An ogive, also known as a cumulative frequency curve, is a graphical representation of cumulative frequency distribution.

It displays the cumulative frequency of a dataset at or below certain values. Ogives are useful for visualizing the distribution of data and identifying patterns in the cumulative frequencies.

Here’s a step-by-step method for constructing an ogive:

**Method of Constructing Ogives:**

**Prepare the Frequency Distribution Table:**

- Begin with a frequency distribution table that includes columns for the data intervals (classes), frequencies, and cumulative frequencies. Class Interval Frequency (f) Cumulative Frequency (CF) 0-10 5 10-20 8 20-30 12 30-40 7 40-50 10

**Calculate Cumulative Frequencies (CF):**

- Determine the cumulative frequency for each class interval by adding the frequency of that interval to the cumulative frequency of the previous interval.
- For the given example:
- ( CF_{10} = 5 )
- ( CF_{20} = CF_{10} + 8 = 13 )
- ( CF_{30} = CF_{20} + 12 = 25 )
- ( CF_{40} = CF_{30} + 7 = 32 )
- ( CF_{50} = CF_{40} + 10 = 42 )

**Plotting the Ogive:**

- Use a graph paper and draw a set of axes. The x-axis represents the data intervals, and the y-axis represents cumulative frequencies.
- Mark the lower class boundaries (midpoints) on the x-axis and the corresponding cumulative frequencies on the y-axis.

**Drawing the Ogive:**

- Starting from the first point (0, 0), plot the points for each class interval using the lower class boundaries and the corresponding cumulative frequencies.
- Connect the points using straight lines to form the ogive.

**Example:**

Suppose you have the following frequency distribution table for exam scores:

```
Class Interval Frequency
0-10 5
10-20 8
20-30 12
30-40 7
40-50 10
```

Now, let’s construct the ogive:

- Calculate Cumulative Frequencies:

```
CF_10 = 5
CF_20 = CF_10 + 8 = 13
CF_30 = CF_20 + 12 = 25
CF_40 = CF_30 + 7 = 32
CF_50 = CF_40 + 10 = 42
```

- Plotting the Ogive:

```
Data Intervals Cumulative Frequencies
0-10 | 5
10-20 | 13
20-30 | 25
30-40 | 32
40-50 | 42
```

- Drawing the Ogive:

- Plot the points (10, 5), (20, 13), (30, 25), (40, 32), and (50, 42).
- Connect the points with straight lines to form the ogive.

The resulting graph will display the cumulative distribution of exam scores, providing insights into the overall performance of the dataset.