Scatter Plots and Correlations

What is a Scatter Plot?

A scatter plot is a graph showing the relationship between two variables.

Each point represents one data pair (x, y).

Purpose: Identify patterns and correlations between variables

Example data: Study hours vs. test scores for 5 students:

• (1, 65): 1 hour study, score 65
• (2, 70): 2 hours study, score 70
• (3, 80): 3 hours study, score 80
• (4, 85): 4 hours study, score 85
• (5, 95): 5 hours study, score 95

Plot these points to see the relationship!

Creating a Scatter Plot

Steps:

1. Draw axes (x-axis = independent variable, y-axis = dependent variable)
2. Label axes with variable names and units
3. Choose appropriate scale
4. Plot each ordered pair as a point
5. Title the graph

Example: Height vs. Shoe Size

Data:

• Height 60 inches, Size 7
• Height 64 inches, Size 8
• Height 68 inches, Size 9
• Height 72 inches, Size 10

Plot each point: (60, 7), (64, 8), (68, 9), (72, 10)

Types of Correlation

Positive Correlation:

• As x increases, y increases
• Points trend upward from left to right
• Slope is positive

Example: Study time vs. test score More studying → higher scores

Negative Correlation:

• As x increases, y decreases
• Points trend downward from left to right
• Slope is negative

Example: Speed vs. travel time Higher speed → less time

No Correlation:

• No clear pattern
• Points scattered randomly
• No relationship between variables

Example: Shoe size vs. math score No connection!

Strength of Correlation

Strong Correlation:

• Points close to a line
• Clear, tight pattern
• Strong relationship

Moderate Correlation:

• Points somewhat close to a line
• Pattern visible but not tight
• Moderate relationship

Weak Correlation:

• Points loosely scattered
• Vague pattern
• Weak relationship

Measuring strength: How close points are to forming a line

Correlation vs. Causation

Correlation: Two variables related (change together)

Causation: One variable CAUSES the other to change

KEY POINT: Correlation does NOT prove causation!

Example 1: Ice cream sales and drowning deaths

Correlation: Both increase in summer Causation? NO! Heat causes both, not each other

Example 2: Study time and test scores

Correlation: Positive Causation? Likely YES! Studying helps scores

Example 3: Shoe size and reading ability in children

Correlation: Positive (both increase with age) Causation? NO! Age causes both

Critical thinking: Always ask "Could there be another factor?"

Line of Best Fit (Trend Line)

A line that best represents the data pattern.

Purpose: Make predictions from data

Characteristics:

• Passes through or near most points
• Equal points above and below line (balanced)
• Minimizes distance to all points

Drawing by hand:

1. Identify the pattern (positive/negative)
2. Draw a line through the "middle" of points
3. Balance points above and below

Example: Study hours vs. scores

Points: (1, 65), (2, 70), (3, 80), (4, 85), (5, 95)

Line approximately: y = 8x + 57

Making Predictions

Use the line of best fit to predict values.

Interpolation: Predict within data range

Example: If line is y = 8x + 57 Predict score for 2.5 hours study: y = 8(2.5) + 57 = 20 + 57 = 77

Expected score: 77

Extrapolation: Predict outside data range

Example: Predict score for 10 hours study: y = 8(10) + 57 = 137

Warning: Extrapolation less reliable! (Can't score 137!)

Outliers in Scatter Plots

Outlier: Point far from the general pattern

Example: Study hours vs. scores

Most points follow pattern, but one student: (4, 50) - studied 4 hours but scored only 50

This is an outlier!

Reasons for outliers:

• Measurement error
• Unusual circumstance (student was sick)
• Special case
• Data entry mistake

Effect on correlation:

• Outliers can weaken correlation
• May affect line of best fit
• Consider removing if justified

Linear vs. Nonlinear Patterns

Linear pattern: Points form roughly a straight line

Nonlinear patterns:

Quadratic: Points form a curve (parabola) Example: Height of thrown ball vs. time

Exponential: Points show rapid increase/decrease Example: Bacterial growth vs. time

No pattern: Random scatter Example: Phone number vs. height

For Algebra 1: Focus on linear patterns!

Correlation Coefficient (r)

Measures strength and direction of linear correlation

Range: -1 to +1

r = +1: Perfect positive correlation (all points on upward line) r = 0: No correlation r = -1: Perfect negative correlation (all points on downward line)

Interpreting r:

• 0.8 to 1.0 or -0.8 to -1.0: Strong
• 0.5 to 0.8 or -0.5 to -0.8: Moderate
• 0 to 0.5 or 0 to -0.5: Weak

Example: r = 0.92 → Strong positive correlation r = -0.73 → Moderate negative correlation r = 0.15 → Weak positive correlation

Real-World Examples

Example 1: Temperature vs. Ice Cream Sales

Data shows positive correlation: Higher temperature → more sales

Makes sense! (causation likely)

Example 2: Car Age vs. Value

Negative correlation: Older car → lower value

Clear causation!

Example 3: Height vs. Arm Span

Strong positive correlation: Taller people have longer arms

Biological relationship!

Example 4: TV Hours vs. GPA

Negative correlation: More TV → lower GPA

Correlation? Yes. Causation? Maybe! (could be other factors)

Analyzing Scatter Plots

Questions to ask:

1. Is there a correlation? (positive/negative/none)
2. How strong is the correlation? (strong/moderate/weak)
3. Are there outliers?
4. Is the pattern linear or nonlinear?
5. Could there be causation?
6. Are there lurking variables?

Lurking variable: Hidden factor affecting both variables

Example: Study time and scores both affected by student motivation

Creating Scatter Plots from Data Tables

Example: Hours of sleep vs. energy level (1-10 scale)

Data: Student A: 5 hours, energy 4 Student B: 7 hours, energy 7 Student C: 6 hours, energy 5 Student D: 8 hours, energy 8 Student E: 4 hours, energy 3

Points: (5, 4), (7, 7), (6, 5), (8, 8), (4, 3)

Pattern: Positive correlation More sleep → more energy

Using Technology

Graphing calculators:

• Enter data in lists
• Create scatter plot
• Calculate correlation coefficient
• Find line of best fit equation

Example: TI-84

1. STAT → Edit → Enter data in L1 and L2
2. 2nd → STAT PLOT → Turn on Plot1
3. ZOOM → ZoomStat
4. STAT → CALC → LinReg (calculate r and equation)

Spreadsheet software:

• Excel, Google Sheets
• Create chart → Scatter plot
• Add trendline
• Display equation and R² value

Common Mistakes to Avoid

1. Confusing correlation with causation Just because two things correlate doesn't mean one causes the other!

2. Ignoring outliers Outliers can significantly affect the line of best fit

3. Extrapolating too far Predictions far outside data range are unreliable

4. Wrong variable on wrong axis Independent variable (x) should be on horizontal axis

5. Poor scale choices Scale should show pattern clearly without distortion

6. Not labeling axes Always label with variable names and units!

Interpreting Slope in Context

The slope of the line of best fit has real meaning!

Example: Study hours vs. test scores Line: y = 8x + 57

Slope = 8 means: Each additional hour of study increases score by about 8 points

Example: Temperature vs. ice cream sales Line: y = 12x + 50

Slope = 12 means: Each degree increase in temperature adds about 12 sales

Y-Intercept in Context

Example: y = 8x + 57 (study hours vs. score)

Y-intercept = 57: Predicted score with 0 hours study

Does this make sense? Maybe! (represents prior knowledge)

Example: y = 12x + 50 (temperature vs. sales)

Y-intercept = 50: Predicted sales at 0°F

May not make sense! (shop might be closed)

Lesson: Interpret intercept carefully in context!

Quick Reference

Positive correlation: Both variables increase together

Negative correlation: One increases, other decreases

No correlation: No relationship

Strong: Points close to line

Weak: Points scattered loosely

Outlier: Point far from pattern

Line of best fit: Line representing overall trend

Interpolation: Predict within data range

Extrapolation: Predict outside data range

Correlation ≠ Causation!

Practice Tips

• Always label axes with variable names and units
• Start with independent variable on x-axis
• Look for overall pattern before drawing line
• Balance points above and below the line
• Identify and note outliers
• Consider whether correlation makes sense
• Don't assume causation without evidence
• Practice reading scatter plots from different contexts
• Use technology to check your work
• Think critically about lurking variables
• Apply to real-world situations
• Understand the difference between interpolation and extrapolation
• Remember: Scatter plots are about relationships!

Scatter plots are powerful tools for visualizing relationships between variables. Master this skill and you'll be able to analyze data in science, social studies, business, and everyday life!