Ogive vs Cumulative Relative Frequency

An ogive plots the running proportion of the data against the class boundaries: above the top of each class sits the share of all observations at or below it. Because a running total can only grow, an ogive never decreases. It starts at 0 at the bottom boundary and finishes at exactly 1 at the top, and the height at any plotted point answers "what fraction is at or below here".

Take 50 commute times grouped into 10-minute classes, with 6, 15, 17, 8 and 4 commutes in them. The running counts are 6, 21, 38, 46 and 50, so dividing by 50 puts the curve through $(10, 0.12)$, $(20, 0.42)$, $(30, 0.76)$, $(40, 0.92)$ and $(50, 1.00)$.

The misreading to name is this one: "the curve is at 0.76 above 30 minutes, so 76 percent of commutes took between 20 and 30 minutes." The height is cumulative. It counts all 38 commutes of 30 minutes or less, not just the ones in the last class. The share inside the 20 to 30 class is a difference of heights, $0.76 - 0.42 = 0.34$, which is 17 commutes. On an ogive, "at or below" questions are one reading and "inside this class" questions are always a subtraction.

Anything read between two plotted points is an estimate, because grouping threw away the individual values inside each class. The curve crosses 0.500 between $(20, 0.42)$ and $(30, 0.76)$, and straight-line interpolation puts the median near $20 + \frac{0.50 - 0.42}{0.76 - 0.42} \times 10 = 22.35$ minutes. That is an estimate from grouped data, not the median of the raw 50 values. A flat stretch means an empty class, not a gap in the record.

The Fall 2026 AP Statistics course framework does not name the ogive or the cumulative relative frequency graph among its representations, so you are unlikely to be asked to build one, though reading one you are handed is fair game.

Cumulative relative frequency is a running total turned into a proportion. Work through the ordered values or classes and at each one report $\frac{\text{cumulative count}}{n}$, the number of observations at or below that point divided by the total $n$. Two properties fall straight out of that definition, and both are worth checking every time: the column can never decrease, and its final entry must be exactly 1.

Twenty-five students report how many pets they own. Nine own none, eight own one, five own two, two own three, and one owns four. The cumulative counts run 9, 17, 22, 24, 25, so the cumulative relative frequencies are 0.36, 0.68, 0.88, 0.96, and 1.00. That 0.68 says 17 of the 25 students own one pet or fewer.

"The cumulative relative frequency at one pet is 0.68, so 68 percent of the class owns one pet." Exactly one pet is $8/25 = 0.32$. The 0.68 counts everyone at or below one, meaning the nine students with none plus the eight with one. Relative frequency answers how much of the data sits in a category; cumulative relative frequency answers how much sits at or below it. Reading one column as though it were the other is the way this topic usually goes wrong.

The running total only means something when the categories carry an order. Eye color has no notion of at or below, so a cumulative column across blue, brown, and green would report a different answer every time you reordered the rows. Quantitative classes are fine, and so are ordered categories such as never, sometimes, often, always.

The Fall 2026 course framework does not name cumulative relative frequency or the ogive among its representations, so building one is unlikely to be asked. The same arithmetic does appear as percentiles and quartiles in topic 1.7, and reading a cumulative graph handed to you in a stimulus is fair game.