Linear Regression and Correlation

📈 Linear Regression and Correlation

📈 Linear Regression and Correlation

📈 Linear Regression and Correlation

Part 1: Scatterplots & Describing a Relationship

Bivariate Data & Scatterplots

Direction, Form, and Strength

Negative Relationships

Why This Matters

Part 2: The Correlation Coefficient r

What $r$ Tells You

Part 3: The Line of Best Fit

Part 4: Predictions, Residuals & Cautions

Part 5: Mixed Practice & Mastery Check

Quick Reference

Computing $r$ by Hand

The Coefficient of Determination $r^2$

The Least-Squares Regression Line

Building the Line: Worked Example

The Two-Step Recipe

Making Predictions

Residuals: How Far Off Was the Prediction?

Correlation Is Not Causation

One Full Problem, Start to Finish

You're Ready

Part 1: Scatterplots & Describing a Relationship

Bivariate Data & Scatterplots

Direction, Form, and Strength

Negative Relationships

Why This Matters

Part 2: The Correlation Coefficient r

What rr Tells You

Part 3: The Line of Best Fit

Part 4: Predictions, Residuals & Cautions

Part 5: Mixed Practice & Mastery Check

Quick Reference

Computing rrr by Hand

The Coefficient of Determination r2r^2r2

The Least-Squares Regression Line

Building the Line: Worked Example

The Two-Step Recipe

Making Predictions

Residuals: How Far Off Was the Prediction?

Correlation Is Not Causation

One Full Problem, Start to Finish

You're Ready

What $r$ Tells You

Computing $r$ by Hand

The Coefficient of Determination $r^2$

Topics in This Part

Worked Example

Interpolation vs. Extrapolation

Worked Example

Topics in This Part

Worked Example

Interpolation vs. Extrapolation

Worked Example

🎯⭐ INTERACTIVE LESSON

Learn step-by-step with interactive practice!

← Back to Standard Lesson

Part 1 of 5 — Scatterplots & Describing a Relationship

Section
Bivariate Data & Scatterplots
Direction, Form, and Strength
Explanatory vs. Response Variables

🔑 Key Concept: When two numerical variables are measured on the same individuals — like hours studied and test score — we plot the pairs as points. The picture that emerges (a scatterplot) tells us whether the variables move together, and how tightly.

Bivariate means two variables. Each individual contributes an ordered pair $(x, y)$ .

Hours studied ( $x$ )	Test score ( $y$ )
1	50
2	65
3	70
4	80
5	90

Plotting these five points gives a scatterplot. Here the points climb from lower-left to upper-right — more hours tends to mean a higher score.

We always put the explanatory variable (the one we think does the explaining) on the horizontal $x$ -axis and the response variable (the one we think responds) on the vertical $y$ -axis.

💡 In "hours studied vs. test score," hours is explanatory ( $x$ ) and score is response ( $y$ ). Ask yourself: which one would I change to affect the other? That one is $x$ .

Concept Check 🎯

Describe every scatterplot with three words:

Direction — Positive (points rise left-to-right) or negative (points fall).
Form — Linear (points follow a straight-line pattern) or nonlinear (a curve).
Strength — How tightly the points hug the pattern: strong, moderate, or weak.

Picture	Direction	Form	Strength
Points rise tightly along a line	Positive	Linear	Strong
Points fall in a loose cloud	Negative	Linear	Weak
Points trace a U-shape	(curved)	Nonlinear	—

⚠️ Always look at the scatterplot first. Everything else in this lesson — the correlation coefficient, the line of best fit — only makes sense when the form is linear. A strong curved pattern can fool a number into looking weak.

Describe the Scatterplot 🔽

A scatterplot of outdoor temperature vs. ice cream sales shows points climbing steadily from lower-left to upper-right, clustered tightly around a straight line.

A negative relationship is just as common: as one variable goes up, the other goes down, so the points fall from upper-left to lower-right.

Explanatory ( $x$ )	Response ( $y$ )	Likely direction
Hours of TV watched	Test grade	Negative
Outdoor temperature	Heating bill	Negative
Age of a used car	Resale price	Negative
Hours exercised	Resting heart rate	Negative

💡 A negative relationship can still be strong — strength is about how tightly the points hug the line, not which way the line tilts. A tightly falling cloud of points is strong and negative.

Concept Check 🎯

Once we know a relationship is linear, two powerful tools open up:

The correlation coefficient $r$ — a single number measuring direction and strength (Part 2).
The line of best fit $\hat{y} = a + bx$ — an equation that lets us predict (Parts 3–4).

You can now read a scatterplot and describe it in three words. Next we turn "strong/moderate/weak" into an exact number.

Part 2 of 5 — The Correlation Coefficient $r$

🔑 The Idea: The correlation coefficient $r$ squeezes the direction and strength of a linear relationship into one number between $-1$ and $+1$ .

Part 3 of 5 — The Line of Best Fit

🔑 The Goal: Find the single straight line $\hat{y} = a + bx$ that comes closest to all the points. We call it the least-squares regression line, and it lets us predict $y$ from .

Part 4 of 5 — Predictions, Residuals & Cautions

🔑 The Payoff: With the line $\hat{y} = a + bx$ in hand, plug in any $x$ to predict . But predictions come with rules — and a famous warning about .

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) describe a scatterplot, (2) interpret $r$ and $r^2$ , (3) build the line of best fit, and (4) predict, find residuals, and avoid the causation trap. Let's put it all together.

Goal	Key move
Describe a scatterplot	direction (pos/neg) · form (linear?) · strength
Measure strength + direction	$r$ , with

The sign of $r$ = the direction. Positive $r$ → positive (rising) trend; negative $r$ → negative (falling) trend.
The size of $r$ (its distance from $0$ ) = the strength.

Value of $r$	Meaning
$r = +1$	Perfect positive line (every point on it)
$r = -1$	Perfect negative line
$r \approx 0$	No linear relationship
$\lvert r \rvert \ge 0.8$	Strong
$0.5 \le \lvert r \rvert < 0.8$	Moderate
$\lvert r \rvert < 0.5$	Weak

💡 $r = -0.9$ is stronger than $r = +0.6$ . Strength depends on how far $r$ is from $0$ , not on its sign. The sign only tells direction.

Concept Check 🎯

For a small data set you can compute $r$ from sums of squares:

$r = \frac{S_{xy}}{\sqrt{S_{xx}\, S_{yy}}}, \quad\text{where}\quad S_{xy} = \sum (x-\bar{x})(y-\bar{y})$

and $S_{xx} = \sum (x-\bar{x})^2$ , .

Data: $x = 0,1,2,3,4$ and $y = 1,3,2,5,4$ . Means: , .

$x$	$y$	$x-\bar{x}$	$y-\bar{y}$

$r = \frac{8}{\sqrt{10 \cdot 10}} = \frac{8}{10} = 0.8$

✅ A moderate-to-strong positive correlation. In practice you'll usually let a calculator do this — but doing it once shows you exactly what $r$ measures.

Compute $r$ 🧮

A different data set gives these sums: $S_{xx} = 16,\quad S_{yy} = 25,\quad S_{xy} = 18$

1) Find $\sqrt{S_{xx}\, S_{yy}} = \sqrt{16 \cdot 25} = \,?$ Find

Square the correlation to get $r^2$ (always between $0$ and $1$ ). It has a concrete meaning:

🔑 $r^2$ is the proportion of the variation in $y$ that is explained by the linear relationship with $x$ .

If the Hours vs. Score data gives $r = 0.99$ , then $r^2 = (0.99)^2 \approx 0.98 = 98\%.$ So about is explained by hours studied; the remaining 2% is due to everything else.

⚠️ Because $r^2$ comes from squaring, it loses the sign — it cannot tell you the direction. A correlation of $r=-0.6$ and one of $r=+0.6$ both give .

Concept Check 🎯

Out of all possible lines, the least-squares line is the one that makes the total of the squared vertical distances from the points to the line as small as possible.

$\hat{y} = a + bx$

The hat on $\hat{y}$ (" $y$ -hat") signals a predicted value, not an observed one.

$b$ = slope — how much $\hat{y}$ changes for each +1 increase in $x$ .
$a$ = $y$ -intercept — the predicted when .

The slope and correlation are linked through the standard deviations $s_x$ and $s_y$ :

$b = r \cdot \frac{s_y}{s_x}, \qquad a = \bar{y} - b\,\bar{x}$

💡 The slope $b$ always has the same sign as $r$ : a positive correlation gives an upward-sloping line, a negative correlation a downward-sloping one.

Concept Check 🎯

Use the Hours vs. Score data: $x = 1,2,3,4,5$ and $y = 50,65,70,80,90$ .

Step 1 — Means. $\bar{x} = 3$ , $\bar{y} = 71$ .

Step 2 — Sums of squares. Working from the deviations gives $S_{xx} = 10, \qquad S_{xy} = 95.$

Step 3 — Slope. The slope can be found directly as $b = \dfrac{S_{xy}}{S_{xx}}$ :

Step 4 — Intercept. Use $a = \bar{y} - b\,\bar{x}$ : $a = 71 - 9.5(3) = 71 - 28.5 = 42.5.$

The line: $\hat{y} = 42.5 + 9.5x$

✅ Check: the regression line always passes through the point of averages $(\bar{x}, \bar{y}) = (3, 71)$ . Test it: $42.5 + 9.5$ ✓

Order the Steps 🔽

You're finding the regression line for a data set with $\bar{x} = 4$ , $\bar{y} = 20$ , $S_{xx} = 8$ , and $S_{xy} = 24$ .

Every regression problem in Algebra 1 boils down to the same two moves, in order:

$\boxed{\;b = \frac{S_{xy}}{S_{xx}}\;}\qquad\text{then}\qquad\boxed{\;a = \bar{y} - b\,\bar{x}\;}$

You must find the slope $b$ first, because the intercept formula needs it. Then assemble $\hat{y} = a + bx$ .

🔑 Sanity check every line you build: plug in $\bar{x}$ and you should get exactly $\bar{y}$ , because the line always passes through $(\bar{x}, \bar{y})$ .

Build the Line 🧮

A data set has $\bar{x} = 5$ , $\bar{y} = 10$ , $S_{xx} = 20$ , $S_{xy} = 30$ .

1) Slope $b = \dfrac{S_{xy}}{S_{xx}} = \,?$ Intercept

To predict, substitute the $x$ -value into the line.

Using $\hat{y} = 42.5 + 9.5x$ (Hours vs. Score):

Predict the score for 6 hours of study: $\hat{y} = 42.5 + 9.5(6) = 42.5 + 57 = 99.5$

Predict the score for 2.5 hours: $\hat{y} = 42.5 + 9.5(2.5) = 42.5 + 23.75 = 66.25$

Interpolation — predicting inside the range of the data ( $x$ between 1 and 5 here). Generally safe.
Extrapolation — predicting outside the data range (e.g. $x = 20$ hours). Risky — the linear pattern may not hold out there.

⚠️ Extrapolation is the #1 prediction trap. Our line would "predict" a score of $42.5 + 9.5(20) = 232.5$ for 20 hours — impossible on a 100-point test. Never trust a prediction far outside the data.

Predict with the Line 🧮

A delivery company models cost ( $y$ , dollars) against distance ( $x$ , miles) with $\hat{y} = 3 + 2x.$

1) Predicted cost for a 10-mile trip: $\hat{y} = \,?$ (dollars) 2) Predicted cost for a 25-mile trip: $\hat{y} = \,?$

A residual is the vertical gap between an actual data point and the line:

$\text{residual} = y - \hat{y} = (\text{observed}) - (\text{predicted})$

In the Hours vs. Score data, the student who studied 2 hours actually scored 65. The line predicts $\hat{y} = 42.5 + 9.5(2) = 61.5.$ So the residual is $65 - 61.5 = +3.5.$

A positive residual means the point sits above the line (the model under-predicted). A negative residual means it sits below the line.

💡 The least-squares line is built so the residuals always sum to zero — positives and negatives perfectly cancel. That's what makes it the "best fit."

Compute Residuals 🧮

Use $\hat{y} = 42.5 + 9.5x$ . A student studied 4 hours and scored 80.

1) Predicted score $\hat{y} = 42.5 + 9.5(4) = \,?$ 2) Residual $= y - \hat{y} = 80 - \,?$ — enter the residual (it is negative)

A strong correlation means two variables move together — it does not prove one causes the other.

⚠️ The classic example: Across a town, ice-cream sales and drowning deaths are strongly positively correlated. Ice cream doesn't cause drownings! A hidden lurking variable — hot summer weather — drives both up at once.

To establish causation you need a controlled experiment, not just a high $r$ . Always ask: Could a third variable explain both?

Concept Check 🎯

⚠️ Remember: the sign of $r$ matches the sign of $b$ ; $r^2$ drops the sign; extrapolation is risky; and correlation never proves causation.

Mixed Practice 🎯

A typical exam question hands you the summary numbers and asks for the line and a prediction. The flow is always the same:

Slope: $b = \dfrac{S_{xy}}{S_{xx}}$
Intercept: $a = \bar{y} - b\,\bar{x}$
Assemble: $\hat{y} = a + bx$
Predict: substitute the requested $x$

Try the complete cycle below — find the slope, the intercept, and then make a prediction.

Build & Predict 🧮

A data set has $\bar{x} = 4$ , $\bar{y} = 12$ , $S_{xx} = 10$ , $S_{xy} = 25$ .

1) Slope $b = S_{xy}/S_{xx} = \,?$ 2) Intercept Using , predict when .

You can describe a scatterplot, read $r$ and $r^2$ , build $\hat{y} = a + bx$ , predict and interpret residuals, and steer clear of the causation trap. One last set of three questions seals the lesson.

💡 On each Exit-Quiz item, decide which tool it's testing first — strength of $r$ , a residual, or causation — then apply the matching move.

Exit Quiz ✅

Answer all three to finish the lesson.

\;S_{yy} = \sum (y-\bar{y})^2

S_{yy} = \sum (y - \overset{y}{ˉ})^{2}

r = \dfrac{S_{xy}}{\sqrt{S_{xx} S_{yy}}} = \,?

(decimal, round to 2 places)

98% of the variation in test scores

b = \frac{95}{10} = 9.5.

42.5 + 9.5 (3) = 42.5 + 28.5 = 71

a = \bar{y} - b\,\bar{x} = \,?

a = \overset{y}{ˉ} - b \overset{x}{ˉ} = ?

$S_{xx}=10$

$S_{yy}=10$