Cumulative Frequency
Cumulative frequency is the running total of frequencies up to and including a given value or class, telling how many observations are at or below that point rather than how many sit exactly at it. For exam scores grouped into bands, the cumulative frequency of the band 60 to 69 is the number of students who scored 69 or lower, which is the count in that band plus the counts in every lower band. Ordinary frequency answers "how many in this class?"; cumulative frequency answers "how many up to here?", and the second question is the one that connects data to ranks, medians, and percentiles.
Cumulative frequency is built by a single pass of addition down an ordered table, and its last entry must equal the total number of observations n, which is the standard error check. Each row's cumulative value is the previous row's cumulative value plus the current class frequency, so the column is non-decreasing and can never fall. Dividing every cumulative frequency by n gives the cumulative relative frequency, a number rising from near 0 to exactly 1, which can be read as the proportion of the data at or below each point. Cumulative totals only make sense when the categories have a natural order; summing frequencies across unordered categories such as colours or countries yields a number with no interpretation.
The graph of cumulative frequency is the ogive, a curve or line chart that climbs from zero to n, and its great advantage over a histogram is that medians and other quantiles can be read straight off it. To find the median, locate n divided by 2 on the vertical axis, move across to the curve, and drop down to the horizontal axis; the same procedure at one quarter and three quarters of n yields the quartiles, and any percentile follows the same way, which is the reason the ogive is the traditional graphical tool for grouped data. Francis Galton is credited with introducing the term ogive into statistics in the 1870s, in the course of his work on ranked measurements and percentiles. The steeper a stretch of the ogive, the more observations are packed into that range, so the shape of the histogram can be recovered as the slope of the cumulative curve.
The continuous, unbinned version of cumulative frequency is the empirical cumulative distribution function, a step curve that rises by 1 divided by n at each observed value and is the data-driven counterpart of the theoretical cumulative distribution function of a probability model. Its importance is large: the Kolmogorov-Smirnov test, developed from work by Kolmogorov in 1933 and Smirnov in the following years, compares an empirical cumulative curve with a hypothesised one, or two empirical curves with each other, using the largest vertical gap between them. In practice cumulative counts show up in survival analysis (the share of patients still event-free over time is one minus a cumulative curve), in service-level reporting (the share of requests completed within a given latency, which is how latency percentiles are defined), in Pareto analysis (cumulative share of defects explained by the biggest causes), and in epidemic tracking, where cumulative case counts are plotted next to daily counts.
The main pitfall is mistaking what the cumulative curve hides. Because it only ever rises, it smooths over short-term variation: a cumulative sales chart looks healthy even when recent weekly sales have collapsed, since the line flattens only slowly. A second error is reading a plateau as stability when it means that no new observations arrived. A third is the less-than versus more-than convention: a cumulative frequency can count observations at or below a point (the usual form) or at or above it, and mixing the two when reading a table leads to median estimates that are off. Finally, with grouped data, values read from an ogive assume observations are spread evenly within each class, so they are interpolated estimates rather than exact quantities.
Cumulative frequency is related to, but distinct from, frequency and relative frequency, and from the frequency table that usually holds all three side by side. It is the specifically ordered, accumulating member of the family, and it is the one that links counting to the order-based statistics: quantile, median, and percentile are all, in effect, questions asked of the cumulative curve in reverse, "at what value does the running total reach this share?".
Read cumulative frequency as the count of observations accounted for so far: a running total that always ends at n, that turns the question "how many at this value?" into "how many up to this value?", and that makes medians, percentiles, and distribution comparisons readable straight off a curve.