Scatterplots and Line of Best Fit (SAT Math)

What is a Scatterplot?

Graph showing relationship between two variables

Each point represents:

• x-coordinate: one variable
• y-coordinate: another variable

Example: Height vs. Weight

• Each point = one person
• x = height
• y = weight

Types of Correlation

Positive Correlation

As x increases, y increases

Pattern: Points slope upward (↗)

Examples:

• Study time vs. test scores
• Temperature vs. ice cream sales
• Height vs. shoe size

Negative Correlation

As x increases, y decreases

Pattern: Points slope downward (↘)

Examples:

• Speed vs. travel time
• Price vs. quantity demanded
• Outdoor temperature vs. heating costs

No Correlation

No clear pattern

Pattern: Points scattered randomly

Examples:

• Shoe size vs. test scores
• Height vs. favorite color

Strength of Correlation

Strong Correlation

Points cluster tightly around a line

• Clear pattern
• Easy to predict

Weak Correlation

Points loosely follow pattern

• General trend but lots of variation
• Harder to predict

Perfect Correlation

All points exactly on a line

• Rare in real data
• r = 1 (positive) or r = -1 (negative)

Line of Best Fit (Trend Line)

What Is It?

Line that best represents the data trend

Also called:

• Regression line
• Trend line
• Best-fit line

Equation Form

Usually written as: $y = mx + b$

Where:

• $m$ = slope (rate of change)
• $b$ = y-intercept (starting value)

Or: $\hat{y} = ax + b$ (predicted value)

Interpreting Slope

Slope ($m$) Meaning

Positive slope ($m > 0$):

• Positive correlation
• For every 1 unit increase in x, y increases by $m$

Example: $y = 2x + 10$

• For every 1 hour of study, score increases by 2 points

Negative slope ($m < 0$):

• Negative correlation
• For every 1 unit increase in x, y decreases by $|m|$

Example: $y = -3x + 100$

• For every 1 mph faster, travel time decreases by 3 minutes

Interpreting Y-Intercept

Y-Intercept ($b$) Meaning

Value of y when x = 0

Example: $y = 5x + 20$

• When study time = 0, predicted score = 20

Watch out: Sometimes x = 0 doesn't make sense!

• If x = year (like 2020), y-intercept is for year 0 (not useful!)

Making Predictions

Interpolation

Predicting within the data range

Generally reliable

Example: Data from x = 10 to x = 50

• Predicting at x = 30 → interpolation ✓

Extrapolation

Predicting outside the data range

Less reliable - pattern may not continue!

Example: Data from x = 10 to x = 50

• Predicting at x = 100 → extrapolation ⚠️

Outliers

What is an Outlier?

Point far from the general pattern

Effects:

• Can significantly affect line of best fit
• May indicate error or special case

On SAT:

• Questions may ask about outliers
• "Which point doesn't fit the pattern?"

Correlation vs. Causation

CRITICAL DISTINCTION!

Correlation: Variables are related Causation: One variable CAUSES change in other

Correlation ≠ Causation!

Example:

• Ice cream sales and drowning deaths are correlated
• But ice cream doesn't CAUSE drowning!
• Both are caused by third factor (hot weather!)

SAT Trap: Don't assume correlation means causation!

Correlation Coefficient ($r$)

What is $r$?

Number measuring strength and direction of correlation

Range: $-1 \leq r \leq 1$

$r = 1$: Perfect positive correlation $r = 0.8$: Strong positive correlation $r = 0.5$: Moderate positive correlation $r = 0$: No correlation $r = -0.5$: Moderate negative correlation $r = -0.8$: Strong negative correlation $r = -1$: Perfect negative correlation

Interpreting $r$

Sign (+ or -): Direction

• Positive = positive correlation
• Negative = negative correlation

Magnitude (how close to 1): Strength

• Close to 1 or -1 = strong
• Close to 0 = weak

Residuals

What is a Residual?

Difference between actual value and predicted value

Formula: Residual = Actual - Predicted

Positive residual: Point above line (actual > predicted) Negative residual: Point below line (actual < predicted) Zero residual: Point exactly on line

Residual Plots

Graph of residuals

Random pattern: Good fit Clear pattern: Poor fit (need different model)

SAT Question Types

Type 1: Interpret Slope

"What does the slope represent?"

Answer: Rate of change, change in y per unit change in x

Type 2: Use Equation to Predict

"According to the line, what is y when x = 10?"

Plug in: $y = m(10) + b$

Type 3: Identify Correlation

"Which best describes the relationship?"

Look at: Direction and strength of pattern

Type 4: Find Outlier

"Which point is farthest from the trend?"

Look for: Point that doesn't fit pattern

Type 5: Correlation vs. Causation

"Does x cause y?"

Remember: Correlation doesn't prove causation!

SAT Strategies

Read the Axes!

Always check what variables are being plotted

Look at the Pattern

Upward slope = positive, downward = negative

Use the Equation

Plug in values - don't try to eyeball!

Check Units

Slope units = (y units) per (x unit)

Remember Real-World Context

Does the answer make sense?

Common SAT Patterns

Temperature and Sales

Often positive correlation

• Hot temperature → more cold drinks sold

Time and Distance

Positive correlation for travel

• More time → more distance covered

Price and Demand

Negative correlation

• Higher price → lower demand

Practice and Performance

Positive correlation

• More practice → better performance

SAT Tips

• Positive correlation: Both increase together (upward slope ↗)
• Negative correlation: One increases, other decreases (downward slope ↘)
• No correlation: Random scatter, no pattern
• Strong correlation: Points cluster tightly around line
• Weak correlation: Points loosely follow pattern
• Slope ($m$): Rate of change (rise/run)
• Y-intercept ($b$): Value when x = 0
• Outlier: Point far from pattern
• Interpolation: Predicting within data range (reliable)
• Extrapolation: Predicting outside data range (less reliable)
• Correlation ≠ Causation: Related doesn't mean one causes other!
• Use the equation: Plug in values to predict
• Read axes carefully: Know what x and y represent
• Context matters: Does answer make real-world sense?
• $r$ close to 1 or -1: Strong correlation
• $r$ close to 0: Weak or no correlation