Data Handling
Grouped Data & Histograms
What is a Bar Graph?
A bar graph (or bar chart) is a visual representation of data using rectangular bars of different heights. The height of each bar represents the frequency of that category.
Features of a Bar Graph:
- X-axis (horizontal): Categories (e.g., colors, subjects, months)
- Y-axis (vertical): Frequency or values
- Bars: Separate rectangles with gaps between them
- Title: Describes what the graph shows
- Labels: Name the axes and categories
What is a Histogram?
A histogram is similar to a bar graph but is used for continuous data grouped into class intervals. In a histogram, bars touch each other (no gaps) because the data is continuous.
Bar Graph vs Histogram:
| Feature | Bar Graph | Histogram |
|---|---|---|
| Data type | Categorical/discrete | Continuous |
| Bar spacing | Gaps between bars | Bars touch each other |
| X-axis | Categories (no order) | Number line (ordered) |
| Example | Favorite colors | Heights of students |
Types of Bar Graphs:
| Type | Description |
|---|---|
| Vertical Bar Graph | Bars go upward from horizontal axis |
| Horizontal Bar Graph | Bars go sideways from vertical axis |
| Double/Grouped Bar Graph | Multiple bars per category for comparison |
| Stacked Bar Graph | Bars divided into parts showing components |
Real-life Applications:
- Comparing sales of different products
- Showing population of different cities
- Displaying test scores distribution
- Analyzing weather patterns
The following data shows the number of students in different clubs: Music: 25, Sports: 40, Art: 30, Drama: 20, Science: 35. Draw a bar graph to represent this data.
- Step 1: Draw X-axis and Y-axis
- Step 2: Label X-axis as "Clubs", Y-axis as "Number of Students"
- Step 3: Choose scale: 1 unit = 5 students (up to 45)
- Step 4: Mark clubs: Music, Sports, Art, Drama, Science
- Step 5: Draw bars of heights: Music=25 (5 units), Sports=40 (8 units), Art=30 (6 units), Drama=20 (4 units), Science=35 (7 units)
- Step 6: Add title: "Student Participation in Clubs"
Answer: Bar graph drawn with appropriately scaled bars.
The heights (in cm) of 40 students are grouped as: 150-155: 5 students, 156-161: 12 students, 162-167: 15 students, 168-173: 8 students. Draw a histogram.
- Step 1: Draw X-axis: Heights (continuous intervals)
- Step 2: Draw Y-axis: Frequency (number of students)
- Step 3: Scale Y-axis: 0 to 16 (1 unit = 2 students)
- Step 4: Draw bar for 150-155: height 5
- Step 5: Draw bar for 156-161: height 12 (touching previous bar)
- Step 6: Draw bar for 162-167: height 15 (touching)
- Step 7: Draw bar for 168-173: height 8 (touching)
- Step 8: Title: "Distribution of Student Heights"
Answer: Histogram drawn with touching bars representing continuous data.
A double bar graph shows the number of boys and girls in three classes: Class A: Boys 15, Girls 12; Class B: Boys 18, Girls 14; Class C: Boys 12, Girls 16. What is the total number of boys and girls? Which class has the most students?
- Step 1: Calculate total boys = 15 + 18 + 12 = 45
- Step 2: Calculate total girls = 12 + 14 + 16 = 42
- Step 3: Total students = 45 + 42 = 87
- Step 4: Class A total = 15 + 12 = 27
- Step 5: Class B total = 18 + 14 = 32
- Step 6: Class C total = 12 + 16 = 28
- Step 7: Class B has the most students (32)
Answer: Total boys = 45, Total girls = 42, Total students = 87. Class B has the most students (32).
- Class interval size = upper limit − lower limit.
- Histogram bars touch (no gaps).
- Bar graph displays categorical data with separate bars (gaps between bars)
- Histogram displays continuous grouped data with touching bars
- X-axis shows categories (bar graph) or number line (histogram)
- Y-axis shows frequency or quantity
- Always label axes and give the graph a title
- Choose appropriate scale for clear visualization
- Double bar graphs compare two related data sets side by side
Pie Charts (Using Angles)
What is a Pie Chart?
A pie chart (or circle graph) is a circular chart divided into sectors (slices). Each sector represents a proportion of the whole data set. The entire circle represents 100% or the total frequency.
Calculating Pie Chart Angles:
Each sector's angle = \(\frac{\text{Category Frequency}}{\text{Total Frequency}} \times 360^\circ\)
What is a Line Graph?
A line graph shows data points connected by straight lines. It is ideal for showing trends over time or continuous change.
Features of Line Graph:
- X-axis usually shows time (years, months, days)
- Y-axis shows measured quantity
- Points are plotted and connected
- Multiple lines can compare different data sets
What is a Stem-and-Leaf Plot?
A stem-and-leaf plot is a way to organize data that shows both the shape of the distribution and the actual data values. The "stem" is the leading digit(s), and the "leaf" is the last digit.
Example: Data: 23, 25, 27, 31, 34, 35, 38, 42
| Stem | Leaf |
|---|---|
| 2 | 3, 5, 7 |
| 3 | 1, 4, 5, 8 |
| 4 | 2 |
Comparison of Graph Types:
| Graph Type | Best Used For | Example |
|---|---|---|
| Pie Chart | Showing parts of a whole | Budget allocation |
| Line Graph | Showing trends over time | Temperature changes |
| Stem-and-Leaf | Displaying all data points concisely | Test scores |
In a survey of 40 people about their preferred mode of transport: Car: 16, Bus: 12, Bicycle: 8, Walk: 4. Draw a pie chart. Find the angle for each sector.
- Step 1: Total = 16 + 12 + 8 + 4 = 40
- Step 2: Car angle = (16/40) × 360° = 0.4 × 360° = 144°
- Step 3: Bus angle = (12/40) × 360° = 0.3 × 360° = 108°
- Step 4: Bicycle angle = (8/40) × 360° = 0.2 × 360° = 72°
- Step 5: Walk angle = (4/40) × 360° = 0.1 × 360° = 36°
- Step 6: Check sum: 144° + 108° + 72° + 36° = 360° ✓
Answer: Angles: Car=144°, Bus=108°, Bicycle=72°, Walk=36°
The temperature recorded at 6 AM each day for a week: Mon: 18°, Tue: 20°, Wed: 22°, Thu: 25°, Fri: 24°, Sat: 21°, Sun: 19°. Draw a line graph and find the day with highest temperature.
- Step 1: Draw X-axis: Days (Mon to Sun)
- Step 2: Draw Y-axis: Temperature (0° to 30°)
- Step 3: Plot points: (Mon,18), (Tue,20), (Wed,22), (Thu,25), (Fri,24), (Sat,21), (Sun,19)
- Step 4: Connect points in order with straight lines
- Step 5: Add title: "Weekly Morning Temperature"
- Step 6: Highest point is Thursday at 25°C
Answer: Line graph drawn. Highest temperature: Thursday (25°C).
Prepare a stem-and-leaf plot for the following test scores: 78, 85, 92, 67, 88, 74, 91, 83, 79, 86, 95, 81, 77, 84, 89, 73, 90, 82, 76, 80. Find the median.
- Step 1: Sort the data: 67, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 95
- Step 2: Identify stems (tens digit): 6, 7, 8, 9
- Step 3: Create plot:
| Stem | Leaves |
|---|---|
| 6 | 7 |
| 7 | 3, 4, 6, 7, 8, 9 |
| 8 | 0, 1, 2, 3, 4, 5, 6, 8, 9 |
| 9 | 0, 1, 2, 5 |
- Step 4: Total 20 values, median is average of 10th and 11th values
- Step 5: 10th value = 82, 11th value = 83
- Step 6: Median = (82 + 83) ÷ 2 = 82.5
Answer: Stem-and-leaf plot created. Median = 82.5
- Sector angle = (value ÷ total) × 360°.
- Full circle = 360°.
- Pie chart: Circle divided into sectors; angle = (frequency/total) × 360°
- Line graph: Points connected by lines; shows trends over time
- Stem-and-leaf plot: Organizes data showing all values; stem = leading digit(s), leaf = last digit
- Pie charts work best with 5-7 categories (too many slices become hard to read)
- Line graphs are ideal for time series data
- Stem-and-leaf plots preserve original data while showing distribution
- Always sort leaves in ascending order for easier analysis
Probability
Theoretical probability is favourable outcomes ÷ total outcomes; experimental probability comes from actual trials. For a fair coin P(head) = ½, for a die P(any one number) = 1/6, and probabilities range from 0 to 1.
- Theoretical P = favourable ÷ total.
- Probability ranges from 0 (impossible) to 1 (certain).