Interpolation is a useful mathematical and statistical tool used to estimate values between two points. In this lesson, you will learn about this tool, its formula and how to use it.
What Is Interpolation?
Interpolation is the process of finding a value between two points on a line or curve. To help us remember what it means, we should think of the first part of the word, 'inter,' as meaning 'enter,' which reminds us to look 'inside' the data we originally had. This tool, interpolation, is not only useful in statistics, but is also useful in science, business or any time there is a need to predict values that fall within two existing data points.
Here's an example that will illustrate the concept of interpolation. A gardener planted a tomato plant and she measured and kept track of its growth every other day. This gardener is a curious person, and she would like to estimate how tall her plant was on the fourth day.
Her table of observations looked like this:
Based on the chart, it's not too difficult to figure out that the plant was probably 6 mm tall on the fourth day. This is because this disciplined tomato plant grew in a linear pattern; there was a linear relationship between the number of days measured and the plant's height growth. Linear pattern means the points created a straight line. We could even estimate by plotting the data on a graph.
But what if the plant was not growing with a convenient linear pattern? What if its growth looked more like this?
What would the gardener do in order to make an estimation based on the above curve? Well, that is where the interpolation formula would come in handy.
The interpolation formula looks like this:
Now, let's explore how to use this formula. The two sets of points between which the estimate can be found are:
Going back to the tomato plant example, the first set of values for day three are (3,4), the second set of values for day five are (5,8), and the value for x is 4 since we want to find the height, y, on the fourth day. After substituting these values into the formula, calculate the estimated height of the plant on the fourth day.
Based on the calculations, the estimated height of the plant on the fourth day is 6 mm.
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With interpolation, sometimes a straight line can be drawn through two points that are on a curve. Then that line can be used to approximate the value at other points on the curve. For instance, if the tomatoes did not have a linear growth pattern, a line could be created from the data points for day 3 to the data points for day 5. Based on this line, the plant's height could be projected. On the graph, this is illustrated by the green dotted lines, which show that the height for the fourth day would be 4 mm.
Interpolation is a way to find values between a pair of data points. The interpolation formula can be used to find the missing value. However, by drawing a straight line through two points on a curve, the value at other points on the curve can be approximated. In the formula for interpolation, x-sub1 and y-sub1 represent the first set of data points of the values observed. X-sub2 and y-sub2 represent the second set of data points. The unknown values are found between these two sets of points.
Interpolation - the process of finding a value between two points on a line or curve
Linear pattern - a pattern in which graphed points create a straight line
Accumulate knowledge of statistical interpolation from this lesson and then attempt to:
Provide definitions for interpolation and linear pattern
Calculate the values between a pair of data points using the interpolation formula
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