Cumulative Frequency Distribution
A cumulative frequency distribution is a table that shows the running total of frequencies up to each class interval. It tells us how many observations fall below (or above) a particular value.
Cumulative frequency is essential for finding the median of grouped data and for drawing the ogive (cumulative frequency curve). The NCERT Class 10 syllabus covers two types: less than and more than cumulative frequency distributions.
This topic builds on grouped frequency distributions and is a prerequisite for understanding ogives and graphical determination of the median.
What is Cumulative Frequency Distribution?
Definition: The cumulative frequency of a class interval is the total number of observations that fall in that class and all preceding classes.
Two Types:
- Less Than Cumulative Frequency: Running total from the lowest class upward. The cumulative frequency of a class tells how many observations are less than the upper limit of that class.
- More Than Cumulative Frequency: Running total from the highest class downward. It tells how many observations are greater than or equal to the lower limit of that class.
Key Terms:
- Frequency (fᵢ) — count of observations in a particular class interval
- Cumulative Frequency (cf) — progressive total of frequencies
- N = Σfᵢ — total frequency (sum of all frequencies)
- The last cumulative frequency in a 'less than' table always equals N.
- The first cumulative frequency in a 'more than' table always equals N.
Cumulative Frequency Distribution Formula
Less Than Cumulative Frequency:
CF of class i = f₁ + f₂ + f₃ + ... + fᵢ
(Add frequencies from the first class up to class i)
More Than Cumulative Frequency:
CF of class i = fᵢ + fᵢ₊₁ + ... + fₙ = N − (f₁ + f₂ + ... + fᵢ₋₁)
(Total frequency minus the sum of all frequencies before class i)
Recovering Frequency from Cumulative Frequency:
fᵢ = CFᵢ − CFᵢ₋₁
(Frequency of any class = its cumulative frequency minus the cumulative frequency of the previous class)
Derivation and Proof
How to Construct a Less Than Cumulative Frequency Table:
- List all class intervals in order from lowest to highest.
- For the first class, cumulative frequency = its own frequency.
- For each subsequent class, add its frequency to the cumulative frequency of the previous class.
- The last cumulative frequency must equal the total frequency N.
How to Construct a More Than Cumulative Frequency Table:
- List the lower limits of each class interval.
- The first entry = N (total frequency).
- For each subsequent class, subtract the frequency of the previous class from the previous cumulative frequency.
- The last cumulative frequency equals the frequency of the last class.
Converting Between Types:
- From Less Than to More Than: More than CF of class i = N − Less than CF of class (i − 1)
- From More Than to Less Than: Less than CF of class i = N − More than CF of class (i + 1)
- Individual frequency: fᵢ = (Less than CF of class i) − (Less than CF of class i − 1)
Types and Properties
Comparison of Less Than and More Than Cumulative Frequency:
| Feature | Less Than CF | More Than CF |
|---|---|---|
| Direction | Top to bottom (ascending) | Bottom to top (descending from N) |
| Boundary used | Upper class boundary | Lower class boundary |
| First entry | = f₁ (frequency of first class) | = N (total frequency) |
| Last entry | = N | = fₙ (frequency of last class) |
| Ogive type | Less than ogive (rising curve) | More than ogive (falling curve) |
| Used for | Finding median, percentiles | Finding median, survival analysis |
Uses of Cumulative Frequency:
- Finding the median of grouped data (identify the median class)
- Drawing ogives (cumulative frequency curves)
- Finding the median graphically from the intersection of two ogives
- Determining quartiles, percentiles, and deciles
- Comparing distributions graphically
Solved Examples
Example 1: Constructing Less Than Cumulative Frequency Table
Problem: Construct the less than cumulative frequency table for the following data.
| Marks | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
| Students | 5 | 8 | 15 | 12 | 10 |
Solution:
| Class | Frequency (f) | Cumulative Frequency (cf) |
|---|---|---|
| 0–10 | 5 | 5 |
| 10–20 | 8 | 5 + 8 = 13 |
| 20–30 | 15 | 13 + 15 = 28 |
| 30–40 | 12 | 28 + 12 = 40 |
| 40–50 | 10 | 40 + 10 = 50 |
Interpretation:
- 13 students scored less than 20
- 28 students scored less than 30
- 40 students scored less than 40
Example 2: Constructing More Than Cumulative Frequency Table
Problem: Using the same data, construct the more than cumulative frequency table.
| Marks | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
| Students | 5 | 8 | 15 | 12 | 10 |
Solution:
| More Than | Cumulative Frequency |
|---|---|
| More than or equal to 0 | 50 |
| More than or equal to 10 | 50 − 5 = 45 |
| More than or equal to 20 | 45 − 8 = 37 |
| More than or equal to 30 | 37 − 15 = 22 |
| More than or equal to 40 | 22 − 12 = 10 |
Interpretation:
- 45 students scored 10 or more
- 37 students scored 20 or more
- 22 students scored 30 or more
Example 3: Converting Less Than CF to Frequencies
Problem: The following less than cumulative frequency distribution is given. Find the individual frequencies.
| Marks less than | 20 | 40 | 60 | 80 | 100 |
|---|---|---|---|---|---|
| CF | 8 | 22 | 45 | 70 | 80 |
Solution:
Step 1: Frequency of 0–20 = 8 (first CF)
Step 2: Frequency of 20–40 = 22 − 8 = 14
Step 3: Frequency of 40–60 = 45 − 22 = 23
Step 4: Frequency of 60–80 = 70 − 45 = 25
Step 5: Frequency of 80–100 = 80 − 70 = 10
| Class | Frequency |
|---|---|
| 0–20 | 8 |
| 20–40 | 14 |
| 40–60 | 23 |
| 60–80 | 25 |
| 80–100 | 10 |
Verification: Total = 8 + 14 + 23 + 25 + 10 = 80 ✓
Example 4: Converting More Than CF to Frequencies
Problem: Convert the following more than CF distribution to a frequency table.
| Production ≥ | 50 | 60 | 70 | 80 | 90 |
|---|---|---|---|---|---|
| CF | 100 | 85 | 55 | 30 | 12 |
Solution:
Step 1: Frequency of 50–60 = 100 − 85 = 15
Step 2: Frequency of 60–70 = 85 − 55 = 30
Step 3: Frequency of 70–80 = 55 − 30 = 25
Step 4: Frequency of 80–90 = 30 − 12 = 18
Step 5: Frequency of 90–100 = 12 (last more-than CF)
Verification: 15 + 30 + 25 + 18 + 12 = 100 ✓
Example 5: Using CF to Find Median Class
Problem: From the cumulative frequency table, identify the median class.
| Class | f | cf |
|---|---|---|
| 0–10 | 4 | 4 |
| 10–20 | 9 | 13 |
| 20–30 | 16 | 29 |
| 30–40 | 11 | 40 |
| 40–50 | 5 | 45 |
Solution:
Step 1: N = 45, N/2 = 22.5
Step 2: Look for the first cf that equals or exceeds 22.5.
Step 3: cf = 4 (no), cf = 13 (no), cf = 29 (yes!) → Median class = 20–30
Answer: The median class is 20–30 because its cumulative frequency (29) is the first to exceed 22.5.
Example 6: Both CF Types — Weight Distribution
Problem: The weights of 60 apples are given. Construct both less than and more than cumulative frequency tables.
| Weight (g) | 80–100 | 100–120 | 120–140 | 140–160 | 160–180 |
|---|---|---|---|---|---|
| Apples | 7 | 12 | 20 | 14 | 7 |
Less Than CF:
| Less than | 100 | 120 | 140 | 160 | 180 |
|---|---|---|---|---|---|
| CF | 7 | 19 | 39 | 53 | 60 |
More Than CF:
| More than or equal to | 80 | 100 | 120 | 140 | 160 |
|---|---|---|---|---|---|
| CF | 60 | 53 | 41 | 21 | 7 |
Example 7: Finding Percentile from CF
Problem: From the following data, determine how many students scored below 60 and what percentage of students scored 40 or above.
| Marks | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 |
|---|---|---|---|---|---|
| Students | 10 | 15 | 20 | 30 | 25 |
Solution:
Less Than CF: 10, 25, 45, 75, 100
Step 1: Students scoring below 60 = CF of class 40–60 = 45
Step 2: Students scoring 40 or above = N − CF of class 20–40 = 100 − 25 = 75
Step 3: Percentage scoring 40 or above = (75/100) × 100 = 75%
Example 8: Constructing CF from Unequal Class Widths
Problem: Construct the cumulative frequency table for the following data with unequal class widths.
| Age (years) | 0–5 | 5–10 | 10–20 | 20–40 | 40–60 |
|---|---|---|---|---|---|
| Persons | 8 | 12 | 15 | 20 | 5 |
Solution:
| Class | f | cf (Less than) |
|---|---|---|
| 0–5 | 8 | 8 |
| 5–10 | 12 | 20 |
| 10–20 | 15 | 35 |
| 20–40 | 20 | 55 |
| 40–60 | 5 | 60 |
Note: Cumulative frequency works the same way regardless of whether class widths are equal or unequal. We simply add frequencies progressively.
Real-World Applications
Real-life applications of cumulative frequency:
- Examinations: To determine the number of students scoring below a certain mark (percentile ranks).
- Quality Control: To find what percentage of products weigh below a certain threshold.
- Demographics: Census data uses cumulative frequency to report what fraction of the population is below a certain age.
- Finance: To determine the percentage of employees earning below or above a salary threshold.
- Healthcare: Cumulative frequency of recovery times helps determine what percentage of patients recover within a given period.
- Environmental Science: Rainfall data presented as cumulative frequency helps in flood forecasting and water resource management.
Key Points to Remember
- Cumulative frequency is the running total of frequencies from the beginning (or end) of a distribution.
- Less than CF: Add frequencies from top to bottom. Used with upper class boundaries.
- More than CF: Subtract frequencies from total, going top to bottom. Used with lower class boundaries.
- The last entry in a less than CF table = N (total frequency).
- The first entry in a more than CF table = N.
- Individual frequency = difference of consecutive cumulative frequencies: fᵢ = CFᵢ − CFᵢ₋₁.
- Cumulative frequency is used to identify the median class (the class where CF first ≥ N/2).
- Ogives (cumulative frequency curves) are drawn using CF tables: less than ogive is an ascending curve, more than ogive is a descending curve.
- The x-coordinate of the intersection of the two ogives gives the median.
- CF tables work for both equal and unequal class widths.
Practice Problems
- Construct both less than and more than cumulative frequency tables: Class: 0–5, 5–10, 10–15, 15–20, 20–25; Frequency: 3, 7, 12, 8, 5.
- The less than CF distribution is: Less than 30: 10, Less than 40: 22, Less than 50: 40, Less than 60: 55, Less than 70: 60. Find the individual frequencies.
- From the following data, find how many observations are (a) less than 45, (b) 25 or more: Class: 5–15, 15–25, 25–35, 35–45, 45–55; Frequency: 4, 8, 14, 10, 6.
- Convert the following more than CF to a frequency table: More than 0: 80, More than 10: 68, More than 20: 50, More than 30: 30, More than 40: 12.
- Construct a cumulative frequency table and identify the median class: Class: 100–150, 150–200, 200–250, 250–300, 300–350; Frequency: 6, 12, 18, 10, 4.
- The marks of 90 students are given in cumulative form: Less than 10: 5, Less than 20: 15, Less than 30: 30, Less than 40: 55, Less than 50: 78, Less than 60: 90. What percentage of students scored (a) below 30? (b) 30 or above?
Frequently Asked Questions
Q1. What is cumulative frequency?
Cumulative frequency is the running total of frequencies. For less than type, it counts how many observations fall below the upper boundary of each class. For more than type, it counts how many observations fall at or above the lower boundary of each class.
Q2. What is the difference between less than and more than cumulative frequency?
Less than CF adds frequencies from the first class downward (ascending total) and uses upper class boundaries. More than CF starts from the total and subtracts frequencies going downward (descending total) and uses lower class boundaries.
Q3. How do you find individual frequencies from cumulative frequency?
Subtract consecutive cumulative frequencies: fᵢ = CFᵢ − CFᵢ₋₁. For the first class, the frequency equals its cumulative frequency.
Q4. What is the use of cumulative frequency in finding the median?
To find the median of grouped data, you need the cumulative frequency to identify the median class — the class whose CF first equals or exceeds N/2. The CF of the class before the median class is used in the median formula.
Q5. What is an ogive?
An ogive is a cumulative frequency curve. A less than ogive plots (upper boundary, less than CF) and rises upward. A more than ogive plots (lower boundary, more than CF) and falls downward. The intersection of both ogives gives the median on the x-axis.
Q6. Can cumulative frequency decrease?
No. Less than cumulative frequency always increases or stays the same (it can never decrease because we are adding non-negative frequencies). Similarly, more than CF always decreases or stays the same.
Q7. What if a class has zero frequency?
If a class has zero frequency, the cumulative frequency of that class equals the cumulative frequency of the previous class. The CF value stays unchanged for that class.
Q8. Is cumulative frequency the same for equal and unequal class widths?
Yes. Cumulative frequency is computed the same way regardless of class width — it is simply the progressive sum of frequencies.
Related Topics
- Ogive (Cumulative Frequency Curve)
- Median of Grouped Data
- Grouped Frequency Distribution
- Mean of Grouped Data
- Statistics - Collection and Presentation
- Frequency Distribution Table
- Histogram of Grouped Data
- Frequency Polygon
- Mean of Ungrouped Data
- Median of Ungrouped Data
- Mode of Grouped Data
- Empirical Relationship Between Mean, Median, Mode
- Mean by Assumed Mean Method
- Mean by Step-Deviation Method
