Graphic Presentation — Ogive (Cumulative Frequency Curve)

An ogive is the graph of a cumulative frequency distribution. Because a cumulative frequency can be built up in two directions, there are two ogives:

• 'Less than' ogive — plot the upper class boundary of each class against the cumulative frequency of all values less than that boundary. This curve always rises from left to right.
• 'More than' ogive — plot the lower class boundary of each class against the cumulative frequency of all values more than (or equal to) that boundary. This curve always falls from left to right.

When both ogives are drawn on the same graph, they intersect at a point whose x-coordinate gives the median of the distribution directly — this is one of the classic uses of the ogive, and it can be cross-checked against the formula $\text{Median} = L + \dfrac{\left(\tfrac{N}{2} - c.f.\right)}{f} \times h$, where $L$ is the lower boundary of the median class, $c.f.$ is the cumulative frequency of the class just before the median class, $f$ is the frequency of the median class, $h$ is the median class's width, and $N$ is the total of all frequencies.

Ogives are also used to read off quartiles and percentiles graphically, and to answer questions such as 'how many students scored below 40 marks?' directly from the curve, without recomputing the whole distribution. …