What is cumulative relative frequency?

The running total, then the division

Start with a frequency table: a list of classes and how many observations fall in each. Cumulative relative frequency adds two columns to it.

• Cumulative frequency is the running total of the counts. Each entry is that class's count plus every count before it.
• Cumulative relative frequency divides that running total by $n$, the total number of observations.

Two checks fall out of the definition, and both are worth doing every time. The column can never decrease, because you are only ever adding non-negative counts to a running total. And the last entry must be exactly 1, because by the end you have counted everyone. If your final entry is 0.98 or 1.03, you have a missing class or an arithmetic slip.

The order of the classes matters, which means this only makes sense for data that have an order: quantitative classes such as test-score bands, or an ordered categorical variable such as "never, sometimes, often, always". Running a cumulative total across unordered categories such as eye color produces a number with no meaning, since a different listing order gives a different answer.

How it differs from a relative frequency table

A relative frequency answers "what proportion of the data is in this class?" A cumulative relative frequency answers "what proportion of the data is at or below the top of this class?" Same data, two different questions.

The practical differences follow from that.

Neither one replaces the other. A histogram of relative frequencies shows you at a glance that a distribution is skewed right; an ogive of the same data makes that hard to see, because every ogive is a rising curve. What the ogive gives you instead is direct answers to position questions. For the count-versus-proportion distinction underneath both, see frequency vs relative frequency.

One useful bridge between the two graphs, as long as the classes are equally wide: the steepest part of an ogive sits over the class with the largest relative frequency, because a tall bar in the histogram is a big jump in the running total. A flat stretch in an ogive is a class where almost nothing happened. With classes of unequal width, compare relative frequency per unit of width instead, for the same reason a histogram with unequal bins has to plot density rather than count.

A worked table you can check

Forty students take a 100-point test. Here is the full table, built from the counts in the second column.

Every relative frequency is the count divided by 40: $3/40 = 0.075$, $7/40 = 0.175$, $14/40 = 0.350$, $11/40 = 0.275$, and $5/40 = 0.125$. Those five add to 1.000.

Every cumulative count is the running total: 3, then $3 + 7 = 10$, then $10 + 14 = 24$, then $24 + 11 = 35$, then $35 + 5 = 40$. Every cumulative relative frequency is that running total over 40: $3/40 = 0.075$, $10/40 = 0.250$, $24/40 = 0.600$, $35/40 = 0.875$, and $40/40 = 1.000$.

The first row is the only place the relative frequency column and the cumulative relative frequency column agree, at 0.075, because nothing has been accumulated yet. After that they answer different questions. The relative frequency 0.350 for the 70s says 35 percent of students scored in the 70s. The cumulative relative frequency 0.600 at the top of that class says 60 percent scored below 80. Those are two different sentences about the same row of the table: 0.350 counts only the 14 students in the 70s, while 0.600 counts all 24 students at or below the top of that class. Mixing them up is the main way this topic goes wrong.

Reading percentiles off an ogive

An ogive, also called a cumulative relative frequency graph, plots the cumulative relative frequency against the class boundaries and connects the points. For the table above the points are $(50, 0)$, $(60, 0.075)$, $(70, 0.250)$, $(80, 0.600)$, $(90, 0.875)$, and $(100, 1.000)$. The first point is at height 0 because nobody scored below 50.

Read it in two directions.

• Value to percentile. Go up from a score on the horizontal axis to the curve, then across to the vertical axis. From 80 you land at 0.600, so a score of 80 sits at about the 60th percentile.
• Percentile to value. Go across from a proportion on the vertical axis to the curve, then down. From 0.875 you land at 90, so the 87.5th percentile is a score of about 90.

Subtraction answers range questions. The proportion scoring 80 or above is $1 - 0.600 = 0.400$, which is 16 of the 40 students. The proportion scoring between 70 and 90 is $0.875 - 0.250 = 0.625$, which is 25 students.

Be careful about which of these readings are exact and which are not. The two subtraction results are exact counts, because 70, 80 and 90 are class boundaries: no interpolation is involved, and 16 and 25 can be checked straight against the frequency column. Two things do make a reading approximate. First, grouped data throw away the individual values inside a class, so the curve between two plotted points is a guess and anything read from there, such as a median, is an estimate. Second, the table's classes are half-open, so 0.600 is strictly the proportion below 80, while a percentile is defined as the value at or below which a percentage falls; those differ by however many students scored exactly 80, a number the table does not record. "About the 60th percentile" is the honest phrasing. For the definition itself, see what does percentile mean.

Estimating the median and quartiles from the curve

The median is the value where the ogive crosses 0.500, $Q_1$ is where it crosses 0.250, and $Q_3$ is where it crosses 0.750. When a crossing lands between two plotted points, estimate it with a straight line between them.

$Q_1$ is easy here: the curve is at exactly 0.250 when the score is 70, so $Q_1 \approx 70$ points.

The median falls between $(70, 0.250)$ and $(80, 0.600)$. You need to travel $0.500 - 0.250 = 0.250$ of proportion into a class that supplies $0.600 - 0.250 = 0.350$, so you go $0.250 / 0.350 = 0.7143$ of the way across a 10-point class:

$Q_3$ falls between $(80, 0.600)$ and $(90, 0.875)$. You need $0.750 - 0.600 = 0.150$ out of the $0.875 - 0.600 = 0.275$ that class supplies, so $0.150/0.275 = 0.5455$ of the way across:

That gives an estimated $\text{IQR} \approx 85.45 - 70 = 15.45$ points. These are estimates from grouped data, not exact quartiles. If you have the original 40 scores, compute the quartiles from them instead, following how to find quartiles by hand or the five-number summary calculator. This site uses the median-excluded (TI-84) convention for exact quartiles.

Where this shows up, and where it does not

Cumulative relative frequency is standard in college introductory statistics and in any software that reports percentiles, and it is the everyday logic behind statements like "you scored better than 80 percent of test takers."

On the AP side, be precise about scope. The Fall 2026 AP Statistics course framework does not name the ogive or the cumulative relative frequency graph among its representations, so an exam question is unlikely to ask you to build one. What it does require is closely related: relative frequency tables for a categorical variable in topic 1.3, graphs of one quantitative variable in topic 1.5, and percentiles and quartiles as summary statistics in topic 1.7. If a cumulative relative frequency graph does appear in a stimulus, reading it is the skill being tested, and this page covers that.

One more connection worth making. A cumulative relative frequency is a proportion, so it is also an estimate of a probability: reading 0.600 at a score of 80 estimates that a randomly chosen student from this group scored below 80 with probability 0.60. That is exactly the job a normal distribution's area does for a modeled variable, which is why normalcdf and an ogive answer the same kind of question. The mean of a table like this one is a weighted mean, taking the class midpoints as the values and the class frequencies as the weights, which gives an estimate of the mean rather than its exact value.