Cumulative Frequency: Definition, Table, Graph and Examples

For an interactive graph, try the ogive graph maker or the frequency distribution table maker.

What Is Cumulative Frequency?

The word "cumulative" means increasing by accumulation: each new entry in the table adds to whatever came before it. Think of filling a jar with marbles one layer at a time. After each layer goes in, the cumulative frequency tells you the total number of marbles in the jar so far — not just how many were in that one layer.

Three properties hold without exception. Cumulative frequency never decreases — it either rises or stays flat when a class has zero observations, never falls. The final cumulative frequency always equals N, giving you a built-in check on your arithmetic. And cumulative frequency only makes sense for ordered data (ordinal, interval, or ratio scale). Applying it to nominal categories like eye colour or political party produces a number with no interpretable meaning, because the "order" you impose is arbitrary.

For the broader context, this topic sits within descriptive statistics — the set of techniques for summarizing and describing datasets before making inferences. Cumulative frequency is closely related to relative frequency and to frequency distributions more generally.

Frequency vs. Cumulative Frequency: The Key Difference

Cumulative frequency tables also feed directly into finding the median, quartiles, and percentiles from grouped data. The interpolation formula for the median from a frequency table requires the cumulative frequency up to the median class — you cannot find it without the CF column.

Less Than and Greater Than Cumulative Frequency

There are two forms of cumulative frequency, and they answer opposite questions. Both are built from the same base frequency table; what changes is the direction of accumulation.

Worked Example: Less Than Cumulative Frequency

Marks scored out of 100 by 40 students:

Worked Example: Greater Than Cumulative Frequency

Using the same 40-student dataset:

How to Calculate Cumulative Frequency — Step by Step

Method 1: Ungrouped (Discrete) Data

With ungrouped data, each row in your table represents a single distinct value rather than a class interval.

Reading: "14 students read 2 books or fewer per week."

Method 2: Grouped (Continuous) Data

The process is identical, but you must identify the upper class boundary of each interval — not the class midpoint — before plotting any graph.

How to Draw a Cumulative Frequency Graph (Ogive)

The cumulative frequency graph goes by several names: ogive, cumulative frequency curve, cumulative frequency polygon. The ogive (pronounced "OH-jive") is the standard term in statistics. It is an S-shaped curve — a sigmoid — that rises from the lower-left of the graph to the upper-right.

Why Upper Class Boundaries, Not Midpoints — The Critical Rule

Step-by-Step Ogive Construction

Using the 50-student example from above:

What the Ogive Shape Tells You

A steep section in the ogive means many data points fall within that x-range — the frequencies are high there. A flat section means few observations. The classic S-shape you see with normally distributed data reflects the fact that most observations cluster around the centre, with fewer in each tail. A curve that rises sharply at the left then flattens early indicates a right-skewed distribution.

Using the Ogive to Find Median, Quartiles, and Percentiles

The ogive's practical value is that it lets you read positional statistics — median, quartiles, deciles, percentiles — directly off the graph without sorting raw data or applying formulas. The procedure is always the same: find the relevant value on the y-axis first, then project horizontally to the curve, then drop vertically to the x-axis.

Finding the Median

Finding Quartiles and IQR

The same procedure works for both quartiles. Locate the relevant cumulative frequency on the y-axis, project to the curve, and read off the x-value.

Once you have Q₁ and Q₃ from the ogive, the interquartile range is simply IQR = Q₃ − Q₁. You can also use these three values directly to build a box plot — the five-number summary adds the minimum and maximum.

Finding Percentiles

The P-th percentile is the value below which P% of the data falls. To locate it on the ogive, calculate the y-axis position as (P/100) × N, then follow the same projection procedure.

For more on positional statistics, see the dedicated pages on percentiles and measures of central tendency.

Finding Deciles

Deciles divide data into 10 equal parts. No top-5 result for this keyword covers deciles from cumulative frequency. The D-th decile sits at position (D/10) × N on the y-axis.

Relative Cumulative Frequency and Cumulative Percentage

Relative cumulative frequency scales the running total to a proportion of N, making comparisons between datasets with different sample sizes possible.

Four-Column Frequency Table

The CDF Connection

Relative cumulative frequency is the empirical cumulative distribution function (ECDF) of the dataset. When you plot relative CF on the y-axis (0 to 1) instead of raw CF, you get the empirical CDF — the same concept underlying probability distributions. The value at any x tells you the estimated probability that a randomly chosen observation falls below x. On a relative CF ogive, the median is always at y = 0.50, regardless of sample size. This connection leads naturally into continuous cumulative distribution functions.

All Four Frequency Types: A Summary

Cumulative Frequency in Python, R, and Excel

No top-5 result for this keyword provides working code. This section covers all three common tools.

Python (pandas + numpy + matplotlib)

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

# Raw test score data
scores = [15, 23, 18, 42, 55, 38, 29, 47, 61, 35,
          28, 44, 52, 37, 19, 48, 63, 31, 26, 41]

# Step 1: Define class boundaries
bins = [10, 20, 30, 40, 50, 60, 70]

# Step 2: Build frequency counts
freq, edges = np.histogram(scores, bins=bins)

# Step 3: Assemble the frequency table
df = pd.DataFrame({
    'Class': [f"{int(edges[i])}–{int(edges[i+1])}" for i in range(len(freq))],
    'Upper_Boundary': edges[1:],
    'Frequency': freq
})

# Step 4: Cumulative frequency
df['Cumulative_Freq'] = df['Frequency'].cumsum()

# Step 5: Relative cumulative frequency
n = df['Frequency'].sum()
df['Relative_CF'] = df['Cumulative_Freq'] / n
df['Cumulative_Pct'] = df['Relative_CF'] * 100

print(df.to_string(index=False))

# Step 6: Plot the ogive (starting point at lower boundary = 10)
x = [bins[0]] + list(df['Upper_Boundary'])
y = [0] + list(df['Cumulative_Freq'])

plt.figure(figsize=(8, 5))
plt.plot(x, y, marker='o', color='steelblue', linewidth=2, label='Less-than ogive')
plt.axhline(y=n/2, color='red', linestyle='--', label=f'Median line (N/2 = {n//2})')
plt.xlabel('Test Score')
plt.ylabel('Cumulative Frequency')
plt.title('Cumulative Frequency Ogive')
plt.legend()
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

R

# Raw data
scores <- c(15, 23, 18, 42, 55, 38, 29, 47, 61, 35,
            28, 44, 52, 37, 19, 48, 63, 31, 26, 41)

# Define class boundaries
breaks <- c(10, 20, 30, 40, 50, 60, 70)

# Frequency table
cut_scores <- cut(scores, breaks=breaks, right=TRUE, include.lowest=TRUE)
freq_table  <- table(cut_scores)

# Cumulative frequency (use cumsum)
cum_freq <- cumsum(freq_table)

# Relative cumulative frequency
n <- sum(freq_table)
rel_cum <- cum_freq / n

# Assemble display data frame
df <- data.frame(
  Class         = names(freq_table),
  Frequency     = as.vector(freq_table),
  CumFreq       = as.vector(cum_freq),
  RelativeCF    = round(as.vector(rel_cum), 3),
  CumPct        = round(as.vector(rel_cum) * 100, 1)
)
print(df)

# Plot ogive
x <- c(breaks[1], breaks[-1])
y <- c(0, as.vector(cum_freq))

plot(x, y, type="l", lwd=2, col="steelblue",
     xlab="Test Score", ylab="Cumulative Frequency",
     main="Cumulative Frequency Ogive")
points(x, y, pch=16, col="steelblue")
abline(h=n/2, lty=2, col="red")
grid()

Excel

Key step for the running SUM formula: in cell C2 type =B2. In C3 type =C2+B3. Drag C3 down through the rest of the frequency column. Each cell adds the class frequency to the previous total, giving you the cumulative frequency column automatically.

Pareto Chart: Cumulative Frequency in Business

A Pareto chart puts cumulative frequency to work in quality management and business analysis. It overlays a bar chart of individual frequencies (sorted largest to smallest) with a line chart of the cumulative frequency, making the 80/20 rule visible at a glance.

The rule, named after economist Vilfredo Pareto, states that roughly 80% of effects come from 20% of causes. In a factory tracking defects, a small number of defect types typically account for most of the total defect count. The cumulative frequency line makes it immediate which categories to address first.

For an interactive version, the Pareto chart maker lets you enter your own data and see both the bar chart and the cumulative percentage line together.

Common Cumulative Frequency Mistakes

Exam mark schemes repeatedly cite these eight errors. Each one has a specific correct procedure that makes the mistake impossible once you know to watch for it.

Real-World Applications of Cumulative Frequency

Practice Problems with Worked Solutions

Frequently Asked Questions

Key Takeaways