Frequency Distribution Table
A frequency distribution table is a systematic way of organising raw data by listing each value or class interval along with the number of times it occurs (frequency).
When data sets are large, listing individual observations is impractical. Grouping data into class intervals and recording their frequencies makes it easier to understand patterns, draw graphs, and compute statistical measures.
In NCERT Class 9, students learn to construct both ungrouped and grouped frequency distribution tables, understand class marks, class boundaries, and cumulative frequency.
What is Frequency Distribution Table?
Definition: A frequency distribution table is a table that shows how often each value (or range of values) appears in a data set.
Key Terms:
- Frequency: The number of times a particular value or class interval occurs.
- Class interval: A range of values into which data is grouped (e.g., 10–20, 20–30).
- Lower class limit: The smallest value in a class interval.
- Upper class limit: The largest value in a class interval.
- Class width (size): Upper limit − Lower limit.
- Class mark (mid-value): (Upper limit + Lower limit) / 2.
- Class boundary: Adjusted limits for continuous classes.
Frequency Distribution Table Formula
Key Formulas:
1. Class Width:
Class Width = Upper Limit − Lower Limit
2. Class Mark:
Class Mark = (Upper Limit + Lower Limit) / 2
3. Number of Class Intervals:
- Approximate number = Range / Desired class width
- Range = Maximum value − Minimum value
4. Adjustment for Discontinuous Classes:
- Adjustment factor = (Lower limit of next class − Upper limit of current class) / 2
- New lower boundary = Lower limit − adjustment
- New upper boundary = Upper limit + adjustment
Derivation and Proof
Steps to Construct a Grouped Frequency Distribution Table:
- Find the range: Range = Maximum value − Minimum value.
- Decide the number of classes: Usually 5 to 10 classes. Choose a convenient class width.
- Determine class intervals: Start from a value at or below the minimum. Each class has equal width.
- Tally the data: Go through each observation and place a tally mark in the corresponding class.
- Count frequencies: Convert tally marks to numerical frequencies.
- Verify: Sum of all frequencies must equal the total number of observations.
Continuous vs Discontinuous Classes:
- Continuous (exclusive): 10–20, 20–30, 30–40. The upper limit of one class = lower limit of the next. A value of 20 goes in 20–30.
- Discontinuous (inclusive): 10–19, 20–29, 30–39. There is a gap between classes. Must be converted to continuous before drawing histograms.
Types and Properties
Types of Frequency Distribution:
1. Ungrouped Frequency Distribution
- Each distinct value is listed separately with its frequency.
- Suitable for small data sets with few distinct values.
2. Grouped Frequency Distribution
- Data is divided into class intervals (groups).
- Suitable for large data sets with a wide range of values.
3. Cumulative Frequency Distribution
- Less-than type: Shows the number of observations less than the upper boundary of each class.
- More-than type: Shows the number of observations greater than or equal to the lower boundary.
4. Relative Frequency Distribution
- Frequency of each class divided by total frequency.
- Expressed as a fraction or percentage.
Solved Examples
Example 1: Example 1: Ungrouped frequency table
Problem: The number of books read by 15 students in a month: 2, 3, 1, 2, 4, 3, 2, 1, 3, 2, 5, 3, 2, 1, 4. Prepare an ungrouped frequency table.
Solution:
- 1 → 3
- 2 → 5
- 3 → 4
- 4 → 2
- 5 → 1
- Total = 15 ✓
Answer: Most students read 2 books (frequency 5).
Example 2: Example 2: Grouped frequency table
Problem: The marks of 30 students (out of 100): 45, 52, 67, 38, 72, 58, 41, 85, 63, 49, 55, 77, 34, 60, 48, 70, 92, 53, 66, 42, 80, 57, 35, 74, 61, 46, 88, 50, 69, 43. Prepare a grouped table with class width 10.
Solution:
- 30 – 40: 38, 34, 35 → 3
- 40 – 50: 45, 41, 49, 48, 42, 46, 43 → 7
- 50 – 60: 52, 58, 55, 53, 57, 50 → 6
- 60 – 70: 67, 63, 60, 66, 61, 69 → 6
- 70 – 80: 72, 77, 70, 74 → 4
- 80 – 90: 85, 80, 88 → 3
- 90 – 100: 92 → 1
- Total = 30 ✓
Answer: The 40–50 class has the highest frequency (7 students).
Example 3: Example 3: Find class marks
Problem: For the class intervals 0–10, 10–20, 20–30, 30–40, 40–50, find the class mark of each.
Solution:
- 0–10: Class mark = (0 + 10)/2 = 5
- 10–20: Class mark = (10 + 20)/2 = 15
- 20–30: Class mark = (20 + 30)/2 = 25
- 30–40: Class mark = (30 + 40)/2 = 35
- 40–50: Class mark = (40 + 50)/2 = 45
Answer: Class marks are 5, 15, 25, 35, 45.
Example 4: Example 4: Convert discontinuous to continuous
Problem: Convert the classes 1–10, 11–20, 21–30, 31–40 to continuous form.
Solution:
- Gap = 11 − 10 = 1. Adjustment = 0.5.
- 0.5 – 10.5, 10.5 – 20.5, 20.5 – 30.5, 30.5 – 40.5
Answer: Continuous classes: 0.5–10.5, 10.5–20.5, 20.5–30.5, 30.5–40.5.
Example 5: Example 5: Cumulative frequency table
Problem: Prepare a cumulative frequency table (less-than type) for: Class 0–10 (f=5), 10–20 (f=8), 20–30 (f=12), 30–40 (f=10), 40–50 (f=5).
Solution:
- Less than 10: 5
- Less than 20: 5 + 8 = 13
- Less than 30: 13 + 12 = 25
- Less than 40: 25 + 10 = 35
- Less than 50: 35 + 5 = 40
Answer: Total observations = 40.
Example 6: Example 6: Find the class interval given range
Problem: Data ranges from 12 to 82. If we want 7 class intervals, what should the class width be?
Solution:
- Range = 82 − 12 = 70
- Class width = 70 / 7 = 10
- Classes: 10–20, 20–30, 30–40, 40–50, 50–60, 60–70, 70–80, 80–90
Answer: Class width = 10.
Example 7: Example 7: Relative frequency
Problem: In a survey of 50 families, the number of children per family had frequencies: 0 children (8), 1 child (15), 2 children (18), 3 children (7), 4 children (2). Find the relative frequency for each.
Solution:
- 0: 8/50 = 0.16 (16%)
- 1: 15/50 = 0.30 (30%)
- 2: 18/50 = 0.36 (36%)
- 3: 7/50 = 0.14 (14%)
- 4: 2/50 = 0.04 (4%)
- Total = 1.00 (100%) ✓
Answer: The most common is 2 children (36%).
Example 8: Example 8: Identifying errors in a frequency table
Problem: A student prepared a frequency table for 25 observations with classes 0–10 (4), 10–20 (6), 20–30 (8), 30–40 (5). Is this correct?
Solution:
- Total frequency = 4 + 6 + 8 + 5 = 23
- But the data has 25 observations.
- 23 ≠ 25 — the table is incorrect.
Answer: The table has an error. Two observations are missing or miscounted.
Real-World Applications
Applications:
- Education: Organising exam marks to identify performance patterns.
- Healthcare: Recording patient data in frequency tables for analysis.
- Business: Sales data organised by price ranges for trend analysis.
- Government: Census data organised into age groups and income brackets.
- Sports: Player performance statistics organised by scoring intervals.
Key Points to Remember
- A frequency distribution table organises data by listing values or class intervals with their frequencies.
- Ungrouped tables list individual values; grouped tables use class intervals.
- Class width = Upper limit − Lower limit.
- Class mark = (Upper limit + Lower limit) / 2.
- Sum of all frequencies must equal the total number of observations.
- Discontinuous classes must be converted to continuous before drawing histograms.
- Cumulative frequency is the running total of frequencies.
- Choose 5 to 10 class intervals for most data sets.
- Start the first class at or below the minimum value.
- This is covered in NCERT Class 9, Chapter 14 (Statistics).
Practice Problems
- The daily wages (in Rs) of 20 workers are: 120, 150, 180, 120, 200, 150, 180, 120, 200, 150, 180, 200, 120, 150, 180, 200, 150, 120, 180, 200. Prepare an ungrouped frequency table.
- The heights (in cm) of 30 students range from 130 to 175. Prepare a grouped frequency table with class width 10.
- Convert the classes 11-20, 21-30, 31-40, 41-50, 51-60 into continuous form and find the class mark of each.
- For a grouped table with classes 0-5, 5-10, 10-15, 15-20 and frequencies 4, 7, 10, 9, prepare a less-than cumulative frequency table.
- Data values range from 2 to 98. If 10 class intervals are desired, what class width should be used?
- Find the error: A frequency table for 40 observations shows classes 0-10 (5), 10-20 (12), 20-30 (14), 30-40 (7). Is it correct?
Frequently Asked Questions
Q1. What is a frequency distribution table?
A frequency distribution table organises raw data by listing each distinct value or class interval alongside the number of times it occurs (its frequency).
Q2. What is the difference between grouped and ungrouped frequency distribution?
In ungrouped distribution, each distinct value is listed separately. In grouped distribution, data is organised into class intervals (ranges), each containing multiple values.
Q3. What is class mark?
The class mark is the midpoint of a class interval: (upper limit + lower limit) / 2. It represents the central value of the class.
Q4. Why must we convert discontinuous classes to continuous?
Histograms require continuous classes (no gaps between bars). Converting ensures the bars touch each other, accurately representing the data.
Q5. How do you decide the class width?
Divide the range by the desired number of classes (usually 5 to 10). Round up to a convenient number.
Q6. What is cumulative frequency?
Cumulative frequency is the running total of frequencies up to a certain class. The less-than type gives the count of observations below each upper boundary.
Q7. Can a value fall exactly on a class boundary?
In continuous (exclusive) classes like 10-20, 20-30, a value of 20 is included in the class 20-30 (upper limit excluded, lower limit included).
Q8. Is this topic in the CBSE Class 9 syllabus?
Yes. Frequency distribution tables are a core part of Chapter 14 (Statistics) in NCERT Class 9 Mathematics.
Related Topics
- Statistics - Collection and Presentation
- Histogram of Grouped Data
- Frequency Polygon
- Cumulative Frequency Distribution
- Mean of Ungrouped Data
- Median of Ungrouped Data
- Mean of Grouped Data
- Median of Grouped Data
- Mode of Grouped Data
- Ogive (Cumulative Frequency Curve)
- Empirical Relationship Between Mean, Median, Mode
- Mean by Assumed Mean Method
- Mean by Step-Deviation Method
- Statistics in Real Life
