Not 'how many scored 60-69?' but 'how many scored *under 70*?' Add the frequencies up as you go and you can read off medians, quartiles, and percentiles straight from the curve.
Cumulative frequency is a running total: for each class, it's the count of all data values up to and including that class. Plotted, it forms an S-shaped 'ogive'.
Finding medians and percentiles from grouped data, 'how many below X?' questions, comparing distributions — the cumulative curve answers them directly.
Reading a cumulative frequency curve (ogive)
- It only ever rises (or stays flat) — you can't 'un-count'.
- Final value = total number of data points.
- Median ≈ value at half the total height.
- Q1 / Q3 ≈ values at 25% / 75% of the total height.
- Any percentile = value at that % of the height.
60 students. On the cumulative curve, at what cumulative frequency do you read off the median?
How do you get the 90th percentile from an ogive?
Find 90% of the total count on the y-axis, go across to the curve, drop down to the x-axis. That x-value is the 90th percentile — 90% of the data is below it.
Cumulative frequency never decreases. If your running total goes down, you've made an arithmetic error — each step *adds* the next class's frequency.
Plot cumulative frequency against the *upper boundary* of each class (the '<20' point goes at x = 20), not the midpoint. That's what makes percentile reading accurate.
- Running total of frequencies — always non-decreasing.
- The ogive (S-curve) lets you read medians, quartiles, percentiles directly.
- Read off at 25% / 50% / 75% of the total height for Q1 / median / Q3.