Cumulative frequency diagrams 1 (Edexcel GCSE Statistics): Revision Notes
Cumulative frequency diagrams 1
What is cumulative frequency?
Cumulative frequency represents a running total that builds up as you work through a frequency table. Think of it like adding up all the frequencies as you go along, creating a total that gets bigger with each step. This running total helps us understand how many observations fall below certain values in our data set.
The key idea is that cumulative frequency never decreases - it either stays the same or increases as you move through your data. Each new frequency value gets added to the previous total, creating a "cumulative" or building effect.
When we have continuous data, we can plot these cumulative totals on a graph and connect the points using straight lines or a smooth curve. This creates what we call a cumulative frequency diagram or graph.
Understanding step polygons for discrete data
When working with discrete data (data that can only take specific values, usually whole numbers), we use a special type of diagram called a cumulative frequency step polygon. This creates a stepped appearance rather than a smooth curve.
Car Park Example: Building Cumulative Frequencies
Let's look at an example using car park data:
Original frequency table:
- 11 cars: frequency of 8
- 12 cars: frequency of 11
- 13 cars: frequency of 9
- 14 cars: frequency of 6
- 15 cars: frequency of 4
Cumulative frequency table:
- ≤11 cars: 8 (just the first group)
- ≤12 cars: 8 + 11 = 19 (first two groups combined)
- ≤13 cars: 19 + 9 = 28 (first three groups combined)
- ≤14 cars: 28 + 6 = 34 (first four groups combined)
- ≤15 cars: 34 + 4 = 38 (all groups combined)
The step polygon shows these cumulative totals as horizontal lines that suddenly jump up at each new value, creating the characteristic stepped pattern.
Worked example: Student heights
Let's work through a complete example using the heights of 100 students to understand how to create a cumulative frequency diagram.
Worked Example: Creating a Cumulative Frequency Diagram
Given data:
- 120 < h ≤ 130 cm: 8 students
- 130 < h ≤ 140 cm: 16 students
- 140 < h ≤ 150 cm: 24 students
- 150 < h ≤ 160 cm: 32 students
- 160 < h ≤ 170 cm: 20 students
Step 1: Calculate cumulative frequencies
We build up the running total by adding each frequency to the previous cumulative total:
- Up to 130 cm: 8 students
- Up to 140 cm: 8 + 16 = 24 students
- Up to 150 cm: 24 + 24 = 48 students
- Up to 160 cm: 48 + 32 = 80 students
- Up to 170 cm: 80 + 20 = 100 students
Step 2: Plot the points
Here's the crucial technique: we always plot the cumulative frequency at the upper bound (top end) of each class interval:
- Plot point (130, 8)
- Plot point (140, 24)
- Plot point (150, 48)
- Plot point (160, 80)
- Plot point (170, 100)
Step 3: Join the points
Since this is continuous data (heights can take any value within the ranges), we join successive points with straight lines or draw a smooth curve through them. You can use a ruler to create straight line segments between consecutive points.
The first point starts at (120, 0) because no students have heights at or below 120 cm in our data set.
Key plotting techniques
Critical Plotting Rules:
For discrete data: Use step polygons where the cumulative frequency stays constant until the next value, then jumps up suddenly.
For continuous data: Plot points at the upper boundary of each class interval and join them with straight lines or a smooth curve.
Starting point: Always consider where your cumulative frequency begins - often this will be zero at the start of your first class interval.
Ending point: Your final cumulative frequency should equal the total number of observations in your data set.
Remember that the choice between step polygons and smooth curves depends entirely on your data type. Discrete data requires steps because the values can only jump from one specific number to another, while continuous data allows for smooth transitions between any values within a range.
Key Points to Remember:
- Cumulative frequency is a running total that builds up by adding frequencies together as you move through the data
- For discrete data, use step polygons that create a stepped, jumping pattern
- For continuous data, plot points at the upper bound of each class interval and join with straight lines or curves
- Always check that your final cumulative frequency equals your total sample size
- The cumulative frequency can never decrease - it only stays the same or increases as you move through the data