Cumulative Frequency: Concepts, Formulas and Examples

Cumulative frequency is the running total of frequencies up to a given class. While an ordinary frequency tells you how many observations fall inside one class, the cumulative frequency tells you how many observations are at or below that class — answering questions like "how many students scored below 60?"

It matters because most real questions about data are "less than" or "more than" questions: the number of people earning less than a salary, or players scoring more than a target. Cumulative frequencies convert a distribution into exactly this form.

Key Concepts

Less-than cumulative frequency: frequencies added from the lowest class upwards; the last value equals the total n. Greater-than (more-than) cumulative frequency: frequencies added from the highest class downwards. True upper limit: the class boundary used when plotting — for the class 40–49, plot the cumulative value at 49.5. Ogive: the graph of cumulative frequency against class boundaries. The median and quartiles can be read straight off an ogive.

Formulas

Less-than cumulative frequency for class i: cfᵢ = f₁ + f₂ + … + fᵢ, the sum of all frequencies up to and including class i. Greater-than cumulative frequency for class i: cf'ᵢ = fᵢ + fᵢ₊₁ + … + fₖ, the sum from class i to the last class. Median from an ogive: read the x-value corresponding to a cumulative frequency of n/2. First and third quartiles: the x-values at cumulative frequencies n/4 and 3n/4. These counts are unit-free.

Worked Example

Use the frequency distribution: 40–49: f = 3, 50–59: f = 7, 60–69: f = 6, 70–79: f = 4.

Less-than cumulative: below 39.5: 0; below 49.5: 3; below 59.5: 3 + 7 = 10; below 69.5: 10 + 6 = 16; below 79.5: 16 + 4 = 20. Greater-than cumulative: 40 and above: 20; 50 and above: 20 − 3 = 17; 60 and above: 17 − 7 = 10; 70 and above: 10 − 6 = 4. Interpretation: 10 students scored below 59.5, i.e. half the class scored under 60. Notice how the less-than series (3, 10, 16, 20) and the greater-than series (20, 17, 10, 4) mirror each other.

Common Misconceptions

1. "Cumulative frequency of the last class should exceed n." No — less-than cumulative at the final class equals exactly n; if it does not, an arithmetic error was made. 2. "Greater-than cumulative starts from the first class." It is built from the top class downwards, so its first plotted value is the total n. 3. "You can find the median from ordinary frequencies by inspection." With grouped data the median is estimated from cumulative frequencies (n/2), not by picking a class that "looks middle".

Real-World Applications

Cumulative frequencies are used to report how many customers spent less than ₹1,000, determine cut-off marks so the top 10% qualify, track how many patients recovered within given days, analyse income distributions below the poverty line, and read medians and quartiles directly from ogives in census and survey reports.

Cumulative frequency turns ordinary counts into running totals from below (less-than) or from above (greater-than). It directly answers "how many are below/above this value?" and its graph, the ogive, lets you read the median and quartiles without any calculation.