Cumulative Frequency Curves: Less Than and More Than Ogives for CBSE Class 10

In statistics, numbers organized in tables are precise, but graphs communicate patterns instantly to the human brain. While histograms and frequency polygons illustrate individual class distributions, the cumulative progression of data—such as what percentage of students scored below $60\%$, or how many wage earners make more than ₹$25,000$ per month—is visualized using a smooth, elegant S-shaped curve called an Ogive (pronounced oh-jive).

In CBSE Class 10 Mathematics, Chapter 13 (Statistics), constructing "Less Than Type" and "More Than Type" Ogives and using them to determine the Median graphically represent high-yield 3-mark and 4-mark questions.

What You Will Learn

• What is a cumulative frequency curve (Ogive)?
• Construction of a Less Than Type Ogive (Upper class limits vs. $cf$)
• Construction of a More Than Type Ogive (Lower class limits vs. $cf$)
• Method 1: Determining the Median graphically from a single ogive using $N/2$
• Method 2: Determining the Median from the point of intersection of both ogives
• Step-by-step solved CBSE board examination graphing problems
• Common axis scaling errors and graph presentation rules

1. What is an Ogive?

Formal Definition

An Ogive (or cumulative frequency curve) is a smooth, continuous graphical representation of a cumulative frequency distribution plotted on a Cartesian coordinate plane.

There are two complementary types of ogives:

1. Less Than Type Ogive: Shows cumulative frequency accumulating from lowest to highest values (an ascending S-curve rising from bottom-left to top-right).
2. More Than Type Ogive: Shows cumulative frequency diminishing from highest to lowest values (a descending S-curve falling from top-left to bottom-right).

2. Constructing a "Less Than Type" Ogive

To plot a Less Than Ogive:

1. Identify the Upper Class Limits of each class interval.
2. Calculate the corresponding Less Than Cumulative Frequency ($cf$).
3. Plot points on graph paper with:
   • $x$-coordinate: Upper Class Limit
   • $y$-coordinate: Corresponding Cumulative Frequency
4. Join the plotted points smoothly with a freehand curve (do NOT use a ruler to draw straight segments!).

    Cumulative                        Less Than Ogive (Ascending S-Curve)
    Frequency (cf)                                                * (Upper limit, N)
         ^                                                   *
       N |                                              *
         |                                         *
     N/2 + - - - - - - - - - - - - - - - - - -* (Curve Point)
         |                                  / |
         |                                 /  |
         |                           *    /   |
         |                      *        /    |
         +-----------------+-------------+----+---------------------->
         O                 Lower        MEDIAN  Upper        Limits (x-axis)

3. Constructing a "More Than Type" Ogive

To plot a More Than Ogive:

1. Identify the Lower Class Limits of each class interval.
2. Calculate the corresponding More Than Cumulative Frequency ($cf$) (starting from total $N$ at the lowest boundary and subtracting frequencies sequentially).
3. Plot points on graph paper with:
   • $x$-coordinate: Lower Class Limit
   • $y$-coordinate: Corresponding More Than Cumulative Frequency
4. Join the points with a smooth, freehand descending curve.

4. Two Graphical Methods to Determine the Median

The median can be read directly from ogives without using any algebraic formula!

Method 1: Using a Single Ogive (The $N/2$ Projection Method)

1. Calculate $\frac{N}{2}$ (where $N$ is total frequency).
2. Locate the numerical value of $\frac{N}{2}$ on the $y$-axis.
3. From this point on the $y$-axis, draw a horizontal dashed line parallel to the $x$-axis to intersect the ogive at point $P$.
4. From point $P$, drop a vertical perpendicular line down to the $x$-axis, meeting it at point $M$.
5. The $x$-coordinate of point $M$ is the exact Median of the data!

Method 2: Using Both Ogives Simultaneously (The Intersection Method)

    Cumulative                       Intersection of Both Ogives
    Frequency (cf)
         ^          \                             /  (Less than ogive)
       N |           \                           /
         |            \                         /
     N/2 + - - - - - - \ - - - - - - - - - - - /
         |              \         P           /
         |               \        *          /
         |                \      / \        /
         | (More than)     \    /   \      /
         |                  \  /     \    /
         +-------------------+--------+--+-------------------------->
         O                            M (MEDIAN on x-axis)

1. Plot the "Less Than Ogive" and "More Than Ogive" on the same set of coordinate axes.
2. The two curves will cross and intersect at a unique point $P$.
3. From point $P$, draw a vertical perpendicular line to the $x$-axis, meeting it at point $M$.
4. The $x$-coordinate of the intersection point $P$ gives the Median! (Notice that the $y$-coordinate of the intersection point is always exactly rac{N}{2}!)

5. Solved CBSE Board Examination Problems

Solved Example: Constructing a Less Than Ogive and Finding Median

Problem: The following distribution shows the daily income of 50 workers of a factory:

Convert the distribution to a less than type cumulative frequency distribution, draw its ogive, and locate the median.

Solution:

Step 1: Form the "Less Than" Cumulative Frequency Table:

Step 2: Plotting the Ogive on Graph Paper:

1. Choose appropriate scales:
   • $x$-axis: $1\text{ cm} = ₹\,20$ (use a kink/break symbol from $0$ to $100$).
   • $y$-axis: $1\text{ cm} = 5\text{ workers}$.
2. Plot the five points: $(120, 12), (140, 26), (160, 34), (180, 40), (200, 50)$.
3. Join the points with a smooth, continuous freehand curve.

Step 3: Determining the Median:

1. Total frequency $N = 50 \implies \frac{N}{2} = \frac{50}{2} = \mathbf{25}$.
2. On the $y$-axis, find $25$.
3. Draw a horizontal line from $y = 25$ to intersect the ogive at point $P$.
4. From point $P$, draw a vertical line down to the $x$-axis.
5. The line touches the $x$-axis at approximately $138.6$.
6. Therefore, <u>the median daily income is approximately ₹$138.60$</u>.

6. Summary and Examination Tips

Exam Tip: In board exams, drawing an ogive with a ruler using straight line segments is penalized! An ogive is a smooth curve drawn freehand.

Common Mistake: Plotting class marks ($x_i$) instead of class limits! For a Less Than Ogive, you MUST use the Upper Limits on the $x$-axis. For a More Than Ogive, you MUST use the Lower Limits!