Data Representation#
Data Fundamentals Revision#
Theory#
Data representation builds upon understanding different types of data and the importance of organizing information effectively for analysis. Data can be classified as qualitative (categorical) or quantitative (numerical), with quantitative data further divided into discrete (countable) or continuous (measurable).
Application#
Examples#
Example 1: Basic Data Classification#
Survey data collected from 30 students about favorite subjects: Math (8), Science (7), English (6), History (5), Art (4)
Method 1: Frequency Analysis
\(\text{Total responses} = 8 + 7 + 6 + 5 + 4 = 30 \quad \text{(sum all frequencies)}\)
\(\text{Relative frequency for Math} = \frac{8}{30} = 0.267 \quad \text{(approximately 26.7\%)}\)
\(\text{Modal category} = \text{Math} \quad \text{(highest frequency of 8)}\)
Interactive Visualization: Data Types Explorer#
Multiple Choice Questions#
Data Representation#
Theory#
Foundational Definitions: Data representation involves organizing and displaying information to reveal patterns, trends, and insights. It transforms raw data into visual or tabular formats that facilitate understanding and analysis.
Key Visualization Methods and Their Properties:
Frequency Tables: Organize data into categories with their corresponding counts
• Structure: Categories/Values | Frequency | Relative Frequency | Percentage • Purpose: Summarize data distribution and calculate proportions • Formula: \(\text{Cumulative Frequency}_i = \sum_{j=1}^{i} f_j\)
Bar Charts: Display categorical data using rectangular bars
• Properties: Bars are separated (not touching), height represents frequency • Orientation: Can be vertical (column chart) or horizontal • Special case: Grouped bar charts for comparing multiple datasets • Formula for bar height: \(h_i = k \cdot f_i\) where \(k\) is a scaling constant
Histograms: Show distribution of continuous data using connected bars
• Key distinction: Bars touch (no gaps) representing continuous intervals • Equal-width intervals: \(\text{Width} = \frac{\text{Range}}{\text{Number of intervals}}\) • Unequal-width intervals: Use frequency density = \(\frac{\text{Frequency}}{\text{Interval width}}\) • Area principle: Area of bar ∝ frequency
Pie Charts: Display parts of a whole using circular sectors
\(\text{Central Angle} = \frac{\text{Category Frequency}}{\text{Total Frequency}} \times 360°\)
\(\text{Sector Area} = \frac{\text{Central Angle}}{360°} \times \pi r^2\)
Line Graphs: Show changes over time or relationships between continuous variables
• Properties: Points connected by line segments • Multiple series: Different lines for comparison • Time series: X-axis represents time periods • Interpolation: Estimating values between data points
Stem-and-Leaf Plots: Display data while preserving individual values
• Structure: Stem (leading digits) | Leaf (trailing digits) • Advantage: Shows distribution and retains actual data values • Back-to-back: Compare two datasets • Key: Always specify the unit represented by stem|leaf
Cumulative Frequency Curves (Ogives): Show running totals of frequencies
• Less than ogive: Plots cumulative frequency against upper class boundaries • Greater than ogive: Plots cumulative frequency against lower class boundaries • Median location: At \(\frac{n}{2}\) on the cumulative frequency axis • Quartile locations: \(Q_1\) at \(\frac{n}{4}\), \(Q_3\) at \(\frac{3n}{4}\)
Interactive Visualization: Data Representation Methods#
Application#
Examples#
Example 1: Creating a Frequency Table#
Solve: Organize the following test scores into a frequency table with class intervals of width 10: 45, 67, 72, 89, 56, 91, 78, 83, 62, 75, 88, 93, 71, 85, 79
Method 1: Systematic Organization
\(\text{Range} = 93 - 45 = 48 \quad \text{(find data spread)}\)
\(\text{Number of intervals} = \lceil \frac{48}{10} \rceil = 5 \quad \text{(round up to cover all data)}\)
\(\text{Class intervals: } 40-49, 50-59, 60-69, 70-79, 80-89, 90-99 \quad \text{(establish boundaries)}\)
\(\text{Frequency count: } f_1=1, f_2=1, f_3=2, f_4=5, f_5=4, f_6=2 \quad \text{(tally each interval)}\)
Example 2: Constructing a Histogram#
Solve: Create a histogram for grouped data representing monthly rainfall (mm): 0-20 (3 months), 20-40 (4 months), 40-80 (2 months), 80-120 (3 months)
Method 1: Equal Area Principle
\(\text{Frequency density}_1 = \frac{3}{20} = 0.15 \quad \text{(for interval 0-20)}\)
\(\text{Frequency density}_2 = \frac{4}{20} = 0.20 \quad \text{(for interval 20-40)}\)
\(\text{Frequency density}_3 = \frac{2}{40} = 0.05 \quad \text{(for interval 40-80)}\)
\(\text{Frequency density}_4 = \frac{3}{40} = 0.075 \quad \text{(for interval 80-120)}\)
Example 3: Calculating Pie Chart Angles#
Solve: A survey of 120 students’ transport methods shows: Bus (45), Walk (30), Car (25), Bicycle (20). Calculate the central angles for a pie chart.
Method 1: Proportional Angle Calculation
\(\text{Angle for Bus} = \frac{45}{120} \times 360° = 135° \quad \text{(largest sector)}\)
\(\text{Angle for Walk} = \frac{30}{120} \times 360° = 90° \quad \text{(quarter circle)}\)
\(\text{Angle for Car} = \frac{25}{120} \times 360° = 75° \quad \text{(calculate proportion)}\)
\(\text{Angle for Bicycle} = \frac{20}{120} \times 360° = 60° \quad \text{(smallest sector)}\)
Multiple Choice Questions#
Sector Specific Questions: Data Representation Applications#
Key Takeaways#
Important
Choose appropriate representations: Match the visualization method to your data type and purpose
Frequency density for histograms: When class widths are unequal, use frequency density = frequency/width
Cumulative frequency curves: Useful for finding medians, quartiles, and percentiles
Pie charts show proportions: Calculate angles using (frequency/total) × 360°
Stem-and-leaf plots: Preserve individual data values while showing distribution
Bar charts vs histograms: Bars separated for categorical data, touching for continuous data
Always label clearly: Include titles, axis labels, units, and keys for interpretation