Ogives (Cumulative Frequency Curves) – Less Than & More Than

Ogive (pronounced "Oh-jive") is not a dance! It's a Cumulative Frequency Curve. 📈

What is an Ogive?

[!NOTE] Definition: A curve obtained by plotting Cumulative Frequencies against class limits.

Use: To determine Median, Quartiles, Deciles, Percentiles graphically.

Types of Ogives 🔄

1. Less Than Ogive 📉

• Plot: Upper Class Limits vs Less Than Cumulative Frequency.
• Shape: Rising upwards (S-shape).
• Meaning: Shows how many items are less than a certain value.

Steps:

1. Convert data to "Less Than" series.
2. Take Upper Limits on X-axis.
3. Take Cumulative Frequencies on Y-axis.
4. Plot points and join freehand.

2. More Than Ogive 📈

• Plot: Lower Class Limits vs More Than Cumulative Frequency.
• Shape: Falling downwards (Reverse S-shape).
• Meaning: Shows how many items are more than a certain value.

Steps:

1. Convert data to "More Than" series.
2. Take Lower Limits on X-axis.
3. Take Cumulative Frequencies on Y-axis.
4. Plot points and join freehand.

Finding Median Graphically 🎯

There are two ways to find Median using Ogives:

Method 1: Using One Ogive (Less Than)

1. Draw Less Than Ogive.
2. Calculate $N/2$ (where $N$ = Total Frequency).
3. Locate $N/2$ on Y-axis.
4. Draw a horizontal line to touch the curve.
5. From that point, draw a vertical line down to X-axis.
6. The point on X-axis is the Median.

Method 2: Using Both Ogives (Intersection)

1. Draw Less Than Ogive (Rising).
2. Draw More Than Ogive (Falling) on same graph.
3. Mark the Intersection Point where they cross.
4. Draw a perpendicular from intersection to X-axis.
5. The point on X-axis is the Median.

Example 📊

Data:

Less Than Series:

• Less than 10: 5
• Less than 20: 15 (5+10)
• Less than 30: 20 (15+5)
• Plot Points: (10, 5), (20, 15), (30, 20).

More Than Series:

• More than 0: 20 (Total)
• More than 10: 15 (20-5)
• More than 20: 5 (15-10)
• Plot Points: (0, 20), (10, 15), (20, 5).

Intersection: At X = 15 (Median).

Summary

• Ogive = Cumulative Frequency Curve.
• Less Than Ogive: Upper Limits vs CF (Rising).
• More Than Ogive: Lower Limits vs CF (Falling).
• Intersection: Gives Median.
• N/2 Method: Also gives Median.

The Bottom Line: Ogives are the GPS for finding partition values (Median, Quartiles) without calculation! 📍