Cumulative Frequency Graphs and Quartiles Cheat Sheet

Cumulative frequency graphs show how many data values are less than or equal to each class boundary. Students use them to estimate the median, quartiles, percentiles, and the spread of grouped data. This cheat sheet helps connect frequency tables, cumulative frequency curves, and summary statistics in one clear reference.

It is especially useful when working with large data sets that are grouped into intervals.

The main idea is to add frequencies as you move through the classes, then plot cumulative frequency against the upper class boundary. Quartiles divide an ordered data set into four equal parts, so $Q_1$ , $Q_2$ , and $Q_3$ are found at about $25\%$ , $50\%$ , and $75\%$ of the total frequency. The interquartile range is $IQR = Q_3 - Q_1$ , which measures the spread of the middle half of the data.

These values can also be used to draw and interpret box plots.

Key Facts

Cumulative frequency is found by adding each frequency to the total of all previous frequencies.
For grouped data, plot each point as $\left(\text{upper class boundary},\text{cumulative frequency}\right)$ .
The total frequency is $n$ , which is the final cumulative frequency in the table.
The median position is $\frac{n}{2}$ , so $Q_2$ is read from the graph at cumulative frequency $\frac{n}{2}$ .
The lower quartile position is $\frac{n}{4}$ , so $Q_1$ is read from the graph at cumulative frequency $\frac{n}{4}$ .
The upper quartile position is $\frac{3n}{4}$ , so $Q_3$ is read from the graph at cumulative frequency $\frac{3n}{4}$ .
The interquartile range is $IQR = Q_3 - Q_1$ .
A greater $IQR$ means the middle $50\%$ of the data is more spread out.

Vocabulary

Cumulative frequency: The running total of frequencies up to and including a particular class or value.
Upper class boundary: The largest boundary value for a class interval, used on the horizontal axis of a cumulative frequency graph.
Median: The middle value of an ordered data set, also called $Q_2$ .
Quartile: One of the values that divides an ordered data set into four equal parts.
Interquartile range: The spread of the middle half of the data, calculated using $IQR = Q_3 - Q_1$ .
Box plot: A graph that displays the minimum, $Q_1$ , median, $Q_3$ , and maximum of a data set.

Common Mistakes to Avoid

Plotting frequency instead of cumulative frequency is wrong because the vertical axis must show running totals, not separate class counts.
Using class midpoints instead of upper class boundaries can shift the graph and give inaccurate estimates for quartiles.
Forgetting that the final cumulative frequency equals $n$ is wrong because all quartile positions depend on the total number of data values.
Reading $Q_1$ at $\frac{1}{3}n$ or $Q_3$ at $\frac{2}{3}n$ is wrong because quartiles split data into four equal parts, not three.
Calculating the range instead of the interquartile range is wrong because $IQR = Q_3 - Q_1$ , not $\text{maximum} - \text{minimum}$ .

Practice Questions

1 A grouped frequency table has frequencies $4, 7, 9, 10$ for four consecutive classes. Find the cumulative frequencies and the total frequency $n$ .
2 For a data set with $n = 80$ , find the cumulative frequency positions used to estimate $Q_1$ , the median, and $Q_3$ .
3 A cumulative frequency graph gives $Q_1 = 18$ , $Q_2 = 25$ , and $Q_3 = 37$ . Find the interquartile range.
4 Explain why a cumulative frequency graph is useful for estimating quartiles from grouped data instead of listing every individual value.

Sign in to save

Sign in to save

Cumulative Frequency Graphs and Quartiles Cheat Sheet

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

Key Facts

Vocabulary

Common Mistakes to Avoid

Practice Questions