1. Line Graphs

Definition: A line graph displays data as points connected by lines, showing how one variable changes in relation to another (usually over time).

Key Components:

• X-axis (horizontal): Independent variable (usually time)
• Y-axis (vertical): Dependent variable (what's being measured)
• Data points: Individual values plotted as dots
• Connecting lines: Show trend or change over time

Interpreting Line Graphs:

• Rising line: Value is increasing
• Falling line: Value is decreasing
• Horizontal line: Value stays constant
• Steep slope: Rapid change
• Gentle slope: Gradual change

Creating Line Graphs:

1. Draw and label axes with appropriate scales
2. Plot each data point at the correct coordinates
3. Connect the points with straight lines
4. Add a title describing what the graph shows

2. Scatter Plots

Definition: A scatter plot shows the relationship between two numerical variables using dots plotted on a coordinate plane. Each dot represents one data pair (x, y).

Purpose:

• Show relationships between two variables
• Identify patterns and trends
• Detect outliers
• Make predictions

Interpreting Scatter Plots:

Look at the pattern of points to determine:

• Direction: Positive, negative, or no correlation
• Form: Linear or nonlinear pattern
• Strength: How closely points cluster around a line
• Outliers: Points far from the general pattern

Creating Scatter Plots:

1. Set up coordinate axes with appropriate scales
2. Label axes with variable names
3. Plot each data pair as a point (x, y)
4. Do NOT connect the points (unlike line graphs)
5. Add a descriptive title

3. Identify Trends with Scatter Plots (Correlation)

Types of Correlation:

Positive Correlation (Positive Association)

• As x increases, y increases
• Points trend upward from left to right
• Example: Hours studied vs. test score

Negative Correlation (Negative Association)

• As x increases, y decreases
• Points trend downward from left to right
• Example: Age of car vs. value

No Correlation (No Association)

• No clear pattern
• Points are randomly scattered
• Variables are not related
• Example: Shoe size vs. test score

Strength of Correlation:

4. Make Predictions with Scatter Plots

Using Trends to Predict: If a clear pattern exists, you can predict unknown values by following the trend.

Types of Predictions:

Interpolation: Predicting a value WITHIN the range of data

• More reliable because it's within observed data
• Example: If data ranges from x=0 to x=10, predict at x=5

Extrapolation: Predicting a value OUTSIDE the range of data

• Less reliable because you assume the trend continues
• Example: If data ranges from x=0 to x=10, predict at x=15

Steps to Make Predictions:

1. Identify the pattern/trend in the scatter plot
2. Draw or imagine a line of best fit
3. Locate the given x-value on the axis
4. Follow up or down to the line of best fit
5. Read the corresponding y-value

Example:

A scatter plot shows hours studied (x) vs. test score (y) with a positive correlation. If a student studies 4 hours and the trend suggests scores increase by 5 points per hour of study starting from a base of 60, predict the score:

Prediction: 60 + (4 × 5) = 80 points

5. Outliers in Scatter Plots

Definition: An outlier is a data point that does not fit the general pattern of the scatter plot. It lies far away from most other points.

Identifying Outliers:

• Look for points that are far from the cluster of other points
• Points that don't follow the trend/pattern
• Usually isolated from the main group

Causes of Outliers:

• Measurement error: Data recorded incorrectly
• Data entry error: Typo or mistake in recording
• Genuine unusual case: Real exception to the pattern
• Different subgroup: Belongs to a different category

Effect of Outliers:

Example:

In a scatter plot of age vs. salary, most points show increasing salary with age. One point shows a 25-year-old earning $500,000 while others at that age earn $30,000-$50,000. This is an outlier (possibly a professional athlete or CEO).

6. Line of Best Fit (Trend Line)

Definition: A straight line drawn through the center of a scatter plot that best represents the relationship between the variables. It minimizes the distance between the line and all data points.

Identifying a Good Line of Best Fit:

• Passes through or near most of the data points
• About equal number of points above and below the line
• Points are evenly distributed on both sides
• Follows the general direction/trend of the data
• Minimizes the vertical distances from points to the line

Drawing a Line of Best Fit:

1. Look at the overall pattern of points
2. Ignore outliers when drawing the line
3. Use a ruler to draw a straight line
4. Balance points above and below the line
5. Extend the line across the graph

Properties:

• Also called "trend line" or "regression line"
• Can be used for predictions (interpolation and extrapolation)
• Represents the general trend, not every individual point
• Should only be used when there's a linear pattern

7. Write Equations for Lines of Best Fit

Goal: Find the equation of the line in slope-intercept form: \( y = mx + b \)

Formula:

\( y = mx + b \)

where \( m \) = slope (rate of change), \( b \) = y-intercept (starting value)

Steps to Write the Equation:

1. Draw the line of best fit through the scatter plot
2. Find the slope (m):
   • Choose two points ON the line (preferably at grid intersections)
   • Use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} \)
3. Find the y-intercept (b):
   • Identify where the line crosses the y-axis
   • OR substitute a point and slope into \( y = mx + b \) and solve for b
4. Write the equation: \( y = mx + b \)

Example:

A line of best fit passes through points (2, 5) and (6, 13). Find the equation.

Step 1: Find slope:

\( m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2 \)

Step 2: Find y-intercept using point (2, 5):

\( 5 = 2(2) + b \) → \( 5 = 4 + b \) → \( b = 1 \)

Step 3: Write equation:

\( y = 2x + 1 \)

8. Interpret Lines of Best Fit: Word Problems

Interpreting the Equation \( y = mx + b \):

Slope (m): Rate of change; how much y changes per unit of x

• Units: (y units) per (x units)
• Positive slope: y increases as x increases
• Negative slope: y decreases as x increases

Y-intercept (b): Starting value; value of y when x = 0

• Initial amount or base value
• Where the trend begins

Example 1:

Equation: \( C = 3h + 20 \) where C is total cost ($) and h is hours of work

Slope = 3: The cost increases by $3 per hour

Y-intercept = 20: There's a $20 initial fee (starting cost)

Predict cost for 10 hours: C = 3(10) + 20 = $50

Example 2:

Equation: \( T = -5d + 70 \) where T is temperature (°F) and d is depth (feet) underground

Slope = -5: Temperature decreases by 5°F per foot of depth

Y-intercept = 70: Surface temperature is 70°F

Predict temperature at 8 feet: T = -5(8) + 70 = 30°F

9. Identify Representative, Random, and Biased Samples

Sampling: Selecting a subset of a population to make inferences about the entire population.

Population vs. Sample:

• Population: The entire group you want information about
• Sample: A subset of the population used to represent the whole

Types of Samples:

1. Random Sample

• Definition: Every member of the population has an equal chance of being selected
• Goal: Reduce bias and get fair representation
• Example: Drawing names from a hat containing all students' names
• Example: Using a random number generator to select participants

2. Representative Sample

• Definition: Accurately reflects the characteristics of the entire population
• Goal: Match population proportions
• Example: If school is 60% girls and 40% boys, sample should be similar
• Note: Random samples are usually representative

3. Biased Sample

• Definition: Does NOT fairly represent the population; certain groups are over/under-represented
• Problem: Leads to inaccurate conclusions
• Example: Surveying only students in the library about study habits (excludes non-library users)
• Example: Calling only landlines (excludes cell phone-only users)

Common Sources of Bias:

Evaluation Questions:

To determine if a sample is good, ask:

• Does everyone in the population have an equal chance of being selected?
• Does the sample match the characteristics of the population?
• Is any group excluded or overrepresented?
• Is the sample size large enough?

Examples:

Example 1: To find favorite lunch at school, survey every 10th student entering cafeteria

✓ Random and representative (all students use cafeteria, systematic selection)

Example 2: To find average income in a city, survey people at a luxury mall

✗ Biased (overrepresents wealthy people, excludes lower-income residents)

Example 3: To find students' opinions on homework, randomly select 50 students from entire school roster

✓ Random and likely representative (all students have equal chance)

Quick Reference: Two-Variable Statistics

Line Graphs vs. Scatter Plots:

Correlation Types:

• Positive: Both variables increase together (↗)
• Negative: One increases, other decreases (↘)
• None: No pattern or relationship

Line of Best Fit Equation:

\( y = mx + b \)

• m (slope): Rate of change
• b (y-intercept): Starting value

Good Samples:

• Random: Everyone has equal chance
• Representative: Matches population characteristics
• Unbiased: No systematic errors or exclusions

💡 Key Tips for Two-Variable Statistics

• ✓ Line graphs: connect points; Scatter plots: don't connect
• ✓ Positive correlation: both increase together (upward trend)
• ✓ Negative correlation: one up, one down (downward trend)
• ✓ No correlation: random scatter, no pattern
• ✓ Outliers: points far from the pattern
• ✓ Line of best fit: balance points above and below
• ✓ Slope tells you rate of change; y-intercept tells starting value
• ✓ Interpolation (within data) more reliable than extrapolation (outside data)
• ✓ Random sample: everyone has equal chance of selection
• ✓ Biased sample: some groups over/underrepresented
• ✓ Larger samples usually more reliable
• ✓ Use line of best fit equation to make predictions