Frequency distribution is the systematic arrangement of raw data into classes with their corresponding frequencies.
## Core Concept
A frequency distribution table groups data into mutually exclusive classes and records how many observations fall into each class. This converts raw, unorganized data into a summarized form suitable for analysis.
Key components: - Class interval: Range of values (e.g., 10–20, 20–30) - Class frequency: Count of observations in each class - Class width: Difference between upper and lower class limits - Cumulative frequency: Running total of frequencies
Types: - Ungrouped frequency distribution: Individual values with their frequencies (used when data range is small) - Grouped frequency distribution: Data organized into class intervals (used when data range is large)
## Formula / Rule
For grouped data:
| Class | Frequency (f) | Cumulative Frequency | |-------|---------------|----------------------| | 10–20 | 5 | 5 | | 20–30 | 8 | 13 | | 30–40 | 12 | 25 | | 40–50 | 5 | 30 |
Key calculations:
- Number of classes: K = 1 + 3.322 log(n) [Sturges' Rule]
- Class width: h = (Range) / K, where Range = Max value – Min value
- Relative frequency: f/N (where N = total frequency)
- Frequency density: f/h (used in histograms with unequal class widths)
## Common Exam Applications
1. Construction of frequency distribution: Given: Raw data 12, 15, 18, 12, 20, 18, 15, 22, 20, 18
Classes: 10–15, 15–20, 20–25
| Class | Tally | Frequency | |-------|-------|-----------| | 10–15 | III | 3 | | 15–20 | IIII | 4 | | 20–25 | III | 3 |
2. Cumulative frequency analysis: Used to determine median, quartiles, and percentiles.
3. Class boundaries vs class limits: - Class limits: 10–20 (inclusive values written in the table) - Class boundaries: 9.5–20.5 (actual boundaries for continuous data, preventing gaps)
4. Histogram interpretation: Requires frequency density when class widths are unequal.
## Common Mistakes
- Overlapping classes: Never use 10–20, 20–30. Use 10–< 20, 20–< 30 or 9.5–19.5, 19.5–29.5
- Missing "< less than": For continuous data, class boundaries must be clearly stated (e.g., "inclusive of lower, exclusive of upper")
- Confusing frequency with cumulative frequency: Frequency = count in that class only; Cumulative = running total
- Wrong class width: Ensure all classes have equal width (unless histogram requires frequency density adjustment)
- Ignoring class boundaries: Raw data 20 can fall in class 20–30 or 10–20; use boundaries to avoid ambiguity
- Frequency density error: When class widths differ, plot frequency/width on y-axis, not raw frequency
Exam tip: When constructing frequency distributions, always state whether classes are "inclusive–exclusive" (e.g., 10 ≤ x < 20) or use decimal boundaries (9.5–19.5) for continuous data. This demonstrates precision and prevents marks loss.