Interpolation
Interpolation is the process of estimating an unknown value that falls between two known data points. Instead of guessing, you use the pattern in your existing data to predict what the value should be at a point inside your data range.
Interpolation is a method of constructing new data points within the range of a discrete set of known data points. In its simplest form — linear interpolation — a straight line is drawn between two adjacent known values, and the estimate is read from that line. Because the estimate lies within the observed data range, interpolation is generally more reliable than extrapolation, which estimates values outside that range.
Key Formula
Where:
- = the input value at which you want to estimate the output
- = the coordinates of the known data point just below x
- = the coordinates of the known data point just above x
- = the estimated output value
Worked Example
Problem: A weather station recorded the temperature at 8:00 AM as 12 °C and at 12:00 PM as 20 °C. Estimate the temperature at 10:00 AM using linear interpolation.
Step 1: Identify the known data points. Let x represent the hour of the day.
(x_1, y_1) = (8, 12)$ and $(x_2, y_2) = (12, 20)
Step 2: Set x to the time you want to estimate.
Step 3: Substitute into the linear interpolation formula.
Step 4: Simplify the expression.
Answer: The estimated temperature at 10:00 AM is 16 °C.
Visualization
Why It Matters
Interpolation appears constantly in statistics, science, and engineering whenever you have a table of values and need an estimate between entries. In AP Statistics, you use a regression line to interpolate predicted values within your observed data range. Understanding the difference between interpolation (within the data) and extrapolation (outside the data) is essential for judging how trustworthy a prediction really is.
Common Mistakes
Mistake: Confusing interpolation with extrapolation
Correction: Interpolation estimates within the range of your data; extrapolation estimates beyond it. Extrapolation carries much more risk of being inaccurate because you're assuming the pattern continues outside the observed values.
Mistake: Assuming linear interpolation is always accurate
Correction: Linear interpolation connects two points with a straight line, which works well when data changes gradually. If the underlying relationship is curved, a linear estimate can be noticeably off. Consider whether a higher-order method or a regression model is more appropriate.