Cumulative Frequency

Cumulative frequency is a running total of frequencies that shows how many data points fall at or below each value or class interval. Each entry adds the current frequency to the sum of all previous frequencies.

Cumulative frequency for a given value or class interval is the sum of the frequency of that interval and all preceding intervals. If a data set is arranged in ascending order with frequencies $f_1, f_2, \ldots, f_n$ , the cumulative frequency for the $k$ th interval is the sum of frequencies from the first through the $k$ th interval. The final cumulative frequency always equals the total number of data points.

Key Formula

CF_k = \sum_{i=1}^{k} f_i

Where:

$CF_k$ = the cumulative frequency up to and including the kth class interval
$f_i$ = the frequency of the ith class interval
$k$ = the class interval number (from 1 to n)

Worked Example

Problem: A teacher records test scores for 30 students grouped into class intervals. Find the cumulative frequency for each interval: 0–20 (3 students), 21–40 (5 students), 41–60 (10 students), 61–80 (8 students), 81–100 (4 students).

Step 1: The first cumulative frequency equals the first frequency.

CF_1 = 3

Step 2: Add the second frequency to the running total.

CF_2 = 3 + 5 = 8

Step 3: Continue adding each frequency to the previous cumulative total.

CF_3 = 8 + 10 = 18

Step 4: Repeat for the remaining intervals.

CF_4 = 18 + 8 = 26, \quad CF_5 = 26 + 4 = 30

Step 5: Check: the last cumulative frequency should equal the total number of students.

CF_5 = 30 = 3 + 5 + 10 + 8 + 4 \checkmark

Answer: The cumulative frequencies are 3, 8, 18, 26, and 30. This tells you, for example, that 18 out of 30 students scored 60 or below.

Visualization

Why It Matters

Cumulative frequency is essential for finding medians and percentiles from grouped data, since it tells you where a particular data point falls in the overall distribution. It is also the basis for cumulative frequency graphs (ogives), which are widely used in statistics to visualize how data accumulates. You will encounter this concept frequently in GCSE and AP Statistics courses.

Common Mistakes

Mistake: Restarting the running total instead of adding to it

Correction: Each cumulative frequency must include all previous frequencies, not just the current one. Always add the current frequency to the cumulative total from the row above.

Mistake: Plotting cumulative frequency at the midpoint of each class interval

Correction: When drawing a cumulative frequency curve (ogive), plot each cumulative frequency at the upper class boundary, not the midpoint, because the total applies to all values up to that boundary.

Related Terms

Histogram — Displays the individual frequencies that cumulative frequency sums
Percentile — Found using cumulative frequency data