Cumulative frequency diagrams 1 (AQA GCSE Statistics): Revision Notes
Cumulative frequency diagrams 1
What is cumulative frequency?
Cumulative frequency shows you how many observations fall at or below a certain value by keeping a running total of all the frequencies. Think of it like keeping score in a game - you add up all the points as you go along, rather than just looking at each individual score.
When you have a frequency table, you can create a cumulative frequency table by adding up all the frequencies from the beginning up to each point. This running total helps you understand how your data is distributed and makes it easier to answer questions about proportions and ranges.
The key idea is that cumulative frequency is always increasing (or staying the same) as you move through your data - it never goes down because you're always adding more observations to your total count.
Building cumulative frequency tables
Let's see how this works with a practical example. Starting with a basic frequency table, you can transform this into a cumulative frequency table by calculating running totals.
Building the cumulative frequency table:
Each cumulative frequency value includes all the previous frequencies plus the current one. For example, if the cumulative frequency for "≤13 cars" is 28, this means that on 28 out of the 38 days, there were 13 cars or fewer in the car park.
The process involves systematically adding each frequency to the sum of all previous frequencies, creating a step-by-step accumulation of your data counts.
Cumulative frequency step polygons
When dealing with discrete data (data that can only take specific whole number values), you should use a step polygon to represent your cumulative frequency. This creates a graph that looks like steps going upwards.
Why step polygons for discrete data?
The step polygon shows clearly that the cumulative frequency stays constant until you reach the next data value, then jumps up by the frequency for that value. This accurately represents discrete data because there are no possible values between the whole numbers.
Drawing cumulative frequency diagrams for continuous data
For continuous data (like heights, weights, or temperatures), the process is slightly different.
Critical Rule for Continuous Data:
Always plot your cumulative frequency at the upper bound (top end) of each class interval. This is essential for accurate representation of continuous data.
Worked example: Student heights
Worked Example: Student Heights Cumulative Frequency
Let's work through a complete example using the heights of 100 students:
Key features to notice:
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Table structure: Each row shows a height range (like 120 < h ≤ 130), its frequency, and the cumulative frequency calculated by adding all previous frequencies.
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Plotting points: Each cumulative frequency is plotted at the upper bound of its interval. For example, the first point (130, 8) represents that 8 students had heights up to 130cm.
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Joining the points: Since this is continuous data, the points can be joined with straight lines or a smooth curve using a ruler.
Key techniques for plotting
Here are the essential steps for drawing cumulative frequency diagrams:
Step-by-Step Plotting Process:
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Start at zero: Begin your diagram at the point where cumulative frequency equals zero at the lower bound of your first interval.
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Plot at upper bounds: Always place each cumulative frequency value at the top end of its class interval.
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Join successive points: Connect the points with straight lines (or a smooth curve if instructed).
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Check your final point: Your last cumulative frequency should equal the total number of observations.
For the student heights example, you would:
- Start at (120, 0)
- Plot (130, 8), (140, 24), (150, 48), (160, 80), (170, 100)
- Join these points with straight lines or a smooth curve
Key Points to Remember:
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Cumulative frequency is a running total that shows how many observations fall at or below each value
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For discrete data, use step polygons that create a "stepped" appearance
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For continuous data, always plot cumulative frequencies at the upper bound of each class interval
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The final cumulative frequency should always equal your total number of observations
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Points on continuous data can be joined with straight lines or smooth curves to help with reading values from the graph