Ogives (Cumulative Frequency)

Cumulative frequency is simply a running total. You start a new column next to your frequency table and keep adding. The first cumulative frequency equals the first frequency; after that, each entry is the previous cumulative frequency plus the new class frequency. The very last cumulative frequency must equal n, the total number of items, which is a useful check that your table is correct. Cumulative frequency answers the question 'how many items are less than this value?', so it always rises (or stays level) as you move down the table.

An ogive (also called a cumulative frequency curve) is the graph you get when you plot cumulative frequency on the y-axis against class boundaries on the x-axis and join the points with a smooth free-hand curve. For a 'less than' ogive you plot each cumulative frequency against the UPPER boundary of its class, because that running total includes everything up to that upper limit. The curve is S-shaped and always rises from left to right. A 'more than' ogive plots against lower boundaries and falls instead.

The real power of an ogive is that it gives you the median without a formula. The median is the middle value, so it sits where the cumulative frequency is half the total, n/2. Find n/2 on the y-axis, draw a horizontal line across to the curve, then drop a vertical line down to the x-axis. The x-value where it lands is the median. The same trick gives the lower quartile at n/4 and the upper quartile at 3n/4, and the inter-quartile range is Q3 minus Q1.

An ogive is only as accurate as the cumulative-frequency column behind it. A single arithmetic slip in the running total shifts every point above it, bends the curve, and gives a wrong median. So in the exam, build the cumulative column carefully, check that the final value equals n, choose a sensible scale, and plot against boundaries (not mid-points). Use a sharp pencil and join the points with one smooth curve, never with straight ruler segments.