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### What is Root Mean Square Error RMSE?

Root Mean Square Error (RMSE) measures **how much error there is between two data sets**. In other words, it compares a predicted value and an observed or known value.

It’s also known as **Root Mean Square Deviation** and is one of the most widely used statistics in GIS.

Different than Mean Absolute Error (MAE), we use RMSE in a variety of applications when compare two data sets.

Here’s an example of how to calculate RMSE in Excel with 10 observed and predicted values. But you can apply this same calculation to any size data set.

### Root Mean Square Error Example

For example, we can compare a predicted LiDAR elevation point with a surveyed ground measurement (observed value).

- Predicted value: LiDAR elevation value
- Observed value: Surveyed elevation value

Root mean square error takes the difference for each LiDAR value and surveyed value.

You can swap the order of subtraction because the next step is to take the square of the difference. This is because the square of a negative value will always be a positive value.

But just make sure that you keep the same order through out.

After that, divide the sum of all values by the number of observations. Finally, we get a RMSE value.

Here’s what the **RMSE Formula** looks like:

### How to Calculate RMSE in Excel

Here is a **quick and easy guide to calculate RMSE in Excel**. You will need a set of observed and predicted values:

### 1 Enter headers

In cell A1, type “observed value” as a header. For cell B1, type “predicted value”. In C2, type “difference”.

### 2 Place values in columns

If you have 10 observations, place observed elevation values in A2 to A11. In addition, populate predicted values in cells B2 to B11 of the spreadsheet

### 3 Find the difference between observed and predicted values

In column C2, subtract observed value and predicted value. Repeat for all rows below where predicted and observed values exist.

`=A2-B2`

Now, these values could be positive or negative.

### 4 Calculate the root mean square error value

In cell D2, use the following formula to calculate RMSE:

`=SQRT(SUMSQ(C2:C11)/COUNTA(C2:C11))`

Cell D2 is the root mean square error value. And save your work because you’re finished.

If you have a smaller value, this means that predicted values are close to observed values. And vice versa.

### What’s Next?

RMSE quantifies how different a set of values are. The smaller an RMSE value, the closer predicted and observed values are.

If you’ve tested this **RMSE guide**, you can try to master some other widely used statistics in GIS:

- Use Principal Component Analysis to Eliminate Redundant Data
- How to Build Spatial Regression Models in ArcGIS
- Spatial Autocorrelation and Moran’s I

very good. thanks a lot.!!!!!!!

Hello,

How do we calculate the RMSE with GCPs. What would be the predicted value?

Hello,

How do you interprete the result of RMSE?

can you calculate within arcmap ?

There is no need to create the C column, this Excel formula can calculate the RMSE from the A and B columns only.

=SQRT(SUMXMY2(A2:A11,B2:B11)/COUNTA(A2:A11))

Can we use RMSE to compare land surface temperature from Landsat (predicted value) with surveyed measurment (observed value) of land surface temperature?

Yes, that is a good example of using RMSE

how to improve the RMSE?

Better predicted values that are closer to actual values

What is the unit of MSE and RMSE in observed and satellite precipitation?

How to interpret the result of RMSE?

Because you’re subtracting predicted with actual values… you can interpret it that the closer it is to 0, the closer actual values are to predicted values. That means a lower RMSE, the better or more accurate it is. I can’t think of a circumstance that this isn’t true.

Can we get a distribution of RMSE ? I think I need to know how to properly size the number of error measurements needed of a single design point so that I can have a way of calculating (or measuring) the RMSE at that design point.

This is so clear explanation of RMSE. Thanks a lot 🙂