Cumulative Frequency — Ogive Curve & Percentiles

Grouped-data questions about medians and percentiles almost always come down to one column you can build yourself: the cumulative frequency, or running total of how many observations fall at or below each class boundary. Add that column and an ogive, the graph of that running total, and finding the median, quartiles, and percentiles becomes a procedure you can repeat.

When To Use Cumulative Frequency

Reach for cumulative frequency whenever a question asks for ordered position in a data set rather than a class-by-class count: a median, a quartile, or a percentile. It is the right tool when raw data are large and a grouped table is easier to read than a long list, and it is what an ogive is built from. If you only need to know how many observations land in one class, plain frequency is enough; the moment "at or below" enters the question, switch to cumulative frequency.

The Procedure, Step By Step

If the class frequencies are $f_1, f_2, \dots, f_k$, then the cumulative frequency up to class $k$ is

Each row adds one more class to the total. If the cumulative frequency is $28$ at the end of a class, then $28$ observations are in that class or below it. For ungrouped data, cumulative frequency is just a running count; for grouped data, it is a running count by class interval.

To locate a value, work from the total frequency $N$:

• the median is around the $N/2$th value
• the first quartile is around the $N/4$th value
• the third quartile is around the $3N/4$th value
• the $p$th percentile is around the $(p/100)N$th value

An ogive turns those positions into a reading. Plot the upper class boundary on the horizontal axis and the cumulative frequency on the vertical axis, then join the points; the curve only rises, because cumulative frequency never decreases. Start from a vertical position, move across to the curve, and drop down to the horizontal axis to estimate the value.

A Complete Pass Through One Table

Suppose test scores for $40$ students are grouped like this:

The total frequency is $N = 40$.

The median

The median is the $N/2 = 20$th value. Reading the cumulative frequencies, the total is $16$ up to 20-30 and $28$ up to 30-40, so the $20$th value lies in the $30$-$40$ class. If you want a grouped-data estimate, use interpolation only when it is reasonable to treat the values as spread fairly evenly through that class:

Here $L = 30$ is the lower boundary, $F_{\text{before}} = 16$ is the cumulative frequency before the class, $f = 12$ is the class frequency, and $w = 10$ is the class width. So

That estimate is not exact. It depends on the assumption that the values inside the $30$-$40$ class are spread fairly smoothly.

The 75th percentile

The $75$th percentile is the $(75/100) \cdot 40 = 30$th value. The total is $28$ up to 30-40 and $36$ up to 40-50, so the $30$th value lies in the $40$-$50$ class. Using the same interpolation idea,

On an ogive, you would mark $30$ on the cumulative-frequency axis, move across to the curve, and read down to about $42.5$ on the score axis.

Where Each Step Tends To Break, And How To Check It

Step: building the column. The frequent slip is confusing frequency with cumulative frequency. Frequency counts one class; cumulative frequency counts that class plus all earlier ones. Self-check: the last cumulative entry must equal $N$.

Step: locating the position. The position comes from the total $N$. Use the wrong total and every later step is off. Self-check: re-derive the position from $N$ before reading the table.

Step: reading off the value. A grouped estimate lands inside a class, not on an exact original value, and it depends on how the data are distributed in the interval. Self-check: confirm your answer falls between the class boundaries you used.

Step: drawing the ogive. For grouped data, plot against class boundaries, especially upper class boundaries; plotting against midpoints changes the meaning. Self-check: the leftmost point should sit at the first cumulative frequency, the rightmost at $N$.

Where Cumulative Frequency Shows Up

Cumulative frequency appears in exam-score summaries, income distributions, quality-control data, and any setting where percentiles or medians matter more than individual bin counts. It is especially handy when raw data are large and a grouped table reads more clearly than the full list of observations.

Run The Procedure Once More

Build a cumulative frequency column for a small grouped table with $N = 50$, then locate where the $20$th, $25$th, and $45$th values fall. Draw the ogive, read off the median and one percentile, and compare those readings with the table-based estimates. Matching the two is the cleanest sign the procedure has clicked.