>Introduction to Modeling>What Is The Difference Between Interpolation and Extrapolation?

What Is The Difference Between Interpolation and Extrapolation?

Once you have a data set, you can find a suitable model or function using regression. The range between the lowest and highest observed $x$ -value is called the validity range of the model. Within this area, as long as you have chosen a suitable model or function, you can be confident that your results correspond relatively well with reality. But if you look at values outside this range, the results might deviate a lot from reality.

You can use the model you have chosen to calculate the function value ( $y$ -value) of points other than the specific data points you were given.

Theory

Interpolation

When you find the function value of a point that is within the validity range of your model, you perform an interpolation. Selecting points within the domain of the model will give you the most reliable results.

Theory

Extrapolation

When you find the function value of a point that is outside the validity range, you’re performing an extrapolation. This result can deviate significantly from the real values.

Example 1

Jay-Z weighed $3.2 kg$ when he was born. The table below shows his weight, $y$ kg, $x$ days after his birth.

Age ( $x$ days) Weight ( $y$ kg)
0 $3.2$
7 $3.5$
20 $4.1$
30 $4.5$
57 $5.7$

Use regression to find a linear model for Jay Z’s weight from the data in the table.

From the table you can see that the validity range of the model is determined by the $x$ values in the interval $[0, 57]$ . You use digital tools to find the linear model for Jay Z’s weight $x$ days after his birth. The model is the function

y = 0.04 x + 3.2

Based on the model you found, how much did Jay Z weigh after 14 days?

You insert $14$ for $x$ in the function and find Jay Z’s weight after $14$ days:

\begin{array}{l} Weight & = 0.04 \cdot 14 + 3.2 \\ = 3.76 kg \end{array}

\begin{array}{l} Weight after 14 days = 0.04 \cdot 14 + 3.2 = 3.76 kg \end{array}

According to the model you found above, Jay Z weighed

3.76

14

days after his birth. Because you selected a value within the domain (

x = 14

), you performed an interpolation.

How much did Jay Z weigh after one year (365 days), based on the model you found?

You insert $365$ for $x$ and calculate Jay Z’s weight after $365$ days:

\begin{array}{l} Weight & = 0.04 \cdot 365 + 3.2 \\ = 17.8 kg \end{array}

\begin{array}{l} Weight after 365 days = 0.04 \cdot 365 + 3.2 = 17.8 kg \end{array}

According to the model you found above, Jay Z weighed

17.8

kg one year (

365

days) after his birth.

Because a one-year-old normally weighs between $8.0$ kg and $12.0$ kg, Jay Z was either extremely overweight, or your result is so different due to the value you used is outside the validity range of your data set.

It is more likely that the model’s accuracy is poor outside the validity range, rather than that Jay Z was extremely overweight.

Because you selected a value outside the range ( $x = 365$ ), you performed an extrapolation.

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Age ( $x$ days)	Weight ( $y$ kg)

0	$3.2$

7	$3.5$

20	$4.1$

30	$4.5$

57	$5.7$