Cumulative Relative Frequency vs Relative Frequency

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.

A relative frequency turns a count into a share of the whole: divide the number of observations in a category by $n$, the total number of observations. The result always lands between 0 and 1, and multiplying by 100 reports the same quantity as a percent. When the categories are mutually exclusive and cover every observation, the relative frequencies add to exactly 1.

Forty students report how they get to school: 18 walk, 14 take the bus, 8 come by car. The relative frequencies are $18/40 = 0.45$, $14/40 = 0.35$, and $8/40 = 0.20$, and those add to 1.00. As percentages, 45 percent, 35 percent, and 20 percent.

The mistake relative frequency exists to prevent is comparing raw counts across groups of different sizes. "School B had 30 walkers to our 18, so walking is more common there" collapses if School B has 200 students: $30/200 = 0.15$ against $18/40 = 0.45$. The count is larger and the share is a third as big. A smaller slip is reporting the relative frequency as 45 when the value is 0.45. The proportion and the percent are the same quantity on two scales, so say which one you are using.

The shares only add to 1 when each observation is counted once. Let students name every method they use and someone who walks and buses appears twice, the shares total more than 1, and the denominator has stopped being the number of students. In a two-way table one word covers three different divisions, by the grand total, by a row total, or by a column total, so a relative frequency there means nothing until you state what you divided by.

Relative frequency is also how a probability gets estimated from data: run a random process many times and the share of trials producing an outcome settles near its probability as the number of trials grows. That share is an estimate built from one set of trials, not the probability itself. Summary statistics for one categorical variable are Unit 1 topic 1.3.