Download Article
Download Article
Linear interpolation, also called simply interpolation or “lerping,” [1] X Research source is the ability to deduce a value between two values explicitly stated in a table or on a line graph. While many people can interpolate on an intuitive basis, the article below shows the formalized mathematical approach behind the intuition.
Steps
Download Article
-
Identify the value for which you want to find a corresponding value. Interpolation can be used for such things as finding a logarithm or trigonometric function value or for the corresponding gas pressure or volume for a given temperature in chemistry. [2] X Research source Because scientific calculators have largely replaced logarithmic and trigonometric tables, we’ll use as our example for finding an interpolated value that of finding the pressure of a gas for a temperature whose value isn’t listed in the reference table or as a graph point.- For the equation we’ll derive, we’ll represent the value that we want to find a corresponding value for as ‘’x’’ and the interpolated value we want to find as ‘’y’’. (We’ll use these labels, because on a graph, the value we know would be plotted on the horizontal, or x-axis, and the value we’re trying to find would be plotted on the vertical, or y-axis.)
- Our ‘’x’’ value will be the temperature of the gas, which for this example will be 37 °C (99 °F).
-
Find the closest values below and above the value of x in the table or on the graph. Our reference table doesn’t give a gas pressure for 37 °C (99 °F), but it does list values for 30 °C (86 °F) and 40 °C (104 °F). The gas pressure at 30 °C (86 °F) is 3 kilopascals (kPa) and the pressure at 40 °C (104 °F) is 5 kPa.- Because we’re representing the temperature of 37 °C (99 °F) with ‘’x’’, we’ll represent the temperature of 30 degrees as ‘’x 1
’’ and the value of 40 degrees as ‘’x 2
’’.
- Because we’re representing the pressure were trying to find with ‘’y’’, we’ll represent the pressure of 3 kPa at 30 °C (86 °F) as ‘’y 1
’’ and the pressure of 5 kPa at 40 °C (104 °F) as ‘’y 2
’’.
Advertisement - Because we’re representing the temperature of 37 °C (99 °F) with ‘’x’’, we’ll represent the temperature of 30 degrees as ‘’x 1
’’ and the value of 40 degrees as ‘’x 2
’’.
-
Find the interpolated value mathematically. The equation for finding the interpolated value can be written as y = y 1 + ((x – x 1 )/(x 2 - x 1 ) * (y 2 - y 1 )) [3] X Research source- Plugging in the values for x, x 1
, and x /2
in their places gives (37 – 30)/(40 -30), which reduces to 7/10 or 0.7.
- Plugging in the values for y 1
and y 2
at the end of the equation gives (5 – 3) or 2.
- Multiplying 0.7 by 2 gives a product of 1.4. Adding 1.4 to the value of y 1
, or 3, gives a value of 4.4 kPa. Checking this against our original values, 4.4 falls between 3 kPa at 30 °C (86 °F) and 5 kPa at 40 °C (104 °F), and because 37 is closer to 40 than it is to 30, the result should be closer to 5 kPa than it is to 3 kPa.
- Plugging in the values for x, x 1
, and x /2
in their places gives (37 – 30)/(40 -30), which reduces to 7/10 or 0.7.
Advertisement
Community Q&A
Search
-
QuestionWhat is the interpolation of 27.5?Hibou8Community AnswerWhen you are doing a linear interpolation, you are approximating the function between points as a line. From there, you can find the estimated value at any point between them. Therefore, you cannot ask the interpolation of a single number - rather, you need multiple numbers to determine your line, and from there you can ask what value you would expect at a point.
Ask a Question
200 characters left
Include your email address to get a message when this question is answered.
Submit
Advertisement
Tips
- If you’re good at estimating distances on graphs, you can do a rough interpolation by eyeballing the position of a point against the x-axis to determine the corresponding y-value. [4] X Research source If the example above had been plotted on a graph where the x-axis was marked in units of 10 °C (50 °F) and the y-axis in units of 1 kPa, you could approximate the position of 37 °C (99 °F) and then look at the y-axis to find a point not quite halfway between 4 and 5 kPa. The equation above simply formalizes the thinking process and provides a more precise value.Thanks
- Related to interpolation is extrapolation, which is finding a corresponding value for a given value outside of the range of values listed in a table or concretely plotted on a graph. [5] X Research sourceThanks
Submit a Tip
All tip submissions are carefully reviewed before being published
Name
Please provide your name and last initial
Thanks for submitting a tip for review!
Advertisement
References
- ↑ https://www.khanacademy.org/computing/pixar/animate/parametric-curves/v/animation-5
- ↑ https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/more-significance-testing-videos/v/hypothesis-testing-and-p-values
- ↑ https://www.khanacademy.org/computing/pixar/animate/parametric-curves/v/animation-5
- ↑ https://www.mathsisfun.com/algebra/distance-2-points.html
- ↑ https://www.youtube.com/watch?v=c4_MJg_c49k
About This Article
Thanks to all authors for creating a page that has been read 396,385 times.
Reader Success Stories
- "My boss provided incomplete data for our report and told me to just interpolate it. Last time I encountered interpolation was 12 years ago, so this really helps reminding me how to proceed." ..." more
Advertisement
84%
of readers found this article helpful
.
Click a star to add your vote
% of people told us that this article helped them.
"My boss provided incomplete data for our report and told me to just interpolate it. Last time I encountered interpolation was 12 years ago, so this really helps reminding me how to proceed."
..."
more
"Having not used tables or interpolating for years (something I could do in my sleep previously), your little wikiHow was simple, straightforward and set me straight. Many thanks."
..."
more
"Amazing, well displayed, and easy to grasp steps. wikiHow perfectly displayed how to interpolate."
"Helped me interpolate the activity coefficients for my quantitative analysis course."
"This was a step by step illustrated method of carrying interpolation, simplified!"
