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3.7c · Correlation & Regression · Sub-skill

Prediction

Using a regression line to make predictions, and knowing when those predictions can and can't be trusted.

Build it up, step by step

Understanding prediction

Click each step below to reveal it — work through them in order the first time round.

Step 1 · Making a prediction

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To predict a value of $y$ for a given $x$ (or vice versa), simply substitute the value into the regression equation and calculate. This is one of the main practical uses of finding a regression line.

Step 2 · Interpolation

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Interpolation means predicting a value of $y$ for an $x$ that lies within the range of the original data used to build the regression line. Interpolated predictions are generally considered reasonably reliable, since they're supported by data close to that range.

Step 3 · Extrapolation

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Extrapolation means predicting a value of $y$ for an $x$ that lies outside the range of the original data. Extrapolated predictions are much less reliable — there's no evidence the same linear trend continues beyond the data you actually collected.

Step 4 · Evaluating the reliability of a prediction

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In the exam, you should always state whether a prediction involves interpolation or extrapolation, and comment on its reliability accordingly. Even for interpolation, the regression line is only a model — predictions won't be exact, and the strength of correlation (PMCC) also affects how much you should trust the prediction.

Worked example

A regression line, based on data for x from 0 to 100, is $y=10+2x$. Predict y when x=50, and when x=500. Comment on the reliability of each prediction.

x=50: y=10+2(50)=110. This is interpolation (50 is within the data range 0–100), so reasonably reliable.
x=500: y=10+2(500)=1010. This is extrapolation (500 is far outside the data range), so this prediction is unreliable.

Test yourself

Past-paper style question

A regression line for predicting plant height, y (cm), from age, x (weeks), based on data collected for plants aged 2 to 10 weeks, is $y=4+3x$.

(a) Predict the height of a plant aged 6 weeks.
(b) A gardener uses the same line to predict the height of a plant aged 40 weeks. Explain why this prediction should not be trusted. [4 marks]

Show the answer

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(a) y = 4 + 3(6) = 4+18 = 22 cm. This is interpolation (6 weeks is within the original data range of 2–10 weeks), so it's a reasonably reliable prediction.

(b) 40 weeks is far outside the original data range (2–10 weeks), so this is extrapolation. Plants don't typically keep growing at a constant linear rate forever — growth usually slows as the plant matures — so using the line this far beyond the data would likely produce a wildly inaccurate, unreliable prediction.

Practice

Prediction worksheet

Five short questions on prediction. Work through them, then reveal the mark scheme to check.

  1. Explain the difference between interpolation and extrapolation.
  2. A regression line $y=5+2x$ is based on data where x ranges from 0 to 20. Is predicting y when x = 15 an example of interpolation or extrapolation?
  3. Using the same regression line as Q2, predict y when x = 15.
  4. Explain why extrapolated predictions are generally considered less reliable than interpolated ones.
  5. A company uses a regression line based on sales data from the last 6 months to predict sales 3 years into the future. Explain why this prediction is likely to be unreliable, referring to interpolation/extrapolation in your answer.

Mark scheme

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  1. Interpolation is predicting a value within the range of the original data; extrapolation is predicting a value outside that range.
  2. Interpolation (15 is within the range 0 to 20).
  3. y = 5+2(15) = 5+30 = 35.
  4. Extrapolated predictions assume the same linear relationship continues beyond the range of data actually observed, but there's no evidence this is true — real-world relationships often change or stop being linear outside the range you've tested.
  5. Predicting 3 years ahead from only 6 months of data is a large extrapolation — 3 years is far outside the range of the original data set, so there's no guarantee the same sales trend would continue that far into the future.
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