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 , , and are found at about , , and of the total frequency. The interquartile range is , 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 .
- The total frequency is , which is the final cumulative frequency in the table.
- The median position is , so is read from the graph at cumulative frequency .
- The lower quartile position is , so is read from the graph at cumulative frequency .
- The upper quartile position is , so is read from the graph at cumulative frequency .
- The interquartile range is .
- A greater means the middle 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 .
- 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 .
- Box plot
- A graph that displays the minimum, , median, , 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 is wrong because all quartile positions depend on the total number of data values.
- Reading at or at is wrong because quartiles split data into four equal parts, not three.
- Calculating the range instead of the interquartile range is wrong because , not .
Practice Questions
- 1 A grouped frequency table has frequencies for four consecutive classes. Find the cumulative frequencies and the total frequency .
- 2 For a data set with , find the cumulative frequency positions used to estimate , the median, and .
- 3 A cumulative frequency graph gives , , and . 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.
Understanding Cumulative Frequency Graphs and Quartiles
Grouped data creates an important limitation. The exact value of each observation is no longer known once values have been collected into intervals. A class labelled ten to less than twenty may contain values close to ten, close to twenty, or anywhere between.
For this reason, readings from a cumulative frequency graph are estimates. The curve assumes that values are spread reasonably evenly within each class.
This assumption is often sensible for narrow intervals, but it can be less reliable when intervals are wide or the data bunches near one end. A sensible answer should usually be given to a suitable level of accuracy, rather than pretending the graph gives an exact value.
Class boundaries matter because measurements that look separate can actually join with no gap. If whole-number heights are recorded in classes from one hundred fifty to one hundred fifty nine, then one hundred sixty to one hundred sixty nine, the true boundaries are one hundred forty nine point five to one hundred fifty nine point five, followed by one hundred fifty nine point five to one hundred sixty nine point five. Using boundaries makes the intervals touch correctly.
On the graph, the curve begins at the lower boundary with cumulative frequency zero. It then rises to each upper boundary. This starting point prevents a false jump before the first class and shows the full distribution more accurately.
The shape of the curve gives information beyond the quartiles. A steep section means many observations lie in a small range of values. A flatter section means relatively few observations occur across that range.
If the curve becomes steep near high values, many results are concentrated there. If it rises slowly for a long distance, the data is more thinly spread in that part. Students should read horizontally from a chosen cumulative frequency to the curve, then vertically down to the value axis.
Reading in the opposite direction finds how many observations are at or below a given value. This is useful for estimating a percentile or finding the proportion below a target score.
Quartiles are most useful when comparisons are needed. Two classes may have similar median test marks, yet one class can have a much larger interquartile range. That class has less consistency among the middle half of its results.
A box plot makes this comparison quick because the box runs from the lower quartile to the upper quartile, while the line inside marks the median. The whiskers show the wider spread according to the rule being used.
In many school courses, values more than one and a half times the interquartile range below the lower quartile or above the upper quartile are treated as possible outliers. Outliers deserve checking, since they may be genuine unusual results, recording errors, or measurements from different conditions.
Common mistakes come from mixing positions with values. One quarter of the total frequency is a location on the vertical axis, not the lower quartile itself. The lower quartile is the data value read from the horizontal axis after following the graph.
It is equally important not to use class midpoints for this type of graph, since midpoints belong in frequency polygons or estimated means. Check that cumulative totals never decrease, that the final total matches the number of observations, and that scales are read carefully. A small error in scale reading can change every summary value.