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Correlation and linear regression

Predict in the right direction · Diagrams

Interpolation and extrapolation

For direct prediction of a response y from an explanatory input x, compare the input with its observed range. This source illustration is schematic and supplies no numerical observations.

The dashed guides mark the smallest and largest observed inputs. The coloured bar spans this input range.

Below the observed inputs

Extrapolation. The explanatory input is smaller than every input observed in the sample.

x0<xmin⁡x_0<x_{\min}

Within the observed input range

Interpolation. The input lies between the smallest and largest observed inputs, including the limits.

xmin⁡≤x0≤xmax⁡x_{\min}\le x_0\le x_{\max}

Above the observed inputs

Extrapolation. The explanatory input is larger than every input observed in the sample.

x0>xmax⁡x_0>x_{\max}
  • Check the explanatory-variable range. For a y-on-x prediction, compare the known x input with its observed x range, rather than with the response y range.
  • Being inside the range does not guarantee accuracy. Check that the scatter supports the model; two observations alone cannot establish an underlying linear relationship.
  • Keep the population and measurement context relevant. A strong correlation within a sample does not establish that the relationship continues outside the observed inputs.

Compare with the source range sketch

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Twelve-point source-role rising scatter between two dashed explanatory-variable limits. Predictions inside these limits are interpolation; predictions on either side are extrapolation. Positions are schematic and no numerical observations are supplied. The plotted horizontal limits describe input range, not response range.
Twelve-point source-role rising scatter between two dashed explanatory-variable limits. Predictions inside these limits are interpolation; predictions on either side are extrapolation. Positions are schematic and no numerical observations are supplied. The plotted horizontal limits describe input range, not response range.
Twelve-point source-role rising scatter between two dashed explanatory-variable limits. Predictions inside these limits are interpolation; predictions on either side are extrapolation. Positions are schematic and no numerical observations are supplied. The plotted horizontal limits describe input range, not response range.
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