Interpolation and extrapolation (Edexcel GCSE Statistics): Revision Notes
Interpolation and extrapolation
What are interpolation and extrapolation?
When you have a scatter graph with a line of best fit drawn through the data points, you can use this line to find estimated values. There are two different ways to do this, depending on where the value you're looking for falls in relation to your original data.
Interpolation means finding an estimated value that falls within the range of your original data points. Since you're working within the boundaries of your known information, interpolation tends to give reliable estimates.
Extrapolation means predicting a value that lies outside the range of your original data points. Because you're going beyond what you actually know, extrapolation is generally less reliable than interpolation.
The key difference lies in whether you're working within your known data range (interpolation) or beyond it (extrapolation). This fundamental distinction determines the reliability of your estimates.
The line of best fit and mean point
Before you can use interpolation or extrapolation, you need to understand an important property of the line of best fit: it always passes through the mean point of your data.
The Mean Point Property
The mean point has coordinates that represent the average of all your x-values and the average of all your y-values:
This gives you a reference point to help ensure your line of best fit is positioned correctly.
Worked example: Candle heights and lifetimes
Worked Example: Using Interpolation and Extrapolation with Candle Data
The situation: A scatter diagram shows the relationship between the initial heights and lifetimes of six candles. We need to add one more candle with an initial height of 7cm and a lifetime of 12 hours.
Step 1: Calculate the mean point For all seven candles, the mean height is 9cm and the mean lifetime is 15 hours. So our mean point is at coordinates .
Step 2: Draw the line of best fit The line of best fit must pass through this mean point and follow the general trend of all the plotted points.
Step 3: Use interpolation A candle has an initial height of 15cm. What is its estimated lifetime?
Looking at the line of best fit, when the height is 15cm, the corresponding lifetime is 21.5 hours. This is interpolation because 15cm falls within the range of heights we measured (between about 5cm and 25cm). Since we're estimating within our known data range, this estimate is reliable.
Step 4: Use extrapolation Another candle has an initial height of 22cm. What is its predicted lifetime?
Using the line of best fit, when the height is 22cm, the corresponding lifetime is 29 hours. This is extrapolation because 22cm is outside the range of heights we actually measured. Since we're predicting beyond our known data, this estimate is not as reliable.
Understanding reliability
The key difference between interpolation and extrapolation lies in their reliability:
- Interpolation is reliable because you're making estimates based on the pattern within your known data range
- Extrapolation is less reliable because you're assuming the same pattern continues outside your measured range, which may not always be true
Exam technique tips
When answering questions about interpolation and extrapolation:
- Always identify which method you're using - check whether your value falls inside or outside the data range
- Comment on reliability when asked - interpolation is reliable, extrapolation is not reliable
- Show your working clearly - draw lines on the graph to show how you read off values
- Calculate the mean point first if you need to draw a line of best fit
- Be precise with your reading - use a ruler to help you read values accurately from the graph
Common Exam Trap
Watch out for questions that ask you to comment on the reliability of your estimate. Many students forget to mention whether they used interpolation or extrapolation, and therefore miss explaining why their answer is reliable or unreliable.
Always state which method you used and comment on its reliability!
Practice approach: Car prices example
Here's how to approach a typical exam question:
Scenario: A graph shows the relationship between car ages and prices, with a line of best fit drawn.
For interpolation: If asked about a car that's 2 years old (within the data range):
- Use the line of best fit to read off the price
- State this is interpolation
- Comment that the estimate is reliable
For extrapolation: If asked about a car that's 6.5 years old (outside the data range):
- Use the line of best fit to predict the price
- State this is extrapolation
- Comment that the estimate is not reliable
Key Points to Remember:
- Interpolation = estimating within the data range = reliable
- Extrapolation = predicting beyond the data range = less reliable
- The line of best fit always passes through the mean point of your data
- Always comment on reliability when asked in exam questions
- Use a ruler for accurate readings from graphs