Cumulative Relative Frequency Graphs

Take any set of numbers, sort them, and keep a running percentage as you move left to right: what percent of the data sits at or below each value so far. Plot that running percentage against the value, and you get a curve that starts at 0 and climbs to 100. It bends into an S-shape because most real data piles up in the middle of its range. Few values are very small, so the curve starts flat. Then a dense cluster of typical values pushes the running percent up fast, so the curve steepens. Then it tapers off again as you run out of large values to add.

You build it straight from a histogram. For each bar, add its relative frequency to the running total from all bars before it, then plot a point at the bar's right edge showing that cumulative percent. Connect the points, and the bar chart becomes a smooth rising curve. Because the curve only ever goes up (it never comes back down, since percentages cannot shrink), it's a direct tool for reading percentiles: find a percent on the vertical axis, trace across to the curve, then drop down to read the value. That value is the percentile score for that percent.

This graph is the natural partner to the histogram and the boxplot from earlier in the unit. Where a histogram shows frequency and a boxplot shows five summary numbers, the cumulative graph shows every percentile at once, which is why it's the tool of choice for finding medians, quartiles, or any other percentile precisely.