Distinguish interpolation from extrapolation

Interpolation is the estimation of a value within the range of observed data, while extrapolation estimates a value beyond that range using an assumed pattern or model, often represented in a table, graph, or linear relationship. The distinction includes recognizing that extrapolation is generally less reliable because the observed trend may not continue outside the data, supporting reasoned interpretation of models and predictions without extending to advanced statistical modeling.

Detailed Explanation: Distinguish interpolation from extrapolation

Interpolation and extrapolation both use a pattern in observed data, but the key difference is where the estimate is made:

Interpolation estimates a value inside the range of the data.
Extrapolation estimates a value outside the range of the data.
Extrapolation is usually less reliable because the pattern may not continue.

Worked example

A cyclist records the distance traveled after different numbers of hours:

Time (hours)	Distance (km)
1	50
2	100
3	150
4	200

The data show a pattern: the cyclist travels $50$ km each hour.

1. Estimate the distance after $2.5$ hours

The observed times range from $1$ to $4$ hours. Since $2.5$ is within this range, this is interpolation.

The distance at $2$ hours is $100$ km, and the distance at $3$ hours is $150$ km. Halfway between $100$ and $150$ is

\frac{100+150}{2}=125.

So, the estimated distance after $2.5$ hours is

\boxed{125\text{ km}}.

2. Estimate the distance after $6$ hours

Six hours is beyond the largest observed time, which is $4$ hours. Therefore, this is extrapolation.

If the pattern continues at $50$ km per hour, then

6 \times 50=300.

The estimated distance after $6$ hours is

\boxed{300\text{ km}}.

However, this estimate is less certain. The cyclist might slow down, stop, or face a hill, so the pattern from $1$ to $4$ hours may not continue.

To decide which type of estimate you are seeing, first identify the range of the observed data. An estimate within that range is interpolation; an estimate beyond it is extrapolation.

Learn by doing: Distinguish interpolation from extrapolation

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Graphing - Scatter Plot (Linear) - Graph to Interpolation or Extrapolation

Earned ?