Measures of Dispersion for Grouped Data · Form 5
What does an ogive show?
An ogive is a cumulative frequency curve: it shows a running total of how many data values fall at or below each upper class boundary, so you can read positions like the median and quartiles straight off the graph.
A running total, drawn as a curve
Instead of showing how many values are in each class on its own, an ogive adds them up as you go. Each point plots an upper class boundary against the cumulative frequency, the total number of values at or below that boundary.
Joined up, these points form a smooth curve that usually looks like a stretched letter S.
Reading positions off the graph
The ogive's real use is answering 'where does a value sit?' The median lines up with half the total frequency, the lower quartile with a quarter of it and the upper quartile with three-quarters, each read across to the curve and down to the horizontal axis.
You can also read the other way round, how many values fall below a given mark, which is why the ogive is the natural home for the interquartile range.
It only ever climbs
Because you keep adding, a cumulative frequency curve never falls, it rises to the total frequency and then levels off. Recognise an ogive by its cumulative-frequency axis and upper-boundary points, not midpoints.
The usual mix-up is to confuse it with a frequency polygon (which plots midpoints and can go up and down) or to plot the running total against the wrong x-value.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I read the median off an ogive?
Find the total frequency N, locate N/2 on the cumulative frequency axis, draw a horizontal line to the curve, then drop straight down to the horizontal axis, that value is the estimated median. The same method works for quartiles using N/4 and 3N/4.
Why does the ogive always plot against upper boundaries, not midpoints?
Cumulative frequency counts how many values are at or below a certain point, and that count is only complete up to the upper boundary of each class, plotting against the midpoint would understate how many values have actually been included by that point on the curve.
What mistake do students make when constructing the cumulative frequency table before plotting?
They forget to add each class's frequency to the running total from the previous class, or they plot the cumulative frequency against the wrong boundary. Always start the table with a cumulative frequency of 0 at the lower boundary of the first class.