11.2 Representing Data

Hours studied vs Test score:

Hours	1	2	3	4	5	6
Score	45	55	60	70	75	85

Description:
• Strong positive correlation
• Line of best fit drawn through center of points
• More hours studied → higher scores

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

• 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

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: $144° + 108° + 72° + 36° = 360°$ ✓

Monthly rainfall (mm):

Reading:
• Highest rainfall: May (~140mm)
• Lowest rainfall: March (~50mm)
• Total Jan-Mar: 100 + 80 + 50 = 230mm

• 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

Test scores of 60 students:

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

Finding quartiles from graph:
• Median (Q2): 30th value ≈ 52
• Q1: 15th value ≈ 38
• Q3: 45th value ≈ 70
• IQR = 70 - 38 = 32

Using the five number summary:
Min = 5, Q1 = 38, Median = 52, Q3 = 70, Max = 95

• 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

Ages of people at a cinema:

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

Note: Wider classes have lower bars (density is lower even if frequency is similar)

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)

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

• 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