Ogive (Cumulative Frequency Curve)
An ogive (pronounced oh-jive) is the graphical representation of a cumulative frequency distribution. It is also called a cumulative frequency curve. This topic is covered in Class 10 CBSE Chapter 14 (Statistics).
There are two types of ogives — the less than ogive and the more than ogive. When both are drawn on the same axes, their point of intersection gives the median of the data.
Ogives are useful for finding the median graphically, especially when the exact formula calculation is complex. They also help determine how many observations lie below or above a particular value.
What is Ogive - Cumulative Frequency Curve, Drawing & Finding Median Graphically?
Definition: An ogive is a cumulative frequency curve plotted on a graph. It shows the running total of frequencies as we move through successive class intervals.
Types:
- Less than ogive — plots cumulative frequency against the upper class boundary. The curve rises from left to right.
- More than ogive — plots cumulative frequency against the lower class boundary. The curve falls from left to right.
Key Terms:
- Cumulative frequency (cf) — the running total of all frequencies up to (or from) a given class
- Less than cf — total frequency of all classes up to and including the current class
- More than cf — total frequency of all classes from the current class onward
- Upper class boundary — the upper limit of a class interval
- Lower class boundary — the lower limit of a class interval
Ogive (Cumulative Frequency Curve) Formula
Steps to Draw a Less Than Ogive:
- Construct a "less than" cumulative frequency table.
- Plot points: (upper class boundary, cumulative frequency).
- Join the points with a smooth freehand curve.
- The first point starts at (lower boundary of first class, 0).
Steps to Draw a More Than Ogive:
- Construct a "more than" cumulative frequency table.
- Plot points: (lower class boundary, cumulative frequency).
- Join the points with a smooth freehand curve.
- The last point ends at (upper boundary of last class, 0).
Finding Median from Ogive:
Method 1: From a single ogive — locate N/2 on the y-axis, draw horizontal line to the curve, drop vertical to x-axis. That x-value is the median.
Method 2: Draw both ogives. Their intersection point's x-coordinate = Median.
Derivation and Proof
Why the intersection of two ogives gives the median:
- The less than ogive at point x gives the number of observations ≤ x.
- The more than ogive at point x gives the number of observations ≥ x.
- At the median value M: the number of observations ≤ M = N/2 and the number ≥ M = N/2.
- So at x = M: less than cf = N/2 and more than cf = N/2.
- The total at any point: less than cf + more than cf = N (but they overlap at the boundary).
- At the intersection, both curves give the same cf value, which occurs exactly at the median.
Constructing Cumulative Frequency Tables:
Example data:
| Class | Frequency | Less than cf | More than cf |
|---|---|---|---|
| 0–10 | 5 | 5 | 50 |
| 10–20 | 8 | 13 | 45 |
| 20–30 | 15 | 28 | 37 |
| 30–40 | 12 | 40 | 22 |
| 40–50 | 10 | 50 | 10 |
Types and Properties
Comparison of the Two Types of Ogives:
| Feature | Less Than Ogive | More Than Ogive |
|---|---|---|
| Plots | (upper boundary, less than cf) | (lower boundary, more than cf) |
| Shape | Rising curve (S-shaped) | Falling curve (inverted S) |
| Starts at | (lower boundary of first class, 0) | (lower boundary of first class, N) |
| Ends at | (upper boundary of last class, N) | (upper boundary of last class, 0) |
| Use | "How many values are ≤ x?" | "How many values are ≥ x?" |
Reading Information from Ogives:
- To find how many students scored less than 60: read the less than ogive at x = 60.
- To find how many students scored more than 40: read the more than ogive at x = 40.
- To find the median: locate N/2 on the y-axis and read the x-value from the curve.
- Quartiles Q₁ and Q₃ can be found at N/4 and 3N/4 on the y-axis.
Solved Examples
Example 1: Drawing a Less Than Ogive
Problem: Draw a less than ogive for the following data and find the median.
| Marks | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
| Students | 4 | 6 | 14 | 10 | 6 |
Solution:
Step 1: Cumulative frequency table:
| Less than | cf |
|---|---|
| 10 | 4 |
| 20 | 10 |
| 30 | 24 |
| 40 | 34 |
| 50 | 40 |
Step 2: Plot points: (10,4), (20,10), (30,24), (40,34), (50,40), starting from (0,0).
Step 3: Join with smooth curve.
Step 4: N/2 = 40/2 = 20. Draw horizontal line from y = 20 to the curve, then drop vertical to x-axis.
The x-value is approximately 26.4.
Answer: Median ≈ 26.4
Example 2: Drawing a More Than Ogive
Problem: Draw a more than ogive for the following data.
| Class | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 |
|---|---|---|---|---|---|
| Frequency | 8 | 12 | 20 | 6 | 4 |
Solution:
Step 1: More than cumulative frequency table:
| More than | cf |
|---|---|
| 0 | 50 |
| 20 | 42 |
| 40 | 30 |
| 60 | 10 |
| 80 | 4 |
Step 2: Plot points: (0,50), (20,42), (40,30), (60,10), (80,4).
Step 3: Join with a smooth falling curve.
Step 4: To find median from this ogive: N/2 = 25. Draw horizontal from y = 25 to curve, read x-value ≈ 45.
Answer: Median ≈ 45
Example 3: Finding Median from Two Ogives
Problem: Using the data below, draw both ogives and find the median from their intersection.
| Class | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 |
|---|---|---|---|---|---|
| Frequency | 5 | 8 | 12 | 10 | 5 |
Solution:
Less than cf: (20,5), (30,13), (40,25), (50,35), (60,40)
More than cf: (10,40), (20,35), (30,27), (40,15), (50,5)
Step: Draw both curves on the same axes. They intersect at approximately x = 36.
Verification by formula:
- N/2 = 20, Median class = 30–40
- Median = 30 + [(20 − 13)/12] × 10 = 30 + 5.83 = 35.83
Answer: Median ≈ 36 (graphical), 35.83 (formula)
Example 4: Less Than Ogive — Daily Wages
Problem: Draw a less than ogive for the daily wages of 50 workers.
| Wages (₹) | 100–200 | 200–300 | 300–400 | 400–500 | 500–600 |
|---|---|---|---|---|---|
| Workers | 6 | 10 | 18 | 12 | 4 |
Solution:
Cumulative frequency:
| Less than | cf |
|---|---|
| 200 | 6 |
| 300 | 16 |
| 400 | 34 |
| 500 | 46 |
| 600 | 50 |
Points to plot: (100,0), (200,6), (300,16), (400,34), (500,46), (600,50)
Median: N/2 = 25. From graph, median ≈ ₹350.
Answer: Median ≈ ₹350
Example 5: Finding Quartiles from Ogive
Problem: From the less than ogive of the data in Example 1, find Q₁ and Q₃.
Solution:
Data: N = 40
- Q₁: at N/4 = 10. From ogive: horizontal from y = 10 meets curve at x ≈ 20. So Q₁ ≈ 20.
- Q₃: at 3N/4 = 30. From ogive: horizontal from y = 30 meets curve at x ≈ 36. So Q₃ ≈ 36.
Interquartile range = Q₃ − Q₁ = 36 − 20 = 16
Answer: Q₁ ≈ 20, Q₃ ≈ 36, IQR ≈ 16
Example 6: Reading Values from a Given Ogive
Problem: A less than ogive for marks of 80 students passes through the points (20,8), (40,22), (60,52), (80,72), (100,80). (a) How many students scored less than 60? (b) How many scored 40 or more? (c) What is the median?
Solution:
(a) Less than 60: Read y-value at x = 60 → cf = 52 students
(b) 40 or more: Total − (less than 40) = 80 − 22 = 58 students
(c) Median: N/2 = 40. Read x-value at y = 40 from the curve → approximately 50.
Answer: (a) 52, (b) 58, (c) Median ≈ 50
Example 7: Constructing Both CF Tables
Problem: Construct both cumulative frequency tables for the following data.
| Class | 5–15 | 15–25 | 25–35 | 35–45 | 45–55 |
|---|---|---|---|---|---|
| f | 3 | 7 | 12 | 5 | 3 |
Solution:
Less than cf:
| Less than | cf |
|---|---|
| 15 | 3 |
| 25 | 10 |
| 35 | 22 |
| 45 | 27 |
| 55 | 30 |
More than cf:
| More than | cf |
|---|---|
| 5 | 30 |
| 15 | 27 |
| 25 | 20 |
| 35 | 8 |
| 45 | 3 |
Example 8: Number of Students Scoring Between Two Values
Problem: From the less than ogive of 60 students: at x = 30, cf = 18; at x = 50, cf = 45. How many students scored between 30 and 50?
Solution:
- Students scoring less than 50 = 45
- Students scoring less than 30 = 18
- Students scoring between 30 and 50 = 45 − 18 = 27
Answer: 27 students
Real-World Applications
Real-life applications of ogives:
- Education: Schools plot ogives of marks to determine percentile ranks and identify the median score of a class.
- Income analysis: Economists use ogives to determine median income, poverty lines, and income distribution patterns.
- Quality control: Manufacturing units plot cumulative defect curves to determine the threshold for acceptable quality.
- Demographics: Census data ogives show population distribution by age — useful for policy planning.
- Medicine: Cumulative frequency of patient recovery times helps in treatment planning.
Key Points to Remember
- An ogive is a cumulative frequency curve.
- Less than ogive: plot (upper boundary, less than cf) — curve rises.
- More than ogive: plot (lower boundary, more than cf) — curve falls.
- Median from single ogive: locate N/2 on y-axis → horizontal to curve → vertical to x-axis.
- Median from two ogives: x-coordinate of their intersection point.
- Ogives are S-shaped (sigmoid) curves.
- The ogive starts at cf = 0 and ends at cf = N (for less than type).
- Quartiles Q₁ and Q₃ can be found at N/4 and 3N/4 on the y-axis.
- The graphical median from the ogive is an approximation; the formula method gives an exact value.
- To find how many values lie between a and b: read cf at b minus cf at a from the less than ogive.
Practice Problems
- Draw a less than ogive for: Class: 0–10, 10–20, 20–30, 30–40, 40–50; Frequency: 7, 10, 15, 8, 10. Find the median graphically.
- Draw a more than ogive for: Class: 10–20, 20–30, 30–40, 40–50, 50–60; Frequency: 3, 6, 12, 8, 1. Find the median.
- Draw both ogives for: Class: 50–60, 60–70, 70–80, 80–90, 90–100; Frequency: 8, 12, 20, 10, 5. Find the median from their intersection.
- From a less than ogive of 100 students: at x = 40, cf = 30; at x = 60, cf = 70. How many scored between 40 and 60?
- Find Q₁ and Q₃ from the ogive of 80 observations with points: (10,5), (20,18), (30,40), (40,62), (50,80).
- A less than ogive passes through (25,10), (50,35), (75,60), (100,80). What percentage of observations are less than 50?
Frequently Asked Questions
Q1. What is an ogive?
An ogive is a cumulative frequency curve. The less than ogive plots cumulative frequency against upper class boundaries (rising curve). The more than ogive plots against lower class boundaries (falling curve).
Q2. How do you find the median from an ogive?
From a less than ogive: locate N/2 on the y-axis, draw a horizontal line to the curve, then drop a vertical to the x-axis. That x-value is the median. From two ogives: the x-coordinate of their intersection is the median.
Q3. Why does the intersection of two ogives give the median?
At the median, the number of values below it equals the number above it — both equal N/2. The less than ogive reaches N/2 at the same x-value where the more than ogive also reaches N/2, creating the intersection.
Q4. What shape does an ogive have?
An ogive is S-shaped (sigmoid). A less than ogive starts flat, rises steeply in the middle (where frequencies are highest), then flattens again at the top. A more than ogive has the reverse S-shape.
Q5. Can you find mode from an ogive?
No. The mode is best found from a histogram (by drawing diagonals in the modal class bar) or by the mode formula. Ogives are used for finding the median and percentiles.
Q6. What is the difference between a histogram and an ogive?
A histogram shows frequency of each class (bars), while an ogive shows cumulative frequency (running total as a smooth curve). A histogram is used to find mode; an ogive is used to find the median.
Q7. How do you find percentiles from an ogive?
The p-th percentile is found by locating pN/100 on the y-axis and reading the corresponding x-value. For example, the 25th percentile (Q₁) is at N/4 and the 75th percentile (Q₃) is at 3N/4.
Q8. Is the ogive median exact or approximate?
The ogive gives an approximate median because we read it from a hand-drawn curve. The exact median is obtained from the formula: Median = l + [(N/2 − cf)/f] × h.
Related Topics
- Cumulative Frequency Distribution
- Median of Grouped Data
- Frequency Polygon
- Histogram of Grouped Data
- Statistics - Collection and Presentation
- Frequency Distribution Table
- Mean of Ungrouped Data
- Median of Ungrouped Data
- Mean of Grouped Data
- Mode of Grouped Data
- Empirical Relationship Between Mean, Median, Mode
- Mean by Assumed Mean Method
- Mean by Step-Deviation Method
- Statistics in Real Life
