Cumulative Frequency — Ogive Curve & Percentiles

Cumulative frequency is the running total in a frequency table. It tells you how many observations are at or below a value or class boundary, which is why it is useful for finding the median, quartiles, and percentiles.

An ogive is the graph of that running total. Once you can read the table and the graph together, grouped-data questions become much easier.

Cumulative Frequency Definition

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

$$F_k = f_1 + f_2 + \cdots + f_k$$

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.

How an Ogive Helps You Read Percentiles

An ogive plots cumulative frequency against class boundaries. For grouped continuous data, you usually plot:

• the upper class boundary on the horizontal axis
• the cumulative frequency on the vertical axis

Then you join the points with a smooth or piecewise line. The curve rises because cumulative frequency never decreases.

The main use of an ogive is reading positions in the ordered data set. If the total frequency is $N$, then:

• 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

On the graph, you start from that vertical position, move across to the ogive, then drop down to the horizontal axis to estimate the value.

Worked Example: Median and 75th Percentile

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

Score	Frequency	Cumulative frequency
0-10	$2$	$2$
10-20	$5$	$7$
20-30	$9$	$16$
30-40	$12$	$28$
40-50	$8$	$36$
50-60	$4$	$40$

The total frequency is $N = 40$.

Find the median from the table

The median is the $N/2 = 20$th value.

Look at the cumulative frequencies:

• up to 20-30, the total is $16$
• up to 30-40, the total is $28$

So the $20$th value lies in the $30$-$40$ class.

If you want a grouped-data estimate, use interpolation only if it is reasonable to treat the values as spread fairly evenly through that class. Then

$$\text{median} \approx L + \frac{N/2 - F_{\text{before}}}{f} \cdot w$$

Here:

• $L = 30$ is the lower boundary of the class
• $F_{\text{before}} = 16$ is the cumulative frequency before the class
• $f = 12$ is the class frequency
• $w = 10$ is the class width

So

$$\text{median} \approx 30 + \frac{20 - 16}{12} \cdot 10 = 30 + \frac{40}{12} \approx 33.3$$

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

Estimate the 75th percentile

The $75$th percentile is the $(75/100) \cdot 40 = 30$th value.

From the cumulative frequencies:

• up to 30-40, the total is $28$
• up to 40-50, the total is $36$

So the $30$th value lies in the $40$-$50$ class.

Using the same interpolation idea,

$$P_{75} \approx 40 + \frac{30 - 28}{8} \cdot 10 = 42.5$$

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

Common Mistakes with Cumulative Frequency

Mixing up frequency and cumulative frequency

Frequency tells you how many observations are in one class. Cumulative frequency tells you how many observations are in that class and all earlier classes together.

Using the wrong position

For the median or a percentile, the position comes from the total frequency $N$. If you use the wrong total, every later step is off.

Treating grouped estimates as exact

An ogive or interpolation gives an estimate inside a class, not an exact original data value. That estimate depends on how the data are distributed inside the interval.

Plotting the wrong horizontal values

For grouped data, ogives are usually plotted against class boundaries, especially upper class boundaries. Plotting against class midpoints changes the meaning.

When Cumulative Frequency Is Used

Cumulative frequency is used whenever you need ordered position in a data set rather than just class-by-class counts. That includes exam-score summaries, income distributions, quality-control data, and any situation where percentiles or medians matter more than individual bin counts.

It is especially useful when raw data are large and a grouped table is easier to read than a long list of observations.

Try a Similar Cumulative Frequency Problem

Take any small grouped table and add a cumulative frequency column before drawing an ogive. Then read the median and one percentile from the graph and compare them with the table-based estimate.

If you want one more check, try your own version with $N = 50$ and ask where the $20$th, $25$th, and $45$th values would fall. That is a simple way to make the idea stick.