Cumulative frequency diagrams 2 (Edexcel GCSE Statistics): Revision Notes
Cumulative frequency diagrams 2
Understanding cumulative frequency diagrams
Cumulative frequency diagrams are powerful tools that help us interpret data and estimate values from collected information. These graphs show the running total of frequencies up to each point, making it easy to find how many items fall below or above certain values.
The key skill you need to master is reading information from these curves accurately. This involves drawing precise lines and understanding what the plotted points represent.
The curve in a cumulative frequency diagram represents the boundary of your data - all actual values lie on or below this line. This is why we can only make estimates from these diagrams.
Reading values from cumulative frequency curves
There are two main techniques for extracting information from cumulative frequency diagrams. Understanding both methods is essential for success in examinations.
Finding cumulative frequency values
When you need to find how many items have a value up to a certain point:
- Draw a vertical line from your chosen value on the horizontal axis
- Continue this line up until it meets the curve
- From this intersection point, draw a horizontal line across to the vertical axis
- Read off the cumulative frequency value
Finding corresponding values
When you need to find what value corresponds to a particular cumulative frequency:
- Draw a horizontal line from your chosen cumulative frequency on the vertical axis
- Continue this line across until it meets the curve
- From this intersection point, draw a vertical line down to the horizontal axis
- Read off the corresponding value
Drawing Technique Tip Always use a ruler and sharp pencil when drawing these lines. Examiners specifically look for precision in your line work, and small errors can lead to incorrect readings.
Worked example: laptop costs
Let's work through a practical example using laptop cost data to demonstrate these techniques.
The situation: We have a cumulative frequency diagram showing information about laptop costs, with the horizontal axis showing cost in pounds and the vertical axis showing cumulative frequency up to 60.
Worked Example: Finding laptops costing more than £460
Question: Find an estimate for the number of laptops with a cost of more than £460.
Method:
- Draw a vertical line from £460 on the horizontal axis up to the curve
- From where this line meets the curve, draw a horizontal line across to read the cumulative frequency
- This gives us the number of laptops costing £460 or less (which is 44)
- To find laptops costing more than £460, subtract from the total number of laptops
Calculation:
- Total laptops = 57 (the maximum cumulative frequency)
- Laptops costing £460 or less = 44
- Laptops costing more than £460 = laptops
Answer: 13 laptops cost more than £460
Worked Example: Finding maximum cost of cheapest laptops
Question: Find an estimate for the maximum cost of the cheapest 26 laptops.
Method:
- Draw a horizontal line from 26 on the cumulative frequency axis across to the curve
- From where this line meets the curve, draw a vertical line down to the cost axis
- Read off the value on the cost axis
Answer: The maximum cost of the cheapest 26 laptops is £350
Essential exam techniques
Mastering these techniques will significantly improve your performance in cumulative frequency questions. Each technique addresses common areas where students lose marks.
Drawing accurate lines
Always use a ruler and sharp pencil when drawing lines on cumulative frequency diagrams. Examiners look for precision, and small errors in line drawing can lead to incorrect readings.
Common Exam Trap: "More than" vs "Less than"
This distinction catches many students! Remember:
- "More than X" = Total - (cumulative frequency at X)
- "X or less" = Read cumulative frequency directly at X
- "At least X" = Total - (cumulative frequency just below X)
Pay careful attention to the exact wording in exam questions.
Reading percentages
When dealing with percentages, convert them to actual frequencies first:
- If 25% of people are aged 20 or less, and there are 100 people total
- Then 25 people are aged 20 or less
- Find where cumulative frequency = 25 and read across to the age axis
Estimation Accuracy Cumulative frequency diagrams give estimates, not exact values. Your readings should be given to an appropriate degree of accuracy - usually to the nearest whole number for frequencies, and to 1 decimal place for other measurements.
Common exam scenarios
Understanding the types of questions you'll encounter helps you prepare effectively. You'll typically encounter questions asking you to:
- Find how many items are above or below a certain value
- Determine what value corresponds to a particular percentage or cumulative frequency
- Calculate ranges (such as the interquartile range)
- Make comparisons between different parts of the data
The key to success is practising these reading techniques until they become automatic. Remember that the curve represents the boundary of your data - all actual values lie on or below this line.
Key Points to Remember:
- Use rulers - accurate line drawing is essential for correct readings
- "More than" questions require subtraction from the total frequency
- Read carefully - distinguish between "more than", "less than", and "or equal to"
- Cumulative frequency always increases - it never goes down as you move right along the curve
- The final point on the curve gives you the total number of items in your dataset