Grouped Frequency Distribution
When data has many distinct values spread over a wide range, listing individual frequencies becomes impractical. Instead, we group the data into class intervals (also called groups or bins) and count how many values fall into each interval. This is called a grouped frequency distribution.
For example, if the marks of 40 students range from 12 to 98, creating a frequency table for every single mark would have dozens of rows. Grouping into intervals like 0–10, 10–20, 20–30, etc., makes the data manageable and reveals patterns.
Grouped frequency distribution is studied in Class 8 NCERT Maths (Data Handling chapter) and extended in Class 9 with mean, median, and mode of grouped data.
Understanding how to organise raw data into grouped tables is essential for constructing histograms, frequency polygons, and for computing statistical measures.
What is Grouped Frequency Distribution?
Definition: A grouped frequency distribution is a table that organises data into class intervals and shows the frequency (count) for each interval.
Key terms:
- Class interval (class) — a range of values (e.g., 10–20).
- Lower class limit — the smallest value in the interval (e.g., 10 in 10–20).
- Upper class limit — the largest value in the interval (e.g., 20 in 10–20).
- Class width (class size) — upper limit − lower limit (e.g., 20 − 10 = 10).
- Frequency — the number of data values in that interval.
- Class mark (mid-value) — (lower limit + upper limit) / 2.
- Tally marks — a counting method using strokes (||||) to record frequency.
Types of class intervals:
- Exclusive (continuous): 10–20, 20–30. The upper limit of one class = lower limit of next. A value of 20 goes in 20–30.
- Inclusive (discontinuous): 10–19, 20–29. Both limits are included. There is a gap between classes.
Grouped Frequency Distribution Formula
Class width:
Class width = Upper limit − Lower limit
Class mark (mid-point):
Class mark = (Lower limit + Upper limit) / 2
Number of classes (approximate):
Number of classes = Range / Class width
Where Range = Maximum value − Minimum value.
Total frequency:
- Sum of all frequencies = total number of data values.
Converting inclusive to exclusive:
- Adjustment factor = (lower limit of a class − upper limit of previous class) / 2
- New lower limit = old lower limit − adjustment
- New upper limit = old upper limit + adjustment
Derivation and Proof
Why do we group data?
Step 1: Raw data: 23, 45, 67, 12, 89, 34, 56, 78, 41, 63, 29, 50, 71, 38, 85, 17, 52, 44, 60, 75.
Step 2: With 20 values ranging from 12 to 89, an ungrouped table would have up to 20 rows (one per unique value).
Step 3: Grouping into intervals of width 20:
- 0–20: 12, 17 → frequency = 2
- 20–40: 23, 29, 34, 38 → frequency = 4
- 40–60: 45, 41, 50, 52, 44, 56 → frequency = 6
- 60–80: 67, 63, 71, 60, 75, 78 → frequency = 6
- 80–100: 89, 85 → frequency = 2
Step 4: The grouped table has only 5 rows instead of 20, and it clearly shows that most values are in the 40–80 range.
Grouping sacrifices individual data detail but gains clarity and pattern recognition.
Types and Properties
Problems on grouped frequency distribution can be classified as follows:
1. Constructing a grouped frequency table:
- Given raw data, create class intervals and count frequencies using tally marks.
2. Finding class mark / mid-point:
- Calculate (lower limit + upper limit) / 2 for each class.
3. Finding the class width:
- Determine the width of the intervals.
4. Converting inclusive to exclusive intervals:
- Adjust limits so classes become continuous.
5. Drawing a histogram from the table:
- Use the grouped frequency table to draw a histogram.
6. Finding cumulative frequency:
- Running total of frequencies, used for “less than” or “more than” ogive curves.
7. Interpreting grouped data:
- Which class has the highest frequency? What proportion of data lies below a certain value?
Solved Examples
Example 1: Example 1: Constructing a grouped frequency table
Problem: The marks of 20 students are: 35, 42, 28, 55, 61, 48, 73, 39, 50, 66, 44, 31, 57, 82, 47, 25, 63, 52, 70, 41. Prepare a grouped frequency table with class intervals of width 10, starting from 20.
Solution:
- 20–30: 28, 25 → 2
- 30–40: 35, 39, 31 → 3
- 40–50: 42, 48, 44, 47, 41 → 5
- 50–60: 55, 50, 57, 52 → 4
- 60–70: 61, 66, 63 → 3
- 70–80: 73, 70 → 2
- 80–90: 82 → 1
Total frequency: 2 + 3 + 5 + 4 + 3 + 2 + 1 = 20 ✓
Example 2: Example 2: Finding class marks
Problem: Find the class marks for: 0–10, 10–20, 20–30, 30–40, 40–50.
Solution:
- 0–10: (0 + 10)/2 = 5
- 10–20: (10 + 20)/2 = 15
- 20–30: (20 + 30)/2 = 25
- 30–40: (30 + 40)/2 = 35
- 40–50: (40 + 50)/2 = 45
Example 3: Example 3: Converting inclusive to exclusive
Problem: Convert these inclusive class intervals to exclusive: 1–10, 11–20, 21–30, 31–40.
Solution:
- Adjustment = (11 − 10) / 2 = 0.5
- Subtract 0.5 from lower limits, add 0.5 to upper limits:
- 1–10 → 0.5–10.5
- 11–20 → 10.5–20.5
- 21–30 → 20.5–30.5
- 31–40 → 30.5–40.5
Now the classes are continuous (exclusive).
Example 4: Example 4: Choosing appropriate class width
Problem: Data ranges from 15 to 95. How many classes of width 10 are needed?
Solution:
- Range = 95 − 15 = 80
- Start from 10 (nearest round number below 15).
- End at 100 (nearest round number above 95).
- Number of classes = (100 − 10) / 10 = 9 classes
- Classes: 10–20, 20–30, ..., 90–100
Example 5: Example 5: Cumulative frequency
Problem: From the table: 0–10 (3), 10–20 (7), 20–30 (12), 30–40 (8), 40–50 (5). Find the cumulative frequencies.
Solution:
- Less than 10: 3
- Less than 20: 3 + 7 = 10
- Less than 30: 10 + 12 = 22
- Less than 40: 22 + 8 = 30
- Less than 50: 30 + 5 = 35
Total students = 35.
Example 6: Example 6: Identifying the modal class
Problem: Marks: 0–20 (5), 20–40 (12), 40–60 (18), 60–80 (10), 80–100 (5). Which is the modal class?
Solution:
- The modal class is the class with the highest frequency.
- Highest frequency = 18 (class 40–60).
Answer: The modal class is 40–60.
Example 7: Example 7: Finding the class containing a specific data value
Problem: In an exclusive distribution with classes 0–10, 10–20, 20–30, ..., which class does 20 belong to?
Solution:
- In exclusive (continuous) intervals, the lower limit is included and the upper limit is excluded.
- 20 belongs to class 20–30 (not 10–20).
Example 8: Example 8: Proportion of data in a class
Problem: Out of 50 students, 15 scored in the 60–80 range. What percentage of students is this?
Solution:
- Percentage = (15/50) × 100 = 30%
Answer: 30% of students scored between 60 and 80.
Example 9: Example 9: Constructing a table with tally marks
Problem: Heights (in cm) of 15 plants: 24, 31, 18, 42, 27, 35, 22, 38, 45, 29, 33, 20, 41, 26, 36. Group into classes of width 10 starting from 10.
Solution:
- 10–20: 18 → Tally: | → Frequency: 1
- 20–30: 24, 27, 22, 29, 20, 26 → Tally: |||| | → Frequency: 6
- 30–40: 31, 35, 38, 33, 36 → Tally: |||| → Frequency: 5
- 40–50: 42, 45, 41 → Tally: ||| → Frequency: 3
Total = 1 + 6 + 5 + 3 = 15 ✓
Example 10: Example 10: Comparing two distributions
Problem: Section A marks: 0–20 (2), 20–40 (8), 40–60 (15), 60–80 (10), 80–100 (5). Section B marks: 0–20 (5), 20–40 (10), 40–60 (10), 60–80 (8), 80–100 (7). Which section performed better overall?
Solution:
- Section A: students scoring 60+ = 10 + 5 = 15 out of 40 = 37.5%
- Section B: students scoring 60+ = 8 + 7 = 15 out of 40 = 37.5%
- Both have same number scoring 60+, but Section A has more in 40–60 (15 vs 10).
Answer: Both sections have equal high scorers. Section A has a stronger middle range.
Real-World Applications
Statistics: Grouped frequency tables are the starting point for calculating mean, median, and mode of large datasets. All statistical analysis begins with organising data.
Education: Schools use grouped distributions to analyse exam results — how many students scored in each range, which range has the most students, and overall performance patterns.
Census and Surveys: Government data on income, age, population, etc., is always presented as grouped distributions because the raw data involves millions of entries.
Quality Control: Factories group measurements (weights, lengths, etc.) into intervals to check if products meet specifications.
Healthcare: Patient data (blood pressure, BMI, age groups) is grouped for analysis and reporting.
Visualisation: Histograms, frequency polygons, and ogive curves are all drawn from grouped frequency tables.
Key Points to Remember
- Grouped frequency distribution organises data into class intervals.
- Class width = upper limit − lower limit.
- Class mark = (lower limit + upper limit) / 2.
- Range = maximum value − minimum value.
- Number of classes ≈ Range / Class width.
- Exclusive intervals: upper limit not included (10–20 means 10 ≤ x < 20).
- Inclusive intervals: both limits included (10–19 means 10 ≤ x ≤ 19).
- Total frequency = total number of data values.
- The modal class has the highest frequency.
- Cumulative frequency gives a running total of frequencies, used for ogive curves and finding median.
Practice Problems
- The weights (in kg) of 25 students are: 42, 38, 55, 47, 50, 60, 43, 52, 48, 65, 41, 53, 58, 44, 51, 62, 39, 56, 49, 63, 45, 57, 40, 54, 46. Prepare a grouped frequency table with class width 5, starting from 35.
- Find the class marks for: 5–15, 15–25, 25–35, 35–45, 45–55.
- Convert the inclusive intervals 11–20, 21–30, 31–40, 41–50 to exclusive form.
- Data ranges from 3 to 97. What class width should you use to get about 10 classes?
- From a grouped table: 0–20 (8), 20–40 (15), 40–60 (20), 60–80 (12), 80–100 (5), find the cumulative frequency for each class.
- Identify the modal class from: 10–20 (6), 20–30 (14), 30–40 (22), 40–50 (9), 50–60 (4).
- 40 students scored marks: 0–25 (4), 25–50 (10), 50–75 (16), 75–100 (10). What percentage scored above 50?
- Explain why grouped frequency is preferred over ungrouped frequency for large datasets.
Frequently Asked Questions
Q1. What is a grouped frequency distribution?
It is a table that organises data into class intervals (groups) and shows how many data values (frequency) fall into each interval.
Q2. What is a class interval?
A class interval is a range of values. For example, 10–20 is a class interval containing all values from 10 up to (but not including) 20.
Q3. What is the difference between exclusive and inclusive class intervals?
In exclusive intervals (10–20), the upper limit is NOT included (values up to 19.99...). In inclusive intervals (10–19), both limits are included.
Q4. How do you find the class width?
Class width = upper limit − lower limit. For example, for the class 20–30, width = 30 − 20 = 10.
Q5. What is the class mark?
The class mark (mid-point) = (lower limit + upper limit) / 2. For example, class mark of 20–30 = (20+30)/2 = 25.
Q6. What is cumulative frequency?
Cumulative frequency is the running total of frequencies from the first class to the current class. It tells how many values are less than the upper limit of the current class.
Q7. What is the modal class?
The modal class is the class interval with the highest frequency. It indicates the range where data is most concentrated.
Q8. Why do we convert inclusive to exclusive intervals?
Exclusive intervals are needed for drawing histograms (bars must touch). Converting ensures there are no gaps between classes.
Q9. How do you decide the class width?
Divide the range by the desired number of classes (usually 5–10). Round to a convenient number. For example, range = 80, desired classes = 8, width = 10.
Q10. Can grouped frequency distribution show exact individual data?
No. Grouping hides individual values. You can tell that 15 students scored between 40–60, but not their exact marks. This trade-off gives clearer overall patterns.
