How to Interpolate in 5 Steps (With Steps and Examples)

By Indeed Editorial Team

Published April 25, 2022

The Indeed Editorial Team comprises a diverse and talented team of writers, researchers and subject matter experts equipped with Indeed's data and insights to deliver useful tips to help guide your career journey.

Predicting the value between two points, known as interpolation, is a valuable tool in statistics, science, and business. When professionals in these fields collect data, they use a graph or an equation to determine the missing values between the existing values of a data set. Learning how to interpolate is crucial in helping you better understand incomplete data and helps you make detailed interpretations and effective decisions. In this article, we describe how to interpolate, what interpolation is, list some examples, and explain the difference between interpolation and extrapolation.

What is interpolation?

Interpolation estimates the output value between two data sets, allowing for a more accurate prediction than visually predicting the value between two datasets yourself. When collecting and analyzing data, interpolation helps to improve accuracy and expand the data within sets to create more informed conclusions. It can enable businesses to determine more efficient strategies, scientists to discover more effective vaccines, or statisticians to find missing data, for example. Interpolation can help save time from repeating a survey or an experiment to find a missing output value.

How to interpolate

Learning how to interpolate is essential for industry professionals collecting data to determine trends, gain insight, and help make decisions based on the findings. When the data set is incomplete, they use interpolation to find the unknown values between the known ones. Different methods of interpolation include linear, quadratic, and spline interpolation. You can use the linear method if your data set creates a straight line and a calculator or software if the data set creates a more complex shape.

Below are the steps for using linear interpolation:

1. Identify your data

Use a table to list your data. An effective way to do this is by using a spreadsheet program. List your independent variables, or your chosen data set, in one column. Fill the other column with your dependent variables, which comprise the data set you're measuring.

For example, you want to discover the effect of various marketing budgets on the sales of a new product and decide to measure sales based on the following marketing budgets: $1,000, $2,000, $3,000, and $4,000. You can place this data in one column and label it "x." Based on your findings, the numbers of products you sold per marketing budget are 45, 70, 80, and 85. Place this data into the other column and label it "y."

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2. Create a line of best fit

After using the values to plot a graph, you can draw a line of best fit. Plot the graph using the data from the x- and y-axes. You can use a spreadsheet program to create this graph. A line of best fit is a straight line with an equal number of points on either side of it. This line represents the general trend of the data.

3. Determine your value for interpolation

Choose a missing independent value between the existing ones to predict this unknown value using interpolation and find its dependent value. For example, if you want to determine the effect on sales of having a $1,500 marketing budget, use the independent values before and after this value for interpolation. The value preceding your unknown value is a marketing budget of $1,000, leading to the sale of 45 units. The value after it is a marketing budget of $2,000, resulting in 70 units sold.

4. Use the linear interpolation equation

The linear interpolation equation is: y = y1 + (y2-y1) x [(x-x1) / (x2-x1)]. The x value represents the $1,500 marketing budget, while the y value represents the sales resulting from this marketing budget. The x1 and y1 values represent the input and output of the value lower than the unknown value, while the x2 and y2 values represent the input and output of the value higher than the unknown value. The x represents the independent value you want to find the output for, while y represents the output.

Based on the example, here's the formula with the values: y = 45 + (70-45) x [(1,500-1,000) / (2,000-1,000)].

5. Solve the equation

Use the following order of operations to solve the equation:

y = 45 + (70-45) x [(1,500-1,000) / (2,000-1,000)]

y = 45 + (35) x [(1,500-1,000) / (2,000-1,000)]

y = 45 + (35) x [(500) / (1,000)]

y = 45 + (35) x [0.5]

y = 45 +17.5

y = 62.5

According to the linear method of interpolation, a marketing budget of $1,500 leads to the sale of 62.5 units.

Examples of interpolation

Industry professionals use interpolation for different reasons, depending on their industry and the type of data they are trying to uncover. Below are some examples of various industries using interpolation for different reasons:

Investing

Investors and financiers depend on predictions to make financial decisions for themselves and their customers. These professionals create a line chart with interpolated data points to help them visualize the trends in specific stocks and bonds. When they see a positive growth trend, they know this is a profitable investment and suggest to others to invest more. If the trend indicates a negative growth, they advise others to sell their stocks and bonds before the value decreases further. This allows for effective investing and often ensure repeat business if predictions are correct.

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Science

Science involves substantial research and the accumulation of data, so interpolation helps scientists enrich their data sets by adding more values. This can help them save on financial resources and time constraints by reducing the experimentation required to reach a conclusion. In addition, scientists may find scattered data when researching issues such as infection rates and use interpolation to discover more precise correlations and fill in the gaps to help them make those determinations. The interpolation method can offer significant benefits to scientists, including epidemiologists, geologists, climatologists, geneticists, and physiologists.

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Photography

After taking a picture, a photographer cannot simply expand the image without significantly reducing its quality. The only way a photographer can increase the size of the image while maintaining the resolution is by adding more pixels. Photographers use interpolation software to spread the existing pixels into the new image size and create a function that examines the missing pixel points, populating them based on existing pixels. Interpolation helps the photographer enlarge their images without losing the original image quality. This may be important for businesses such as newspapers or people who want to take photos for personal reasons.

Related: 15 Essential Photographer Skills for Your Resume

Engineering

Engineers can use interpolation to increase the precision of deciding which materials to use in construction or to develop software to create enriched content, like animation. By helping to estimate the outcome of the in-between sample data points, engineers use interpolation in their software tools to make more efficient decisions and build more precise models. Software engineers, mechanical engineers, structural engineers, productions engineers, and civil engineers can all benefit from interpolation.

Marketing

Professionals in the marketing field can use interpolation to collect data from customer surveys or analyze their marketing strategies and budgets. Interpolation helps them gather specific information about how certain ranges of activities have affected sales or how certain types of customers react to different marketing strategies. Marketers can use complex statistical software or create graphs and draw appropriate lines of best fit to interpolate.

What is the difference between interpolation and extrapolation?

While interpolation is finding an unknown value within a sequence based on the known values in the sequence, extrapolation uses the known values in a sequence to determine unknown values outside of it. Both techniques use different formulas and seek different outcomes. Interpolation can help professionals find missing values within a data set to add more value to the existing data, while extrapolation can help professionals make predictions about future values. Extrapolation is an effective tool for analyzing future trends and for determining what might happen if the independent value continues to increase.

For example, an investor who wants to make a prediction about a specific return on investment within an investment portfolio can use interpolation to determine these missing values. If the investor wants to advise their client on the portfolio's performance for the next quarter, they can use extrapolation with the current data sets and make an assumption about its future performance. Just as they could create a line of best fit for interpolation, they could also use a line of best fit and extend it further, establishing a visualization of the future trend.

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