What Is Cumulative Frequency?
Cumulative frequency is a running total of frequencies. For grouped data, each cumulative frequency value tells you how many data points fall at or below the upper boundary of that class interval. This concept is fundamental to IGCSE statistics because it allows you to estimate the median, quartiles, percentiles, and interquartile range from grouped data, which cannot be done directly from a standard frequency table.
Cambridge examiners test cumulative frequency regularly on Paper 2 and Paper 4, and these questions typically carry substantial marks. Understanding the process thoroughly is well worth the effort.
Building a Cumulative Frequency Table
Suppose you have the following grouped data showing the times (in minutes) taken by 80 students to complete a test:
- 10 < t ≤ 20: frequency 5
- 20 < t ≤ 30: frequency 12
- 30 < t ≤ 40: frequency 20
- 40 < t ≤ 50: frequency 25
- 50 < t ≤ 60: frequency 13
- 60 < t ≤ 70: frequency 5
To create the cumulative frequency column, add each frequency to the running total:
- Up to 20 minutes: 5
- Up to 30 minutes: 5 + 12 = 17
- Up to 40 minutes: 17 + 20 = 37
- Up to 50 minutes: 37 + 25 = 62
- Up to 60 minutes: 62 + 13 = 75
- Up to 70 minutes: 75 + 5 = 80
The final cumulative frequency should always equal the total number of data points. If it does not, check your arithmetic.
Drawing the Cumulative Frequency Curve
When plotting the cumulative frequency curve (also called an ogive), there are important rules to follow:
- Plot points at the upper class boundaries, not at the midpoints. For the class 10 < t ≤ 20, the upper boundary is 20.
- Include a starting point at the lower boundary of the first class with a cumulative frequency of zero. In our example, plot (10, 0).
- Join the points with a smooth curve, not straight lines. Cambridge mark schemes specifically penalise joining points with a ruler.
- Label your axes clearly. The horizontal axis shows the variable (e.g., time in minutes), and the vertical axis shows cumulative frequency.
The resulting S-shaped curve allows you to read off statistical values by drawing horizontal lines from the cumulative frequency axis to the curve and then dropping vertically to the horizontal axis.
Finding the Median from a Cumulative Frequency Curve
The median is the middle value, found at the n/2 position on the cumulative frequency axis, where n is the total number of data points. For 80 students, the median position is at cumulative frequency 40.
Draw a horizontal line from 40 on the vertical axis to the curve. Then drop a vertical line to the horizontal axis. The value you read off is the estimated median.
Note the word “estimated.” Because the data is grouped, we cannot find the exact median. The cumulative frequency curve gives an approximation, and examiners accept reasonable readings with a tolerance of about half a small square.
Finding Quartiles and the Interquartile Range
The lower quartile (Q1) is found at the n/4 position. For n = 80, this is at cumulative frequency 20. The upper quartile (Q3) is at the 3n/4 position, which is cumulative frequency 60.
Read off Q1 and Q3 from the curve using the same horizontal-then-vertical technique as for the median. The interquartile range (IQR) is then calculated as Q3 − Q1.
The IQR is a measure of spread that is not affected by extreme values, making it more reliable than the range. Cambridge frequently asks you to compare distributions using the median and IQR, so practise writing comparison statements such as: “Distribution A has a higher median, indicating that on average the values are larger. Distribution B has a smaller IQR, indicating that its values are more consistent.”
Finding Percentiles
Occasionally, Cambridge asks for a specific percentile. The 90th percentile, for instance, is found at the 0.9n position. For n = 80, that is cumulative frequency 72. Read off the value from the curve as before.
Percentile questions might also be phrased differently. For example, “estimate the number of students who took more than 55 minutes.” Draw a vertical line from 55 on the horizontal axis up to the curve, then read across to the cumulative frequency axis. Suppose you read 70. Then the number who took more than 55 minutes is 80 − 70 = 10.
Common Mistakes in Cumulative Frequency Questions
Many students lose marks unnecessarily on these questions. Watch out for:
- Plotting points at class midpoints instead of upper boundaries. This is the single most common error and will cost you all the accuracy marks.
- Joining points with straight lines instead of a smooth curve. The mark scheme typically states “smooth curve” as a requirement.
- Not including the starting point at (lower boundary, 0). Without this point, your curve will be incomplete.
- Using n/2 + 1 for the median. At IGCSE level with grouped data and large sample sizes, use n/2. The “+1” formula is for ungrouped data with small samples.
- Reading the curve inaccurately. Take your time drawing horizontal and vertical lines. Use a ruler for the straight portions and read values carefully.
- Forgetting to subtract when asked for “more than” a value. If the cumulative frequency up to 50 is 62, then the number greater than 50 is 80 − 62 = 18.
Cumulative Frequency and Box Plots
Cambridge sometimes asks you to draw a box-and-whisker plot from your cumulative frequency data. For this, you need five values: the minimum, Q1, the median, Q3, and the maximum. The minimum and maximum come from the data table (the lower boundary of the first class and the upper boundary of the last class), while Q1, the median, and Q3 come from your cumulative frequency curve.
Draw the box from Q1 to Q3, with a line inside at the median. Extend whiskers from the box to the minimum and maximum values. This visual representation makes it easy to compare distributions side by side.
Practice Tips
To improve at cumulative frequency questions, practise both the computational and graphical elements:
- Create cumulative frequency tables from raw frequency tables to build fluency
- Draw curves on graph paper using past paper data sets
- Practise reading values from printed cumulative frequency curves in past papers
- Write comparison statements using the correct mathematical vocabulary
Speed and accuracy in reading graphs come with repeated practice, and these skills transfer to other graph-based questions in the statistics section.
Get Expert Help with IGCSE Maths
Need help with statistics topics like cumulative frequency, histograms, or probability? Our IGCSE Maths tutors specialise in making data handling clear and approachable, with plenty of exam-focused practice.
Need Help With IGCSE Maths?
Book a free 60-minute trial class with Teacher Rig and get personalised guidance for your IGCSE Maths preparation.