Linear interpolation in Excel is a powerful method for estimating unknown values between two data points within a dataset. Imagine working with financial models, engineering calculations, or scientific data where you need to predict or fill in missing values with precision. Linear interpolation allows you to bridge the gap between known values by calculating intermediate points based on a straight-line relationship. Using formulas or built-in functions, Excel makes it easy to perform these calculations and apply them to a range of data. This guide will walk you through the step-by-step process of performing linear interpolation in Excel, complete with practical examples to help you master this essential technique for data analysis.
Linear Interpolation in ExcelWhat is Linear Interpolation
Linear interpolation is a method used in Excel to estimate a value along a straight line within a set of known data points. It is commonly used in mathematics, finance, science, and engineering to predict or calculate a value based on two known values at other points along the same line or interval. Essentially, linear interpolation allows you to fill in the gaps between two known values.
How Linear Interpolation Works
Linear interpolation assumes that the change between two values is linear and that this linear trend can be used to estimate intermediate values. In Excel, this is typically done using a formula based on the slope of the line connecting two points.
In Excel, there are various ways to calculate linear interpolation depending on the complexity of your data and specific requirements. Here are some common methods you can use to implement a linear interpolation formula in Excel:
This is the standard approach when you know two points and want to interpolate a value between them:
𝑦=𝑦1+(𝑦2-𝑦1)/(𝓍2-𝓍1)x(𝓍-𝓍1)
The above Linear Interpolation Equations means:
- 𝓍1 and 𝑦1 are the first coordinates, and
- 𝓍2 and 𝑦2 are the second coordinates
- 𝓍 is the point to perform the interpolation
- 𝑦 is the interpolated value
In Excel, if your data is organized such that 𝓍1, 𝑦1 are in cells A2 and B2 respectively, and 𝓍2, 𝑦2 are in cells A3 and B3, with 𝓍 (the value for which you want to interpolate) in cell A4, the formula in B4 would be:
=B2 + ((B3 - B2) / (A3 - A2)) * (A4 - A2)
We'll use a simple example where you need to interpolate data, which is a common task for estimating values between two known points using the below scenario to explain all the methods and ways to perform Linear Interpolation in Excel.
Example: Suppose you are analyzing temperatures at various depths in a lake and have measurements at 10 meters and 20 meters, but you need to estimate the temperature at 15 meters.
Data Setup:
- Depth 10m: Temperature is 15°C
- Depth 20m: Temperature is 10°C
This method involves manually entering a formula in Excel to interpolate between two points.
Step 1: Enter Your Data
Open Excel and create a new workbook and Input Your Known Data Points:
- In cell A1, type
Depth (meters). - In cell B1, type
Temperature (°C). - Enter the depths and temperatures as follows: A2 as
10, B2 as 15, A3 as 20, B3 as 10.
Enter Your DataStep 2: Calculate Interpolation
- Identify the Depth for Interpolation: Decide which depth you want to interpolate the temperature for (e.g., 15 meters). Enter
15 in cell A4. - Enter the Interpolation Formula in B4:
=B2 + ((A4 - A2) * (B3 - B2) / (A3 - A2))
This formula calculates the temperature at a depth of 15 meters based on the known points.
Calculate Interpolation- Press Enter in cell B4 to see the interpolated temperature.
Important Notes:
- Make sure your lookup value (in this case, depth for interpolation) is between your known data points.
- Excel calculations assume a linear relationship between points.
Execute the FormulaThe FORECAST.LINEAR function is used for predicting a future value along a linear trend. The FORECAST function is ideal for making predictions based on linear relationships in historical data. It's commonly used in financial forecasting and inventory planning where trends are expected to continue.
Note:
- If you are looking to Perform Linear Interpolation Using the FORECAST Function, The FORECAST function is older and mainly available for compatibility with earlier versions of Excel. In Excel 2016 and newer versions, the FORECAST function is replaced with FORECAST.LINEAR. The syntax and usage of the two functions are the same.
- Since FORECAST assumes a linear trend, it may not provide accurate predictions for data with non-linear characteristics or with significant fluctuations.
Example: Suppose you are analyzing temperatures at various depths in a lake and have measurements at 10 meters and 20 meters, but you need to estimate the temperature at 15 meters.
Data Setup:
- Depth 10m: Temperature is 15°C
- Depth 20m: Temperature is 10°C
Step 1: Enter Data
- Follow the same data entry steps as in Method 1.
Step 2: Use FORECAST.LINEAR
- Click on Cell B4: This is where you will see the interpolated result.
- Enter the FORECAST.LINEAR Formula:
=FORECAST.LINEAR(15, B2:B3, A2:A3)
This function predicts the temperature at 15 meters using linear regression based on your data.
Use FORECAST.LINEAR- Press Enter to calculate and display the interpolated temperature in cell B4.
Important Notes:
- Ensure that your known x-values (depths) and y-values (temperatures) are in separate, adjacent columns.
- FORECAST.LINEAR is best used when you have a larger dataset that follows a linear trend.
Linear InterpolationVisualizing data can help confirm the relationship between variables and provide a visual interpolation.
Example: Suppose you are analyzing temperatures at various depths in a lake and have measurements at 10 meters and 20 meters, but you need to estimate the temperature at 15 meters.
Data Setup:
- Depth 10m: Temperature is 15°C
- Depth 20m: Temperature is 10°C
Step 1: Enter Data
- Use the same dataset as in the previous methods.
Step 2: Create a Scatter Plot
Select Your Data: Highlight cells A1:B3. > Insert Scatter Plot:
- Go to the
Insert tab. - Click on
Insert Scatter (X, Y) or Bubble Chart. - Choose
Scatter.
Create a Scatter PlotStep 3: Add a Trendline
Click on the Chart: This makes the Chart Tools appear in the ribbon. > Add Trendline:
- Go to the
Chart Design tab. - Click
Add Chart Element. - Hover over
Trendline and select Linear. - Check the box for
Display Equation on chart to see the interpolation formula.
Add a TrendlineStep 4: Use the Equation for Interpolation
- The equation displayed on the chart can be used manually to calculate temperatures at any given depth not included in your dataset.
Use the Equation for InterpolationImportant Notes:
- Adding a trendline is useful for seeing the overall trend and can guide in making interpolation decisions.
- The displayed equation on the chart represents the line of best fit based on your data.
Dynamic Range Interpolation with FORECAST, OFFSET, and MATCH
Combining FORECAST with OFFSET and MATCH allows for dynamic interpolation in datasets where the range of data can vary. This is particularly useful in scientific data analysis where experimental data points may not be uniformly spaced. Often used in environmental science to interpolate temperature or pollution levels at unmeasured points between known data samples.
Note:
- This approach leverages both older functions like FORECAST and versatile functions like OFFSET and MATCH, making it adaptable to modern complex datasets.
- This method can be error-prone if MATCH does not find a close enough match within the dataset, potentially leading to inaccurate interpolations.
Example: Suppose you are analyzing temperatures at various depths in a lake and have measurements at 10 meters and 20 meters, but you need to estimate the temperature at 15 meters.
Data Setup:
- Depth 10m: Temperature is 15°C
- Depth 20m: Temperature is 10°C
Step 1: Define Your Data Range
- Input your data as before, potentially with more data points.
Click on Cell B4 where you want the result and Press Enter and check the interpolated value.
- Enter the Formula in your results cell:
=FORECAST(15, OFFSET(A1, MATCH(15, A2:A100, 1), 1, 2, 1), A2:A3)
This formula uses MATCH to find the position closest to the interpolation point, then OFFSET to adjust the range dynamically.
Implement the Combined FormulaUsing the TREND Function for Linear Interpolation
Performing linear interpolation using the TREND function in Excel is particularly useful when you have multiple data points and want to estimate values (interpolate) at specific points based on a linear trend. This function is capable of handling arrays and returning multiple interpolated values, making it vital for detailed data analysis. Commonly used in epidemiology to predict disease spread based on existing case data.
Note:
- TREND is a well-established Excel function, still widely used due to its accuracy in forecasting and analysis of trends in larger or more complex datasets.
- Like FORECAST, TREND assumes a linear relationship and may not be suitable for datasets where trends change direction or are cyclical.
Example: Suppose you're managing production data and need to interpolate daily output levels based on recorded weekly data to better understand daily trends.
Here's an example of how you might have data laid out in Excel:
| Day (Weekly Recorded) | Output (Units) |
|---|
| 1 | 150 |
| 8 | 200 |
| 15 | 250 |
You want to interpolate the output for every day of the week, not just the days you have records for.
- In cell A1, type
Day (Weekly Recorded). - In cell B1, type
Output (Units). - Enter the days in cells A2 through A4 (1, 8, 15).
- Enter the corresponding outputs in cells B2 through B4 (150, 200, 250).
Input DataStep 2: Set Up Interpolation Points
Now you have to Enter Days for Interpolation
- In column C, list all the days you want to interpolate outputs for, say days 1 through 15.
- You can use Excel's auto-fill feature to quickly fill these days: Enter 1 in cell C2, 2 in C3, select both cells, then drag down the fill handle to auto-populate up to 15.
Set Up Interpolation PointsStep 3: Apply the TREND Function
- Select the Range for Output Results: Click on cell D2.
- Enter the TREND Formula:
=TREND(B2:B4, A2:A4, C2:C16)
- B2:B4 is the range of known y-values (outputs).
- A2:A4 is the range of known x-values (days).
- C2:C16 is the range of new x-values where you want to estimate the y-values (outputs for days 1 through 15).
Apply the TREND Function- After entering the formula in cell D2, drag the fill handle down through cell D16 to fill the formula for each day.
Drag the Formula DownImportant Notes:
- Accuracy and Applicability: TREND assumes a linear relationship between the data points. If the actual relationship isn’t linear, this method might not give accurate predictions for days not specifically recorded.
- Data Coverage: Ensure that the data points used (A2:A4 and B2:B4) are representative of the overall trend you expect in the dataset. Sparse or skewed data might lead to misleading interpolations.
- Use of Array Formula: If using older versions of Excel, you might need to enter the TREND function as an array formula by pressing
Ctrl+Shift+Enter instead of just Enter.
Using the INTERCEPT Function for Linear Interpolation
Using the INTERCEPT function in Excel is a robust way to engage in linear interpolation when combined with the SLOPE function. The INTERCEPT function calculates the y-intercept (b) of the best-fit line (y = mx + b) for a dataset based on the linear regression principle. Here’s how to perform linear interpolation using both the INTERCEPT and SLOPE functions for a precise estimation of unknown data points.
Note:
- INTERCEPT is an older statistical function in Excel, still used for its direct approach to finding starting values in linear data models.
- The accuracy of INTERCEPT can be compromised if the data points do not actually conform to a linear model or if outliers are present.
Example: Imagine you're a market analyst tracking the relationship between advertising spend and sales performance. You have data points for several advertising budgets and the corresponding sales outcomes but need to interpolate potential sales for an unrecorded budget amount.
Your data might look like this in Excel:
| Advertising Spend ($'000) | Sales ($'000) |
|---|
| 50 | 300 |
| 100 | 450 |
| 150 | 550 |
You want to interpolate the sales for an advertising spend of $75,000.
Step 1: Enter Your Data
Open Excel and create a new workbook. > Input Your Known Data Points:
- In cell A1, type
Advertising Spend ($'000). - In cell B1, type
Sales ($'000). - Enter the advertising budgets in cells A2 through A4 (50, 100, 150).
- Enter the corresponding sales outcomes in cells B2 through B4 (300, 450, 550).
Enter Your DataStep 2: Calculate the Slope (m) and Intercept (b)
Calculate SLOPE:
Select cell C1 and type Slope > Click on cell C2 and enter the formula:
=SLOPE(B2:B4, A2:A4)
Press Enter. This calculates the slope (m) of the line that best fits your data.
Calculate the SlopeCalculate Intercept:
- Select cell D1 and type
Intercept. - Click on cell D2 and enter the formula:
=INTERCEPT(B2:B4, A2:A4)
- Press Enter. This calculates the y-intercept (b) of the best-fit line.
Calculate InterceptInterpolate Sales for $75k Spend:
- Select cell E1 and type
Estimated Sales for $75k Spend. - Click on cell E2 and enter the formula to calculate the sales based on the linear equation (y = mx + b):
=C2*75 + D2
- Here,
C2 contains the slope, and D2 contains the intercept. The value 75 represents the unrecorded advertising spend. - Press Enter.
Perform Linear InterpolationImportant Notes:
- Accuracy of Interpolation: The INTERCEPT and SLOPE functions assume a linear relationship between the variables. If your actual data relationship is nonlinear, this approach might yield less accurate interpolations.
- Data Suitability: This method is most effective when your data points are reasonably consistent and distributed along a clear trend line. Outliers or highly variable data can significantly affect the slope and intercept calculations, leading to misleading results.
- Use in Decision Making: The calculated interpolated values should be used as estimates within the context of broader data analysis. They are particularly useful for predicting outcomes within the range of existing data but should be validated against additional data or industry benchmarks if possible.
Using XLOOKUP for Interpolation
Using the XLOOKUP function for interpolation in Excel involves a slightly different approach because XLOOKUP itself doesn't calculate between points but can be used to retrieve the closest higher or lower values necessary for a manual interpolation. Below is the method of using XLOOKUP to perform a simple form of interpolation by first retrieving adjacent values and then manually calculating the interpolated value.
Note:
- Accuracy: This method assumes a linear relationship between the data points. If the actual relationship is non-linear, the interpolated value may not be accurate.
- Data Sparsity: The more spaced apart your known data points, the less accurate your interpolation might be. Closer data points generally provide more reliable interpolations.
- Boundary Values: If your interpolation spending target is outside the range of known spending values, this approach will not work properly, and you might need extrapolation or a different method.
Example: Suppose you are analyzing the relationship between advertising spending and sales revenue. You have data for certain spending levels but need to interpolate to estimate revenue at a spending level that isn't directly observed in your dataset.
Let's say you have the following data in Excel:
| Spending ($000) | Revenue ($000) |
|---|
| 50 | 300 |
| 100 | 450 |
| 150 | 550 |
You want to interpolate the revenue for a spending level of $75.
Note: This method is not a true interpolation but can be used for simple lookup.
Step 1: Enter Your Data
Open Excel and create a new workbook and Input Your Known Data Points:
- In cell A1, type
Spending ($000). - In cell B1, type
Revenue ($000). - Enter the spending levels and revenues as follows: A2 as
50, B2 as 300, A3 as 100, B3 as 450, A4 as 150, B4 as 550.
Step 2: Identify Points for Interpolation
You need to find the revenue values for spending just above and just below $75.
Step 3: Use XLOOKUP to Find Adjacent Values
Find Lower Adjacent Value:
- Select cell C2 for the revenue when spending is just below $75.
- Enter the XLOOKUP formula to find the closest lower spending level:
=XLOOKUP(75, A2:A4, B2:B4, , -1)
This function looks for $75 in the spending column. The -1 at the end instructs XLOOKUP to find the exact match or the next smallest value if no exact match is found.
Find Lower Adjacent ValueFind Higher Adjacent Value:
- Select cell C3 for the revenue when spending is just above $75.
- Enter the XLOOKUP formula to find the closest higher spending level:
=XLOOKUP(75, A2:A4, B2:B4, , 1)
This function setup similarly looks for $75, but 1 instructs XLOOKUP to return the next largest value if no exact match exists.
Find Higher Adjacent ValueStep 4: Calculate Interpolation Manually
- Select a Cell for the Interpolated Result: Click cell D2 where you want to display the interpolated revenue.
- Enter the Interpolation Formula:
=C2 + ((75 - 50) / (100 - 50)) * (C3 - C2)
Here, C2 and C3 are the revenues at spending levels $50 and $100, respectively.
Press Enter in cell D2. Excel calculates the interpolated revenue, providing an estimate based on the linear trend between the two adjacent points.
Calculate Interpolation ManuallyInterpolation with a Defined Name Formula in Excel allows you to create custom formulas that you can save and use consistently across different sheets or workbooks without the need to retype or recreate complex calculations. This method is especially beneficial for repeated interpolation tasks within similar datasets. Below is a step-by-step guide on setting up and using a Defined Name for interpolation in Excel:
Example: Imagine you are a data analyst needing to regularly interpolate monthly sales figures based on quarterly reported values. You want a reusable formula that can be easily applied to various datasets within the company.
You might have quarterly sales data as follows:
| Month | Sales |
|---|
| Q1 | 150 |
| Q2 | 200 |
| Q3 | 250 |
You need to interpolate sales figures for the months within these quarters.
Step 1: Enter Your Data
Open Excel and start a new workbook. > Input Your Known Data Points:
- In cell A1, type
Quarter. - In cell B1, type
Sales. - Enter the quarter labels in cells A2 through A4 (Q1, Q2, Q3).
- Enter the corresponding sales figures in cells B2 through B4 (150, 200, 250).
Open the Name Manager:
- Go to the Formulas tab > Click on Name Manager.
- Press New to create a new named formula.
Create a Named Formula:
- In the Name field, type a name for your formula, e.g.,
LinearInterpolation. - In the Refers to field, enter your interpolation formula using Excel formula syntax. For example:
=TREND(INDIRECT("B" & MATCH(Sheet1!$A$6,Sheet1!$A$2:$A$4,0)+1):INDIRECT("B" & MATCH(Sheet1!$A$6,Sheet1!$A$2:$A$4,0)+2), ROW(INDIRECT("1:" & ROWS(Sheet1!$A$2:$A$4))), ROW(Sheet1!$A$6), TRUE)- This formula assumes you have input cells to determine the range for interpolation dynamically based on a specific month in cell A6 of Sheet1.
- Click OK to save the defined formula.
Define Your Interpolation FormulaUse the Named Formula in a Cell:
- Click in the cell where you want the interpolated value to appear.
- Enter
= followed by the name of your defined formula, such as:
=LinearInterpolation
Press Enter
Important Notes:
- Flexibility and Reusability: Once defined, this named formula can be used across different sheets or workbooks without the need to rewrite the interpolation logic. This ensures consistency in how data is interpolated across your analyses.
- Complexity Management: Defined names hide complex formulas behind simple names, making your worksheets cleaner and easier to understand.
- Error Handling: Ensure that the ranges and references within your defined formula are correctly set to prevent errors. Errors in the defined name formula are not always immediately apparent because they don't show up until the formula is used in a sheet.
Using the LINEST Function for Linear Interpolation
Using the LINEST function for interpolation in Excel is particularly powerful when you need to perform linear regression on a set of data and then use the regression line to interpolate or even extrapolate values. The LINEST function returns statistics about the line by fitting a line to the data points using the least squares method, and you can use the output to calculate specific y-values for given x-values.
Suppose you are a financial analyst looking to interpolate the expected revenue for certain investment levels based on historical data points. You have investment amounts and their corresponding returns but need to interpolate potential returns for unrecorded investment amounts.
Here's a simple dataset:
| Investment ($000) | Return ($000) |
|---|
| 100 | 15 |
| 200 | 30 |
| 300 | 45 |
You need to interpolate the return for an investment of $250,000.
Step 1: Enter Your Data
Open Excel and create a new workbook > Input Your Known Data Points:
- In cell A1, type
Investment ($000). - In cell B1, type
Return ($000). - Enter the investment amounts in cells A2 through A4 (100, 200, 300).
- Enter the corresponding returns in cells B2 through B4 (15, 30, 45).
Step 2: Use LINEST to Find the Slope and Intercept
Select Two Cells for the Output: Because LINEST can output multiple statistics, select two adjacent cells horizontally, say C2 and D2.
Enter the LINEST Formula:
- Type the following formula and press
Ctrl+Shift+Enter (as it is an array formula):
=LINEST(B2:B4, A2:A4, TRUE, TRUE)
This will fill C2 with the slope (m) and D2 with the intercept (b).
Enter the LINEST FormulaInterpolate the Return for $250k Investment:
- Select cell E2 where you want to display the interpolated return.
- Enter the formula to calculate the return based on the linear equation y=mx+by = mx + by=mx+b
=C2*250 + D2
Here, C2 contains the slope, and D2 contains the intercept. The value 250 represents the unrecorded investment amount.
Press Enter in cell E2. Excel calculates the interpolated return, providing an estimate based on the linear relationship established from your data.
Perform Linear InterpolationImportant Notes:
- Accuracy of Interpolation: LINEST assumes that the relationship between your variables is linear. If your actual data relationship is nonlinear, this method might not yield accurate interpolations.
- Data Suitability: This method is most effective when your data points are consistent and closely distributed along a clear trend line. Sparse or highly variable data can significantly affect the slope and intercept calculations, leading to misleading results.
- Usage Flexibility: LINEST is particularly useful for datasets where you may also be interested in obtaining detailed regression statistics such as the R-squared value, standard errors, or F-statistics, which can provide deeper insights into the reliability of your regression model.
Linear Interpolation VBA Function in Excel
Using VBA (Visual Basic for Applications) to perform interpolation in Excel allows for creating highly customizable solutions, especially when built-in Excel functions may not directly meet your needs. This method is ideal for complex interpolation scenarios or when automation is required across numerous data sets or repeated tasks.
Example: Imagine you need to interpolate temperature data from a given dataset where measurements are irregularly spaced in time. You want a VBA function that can interpolate values for specific time points that aren't directly measured.
You might have temperature data as follows:
| Time (Hours) | Temperature (°C) |
|---|
| 1 | 22 |
| 3 | 24 |
| 5 | 25 |
You need to interpolate the temperature for time = 2 and 4 hours.
Step 1: Open the VBA Editor
- Open Excel and the workbook where you want to perform interpolation.
- Access the VBA Editor:
Press ALT + F11 to open the VBA Editor.
Step 2: Insert a New Module
- In the VBA Editor, right-click on any of the objects in the Project Explorer.
- Choose
Insert > Module. This action opens a new module window.
Step 3: Write the Interpolation Function
- Type the Following Code into the new module:
Function LinearInterpolation(x As Double, x1 As Double, y1 As Double, x2 As Double, y2 As Double) As Double
' Calculate the slope (m) and the intercept (b)
Dim m As Double
Dim b As Double
m = (y2 - y1) / (x2 - x1)
b = y1 - m * x1
' Calculate the interpolated y value
LinearInterpolation = m * x + b
End Function
- This function takes five parameters: the x value where you want to interpolate (
x), and the coordinates of the two data points between which you want to interpolate (x1, y1 for the first point and x2, y2 for the second point). - The function calculates the slope
m and y-intercept b of the line between the two points, then uses these to calculate and return the interpolated y value at x.
Step 4: Use the Interpolation Function in Excel
- Return to Excel.
- In your worksheet, identify where you want to display the interpolated values.
- Enter the Function Like Any Excel Function:
- For example, to interpolate the temperature at time = 2 hours, enter in a cell:
=LinearInterpolation(2, 1, 22, 3, 24)
Press Enter to see the interpolated temperature.
Important Notes:
- Testing and Validation: It's crucial to thoroughly test the VBA function with known data points to ensure it behaves as expected. Make sure to handle potential errors such as division by zero when x1 and x2 are the same.
- Security Settings: VBA macros can pose security risks, especially when enabled in documents from unknown sources. Always ensure your Excel settings and macros are managed securely.
- Portability: Remember that VBA scripts are part of Excel files and may not run on systems where macros are disabled or unsupported (e.g., some mobile versions of Excel).
- yyy is the interpolated value you want to find.
- xxx is the point at which you want to interpolate.
- x1x_1x1 and x2x_2x2 are the x-values of the two known data points.
- y1y_1y1 and y2y_2y2 are the y-values of the two known data points.
This formula calculates the y-value at point xxx based on a straight line between (x1x_1x1, y1y_1y1) and (x2x_2x2, y2y_2y2).
How to interpolate a line graph in Excel?
Interpolating a line graph in Excel involves creating a chart from your data and then adding a trendline that estimates values between your actual data points. Here’s a quick step-by-step guide:
- Enter Your Data: Input your X values (e.g., time) in one column and Y values (e.g., temperature) in another column.
- Create a Line Graph:
- Highlight your data.
- Go to the Insert tab.
- Choose Line Chart from the Chart options.
- Add a Trendline:
- Click on the line graph to select it.
- Right-click on the data series (the line itself).
- Select Add Trendline from the context menu.
- In the Trendline Options, choose Linear for linear interpolation.
- Check the box for Display Equation on chart if you want the interpolation formula shown on the graph.
This process will visually interpolate between your plotted data points, giving a smooth line that extends across your graph based on the linear trend established by your data.
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