Histograms with unequal class widths (Edexcel GCSE Statistics): Revision Notes
Histograms and area keys
When working with histograms that have unequal class widths, we need special tools to find the actual frequencies. This is where area keys become incredibly useful!
What is an area key?
An area key is a special tool that helps us interpret histograms with unequal class widths. Think of it as a "decoder" that tells us what each small square on the histogram represents in terms of actual numbers.
In a histogram with an area key:
- The histogram is drawn on a grid of small squares
- The area key tells us how many items each small square represents
- We can find frequencies by counting squares and using the key
For example, if the area key says "1 small square represents 2 fish", then 10 small squares would represent 20 fish.
The fundamental relationship
The most important formula to remember when working with histograms and area keys is:
This relationship is crucial because:
- Frequency density is what we read from the y-axis of the histogram
- Class width is the width of each interval (which may be different for each bar)
- Frequency is the actual number of items we're looking for
How to use area keys step by step
Method 1: Using the area directly
- Count the small squares in the bar you're interested in
- Multiply by the area key value to find the frequency
- Add up frequencies if you need totals across multiple bars
Method 2: Using frequency density
- Read the frequency density from the y-axis (the height of the bar)
- Find the class width (the width of the interval)
- Multiply frequency density by class width to get the frequency
Worked example walkthrough
Worked Example: Self-service Checkout Times
Given information:
- Histogram showing checkout times with frequency density on y-axis
- Time intervals: 0 < t ≤ 10, 10 < t ≤ 40, 40 < t ≤ 80, 80 < t ≤ 100
- Area key: 1 small square represents 4 people
Step-by-step solution:
Step 1: Find the frequency density for the 80 < t ≤ 100 interval
- From the histogram, we can read that the frequency density = 0.4
- This goes in our frequency table
Step 2: Calculate frequencies using the formula
-
For 0 < t ≤ 10: Class width = 10, Frequency density = 0.2
- Frequency =
-
For 10 < t ≤ 40: Class width = 30, Frequency density = 0.4
- Frequency =
-
For 40 < t ≤ 80: Class width = 40, Frequency density = 0.6
- Frequency =
-
For 80 < t ≤ 100: Class width = 20, Frequency density = 0.4
- Frequency =
Step 3: Verify using the area key
- Count the squares in each bar and multiply by 4 to check your answers
Common exam traps and tips
Common Exam Traps to Avoid:
Trap 1: Confusing frequency and frequency density
- Remember: Frequency density is on the y-axis, frequency is what you calculate
- Always check which one the question is asking for
Trap 2: Forgetting about unequal class widths
- Tip: Always identify the class width before calculating
- Don't assume all intervals are the same size
Trap 3: Misreading the area key
- Tip: Double-check what each small square represents
- Make sure you're counting squares correctly
Exam technique tips:
- Always show your working: write down the formula before substituting
- Label your final answers clearly (e.g. "Frequency = 24")
- Check your answers make sense in context
- Use the area key method to verify your frequency density calculations
Working backwards
Sometimes you might need to work backwards from frequency to frequency density:
This is useful when:
- You're given frequencies and need to draw a histogram
- You're checking your calculations
- You need to complete missing values in a table
Remember!
Key Points to Remember:
- Area keys help you "unlock" frequencies from histograms with unequal class widths
- The golden formula:
- Always identify the class width first - don't assume all intervals are equal
- You can use two methods: count squares and multiply by the key, or use frequency density calculations
- Show all your working in exams and always check your answers make sense in the real-world context