11.2 Representing Data
Understanding Scatter Graphs
Scatter graphs show the relationship between two variables. They help us see correlation (whether two things are related)
Scatter graphs show the relationship between two variables. They help us see correlation (whether two things are related)
⚡ Types of Correlation:
Positive Correlation
As x increases, y increases
Negative Correlation
As x increases, y decreases
No Correlation
No clear relationship
Example 1: Scatter Graph with Line of Best Fit
Hours studied vs Test score:
Description:
• Strong positive correlation
• Line of best fit drawn through center of points
• More hours studied → higher scores
| Hours | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Score | 45 | 55 | 60 | 70 | 75 | 85 |
• Strong positive correlation
• Line of best fit drawn through center of points
• More hours studied → higher scores
Example 2: Interpreting Correlation
Temperature vs Ice cream sales: Positive correlation
As temperature increases, ice cream sales increase
Age of car vs Value: Negative correlation
As car age increases, value decreases
Shoe size vs IQ: No correlation
No relationship between these variables
Important: Correlation ≠ Causation!
Just because two things correlate doesn't mean one causes the other
As temperature increases, ice cream sales increase
Age of car vs Value: Negative correlation
As car age increases, value decreases
Shoe size vs IQ: No correlation
No relationship between these variables
Important: Correlation ≠ Causation!
Just because two things correlate doesn't mean one causes the other
💡 Scatter Graph Tips:
• Label axes: Always show what each axis represents
• Use scales: Choose sensible scales that use most of the space
• Line of best fit: Should pass through the mean point
• Equal spread: Roughly equal points above and below the line
• Don't force through origin: Unless the data suggests it
• Use scales: Choose sensible scales that use most of the space
• Line of best fit: Should pass through the mean point
• Equal spread: Roughly equal points above and below the line
• Don't force through origin: Unless the data suggests it
🎯 Correlation Practice
Visual Comparisons
Pie charts show parts of a whole. Bar charts compare different categories.
Pie charts show parts of a whole. Bar charts compare different categories.
⚡ Chart Comparison:
Pie Chart
Best for proportions
Bar Chart
Best for comparisons
Example 1: Drawing a Pie Chart
50 students chose their favorite sport:
Football: 20, Basketball: 15, Tennis: 10, Swimming: 5
Calculate angles:
Total = 50 students
One student = $360° \div 50 = 7.2°$
Football: $20 \times 7.2° = 144°$
Basketball: $15 \times 7.2° = 108°$
Tennis: $10 \times 7.2° = 72°$
Swimming: $5 \times 7.2° = 36°$
Check: $144° + 108° + 72° + 36° = 360°$ ✓
Football: 20, Basketball: 15, Tennis: 10, Swimming: 5
Calculate angles:
Total = 50 students
One student = $360° \div 50 = 7.2°$
Football: $20 \times 7.2° = 144°$
Basketball: $15 \times 7.2° = 108°$
Tennis: $10 \times 7.2° = 72°$
Swimming: $5 \times 7.2° = 36°$
Check: $144° + 108° + 72° + 36° = 360°$ ✓
Example 2: Reading a Bar Chart
Monthly rainfall (mm):
Reading:
• Highest rainfall: May (~140mm)
• Lowest rainfall: March (~50mm)
• Total Jan-Mar: 100 + 80 + 50 = 230mm
• Highest rainfall: May (~140mm)
• Lowest rainfall: March (~50mm)
• Total Jan-Mar: 100 + 80 + 50 = 230mm
💡 Chart Tips:
• Pie charts: Angles must sum to 360°
• Bar charts: Equal width bars, gaps between them
• Always label: Title, axes, and units
• Use color effectively: Make it easy to read
• Start bars at zero: Otherwise comparison is misleading
• Bar charts: Equal width bars, gaps between them
• Always label: Title, axes, and units
• Use color effectively: Make it easy to read
• Start bars at zero: Otherwise comparison is misleading
Understanding Data Spread
Box plots (box-and-whisker diagrams) show the spread of data using five key values. Cumulative frequency helps us find these values.
Box plots (box-and-whisker diagrams) show the spread of data using five key values. Cumulative frequency helps us find these values.
⚡ The Five Number Summary:
Key Values:
• Minimum: Lowest value
• Q1 (Lower Quartile): 25% of data below this
• Median (Q2): 50% of data below this
• Q3 (Upper Quartile): 75% of data below this
• Maximum: Highest value
• IQR (Interquartile Range): Q3 - Q1 (spread of middle 50%)
• Minimum: Lowest value
• Q1 (Lower Quartile): 25% of data below this
• Median (Q2): 50% of data below this
• Q3 (Upper Quartile): 75% of data below this
• Maximum: Highest value
• IQR (Interquartile Range): Q3 - Q1 (spread of middle 50%)
Example 1: Drawing a Cumulative Frequency Graph
Test scores of 60 students:
Finding quartiles from graph:
• Median (Q2): 30th value ≈ 52
• Q1: 15th value ≈ 38
• Q3: 45th value ≈ 70
• IQR = 70 - 38 = 32
| Score | Frequency | Cumulative Frequency |
|---|---|---|
| 0 ≤ x < 20 | 5 | 5 |
| 20 ≤ x < 40 | 12 | 17 |
| 40 ≤ x < 60 | 18 | 35 |
| 60 ≤ x < 80 | 15 | 50 |
| 80 ≤ x < 100 | 10 | 60 |
• Median (Q2): 30th value ≈ 52
• Q1: 15th value ≈ 38
• Q3: 45th value ≈ 70
• IQR = 70 - 38 = 32
Example 2: Drawing a Box Plot
Using the five number summary:
Min = 5, Q1 = 38, Median = 52, Q3 = 70, Max = 95
Min = 5, Q1 = 38, Median = 52, Q3 = 70, Max = 95
💡 Box Plot Tips:
• Plot on upper boundaries: For cumulative frequency
• IQR shows spread: Larger IQR = more spread out
• Box shows middle 50%: Of the data
• Compare box plots: Easily see which dataset is more spread out
• Outliers: Values more than 1.5 × IQR from Q1 or Q3
• IQR shows spread: Larger IQR = more spread out
• Box shows middle 50%: Of the data
• Compare box plots: Easily see which dataset is more spread out
• Outliers: Values more than 1.5 × IQR from Q1 or Q3
Histograms with Unequal Class Widths
Histograms look like bar charts but show continuous data. When class widths are unequal, we use frequency density on the y-axis.
Histograms look like bar charts but show continuous data. When class widths are unequal, we use frequency density on the y-axis.
⚡ Key Formula:
Frequency Density
$$\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}$$
Makes bars comparable
Finding Frequency
$$\text{Frequency} = \text{FD} \times \text{Width}$$
Area of bar = frequency
Example 1: Drawing a Histogram
Ages of people at a cinema:
Note: Wider classes have lower bars (density is lower even if frequency is similar)
| Age | Frequency | Class Width | Frequency Density |
|---|---|---|---|
| 0-10 | 20 | 10 | 2.0 |
| 10-20 | 40 | 10 | 4.0 |
| 20-30 | 35 | 10 | 3.5 |
| 30-50 | 30 | 20 | 1.5 |
| 50-80 | 15 | 30 | 0.5 |
Example 2: Reading a Histogram
From the histogram above, estimate how many people were aged 30-50:
Method:
Frequency = Frequency Density × Class Width
Frequency = 1.5 × 20 = 30 people
Or: Count the area of that bar (width × height)
Method:
Frequency = Frequency Density × Class Width
Frequency = 1.5 × 20 = 30 people
Or: Count the area of that bar (width × height)
Example 3: Comparing Bar Charts and Histograms
Bar Chart:
• Gaps between bars
• Represents categories (discrete data)
• Height = frequency
• Bars can be reordered
Histogram:
• No gaps between bars
• Represents continuous data
• Height = frequency density (if widths vary)
• Bars must be in order
• Area = frequency
• Gaps between bars
• Represents categories (discrete data)
• Height = frequency
• Bars can be reordered
Histogram:
• No gaps between bars
• Represents continuous data
• Height = frequency density (if widths vary)
• Bars must be in order
• Area = frequency
💡 Histogram Tips:
• Check class widths: If unequal, use frequency density
• No gaps: Bars touch in histograms (continuous data)
• Area matters: Area of bar = frequency
• Finding frequency: FD × width, or count area
• Be careful with boundaries: Use correct upper boundaries
• No gaps: Bars touch in histograms (continuous data)
• Area matters: Area of bar = frequency
• Finding frequency: FD × width, or count area
• Be careful with boundaries: Use correct upper boundaries
🧮 Frequency Density Calculator:
Real Life Uses of Data Representation:
• Business: Sales trends, market analysis
• Science: Experimental results, data distribution
• Medicine: Patient data, treatment outcomes
• Education: Test score distributions
• Government: Census data, economic indicators
• Sports: Performance statistics, rankings
• Business: Sales trends, market analysis
• Science: Experimental results, data distribution
• Medicine: Patient data, treatment outcomes
• Education: Test score distributions
• Government: Census data, economic indicators
• Sports: Performance statistics, rankings