# Root Mean Square Error RMSE in GIS

Last Updated: Aug 11, 2017

### What is Root Mean Square Error RMSE?

Root Mean Square Error (RMSE) (also known as Root Mean Square Deviation) is one of the most widely used statistics in GIS.

We use RMSE in a variety of applications when we need to compare two data sets.

RMSE measures how much error there is between two datasets. In other words, it compares a predicted value and an observed or known value.

Let’s examine RMSE with a bit more detail.

### 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.

RMSE Formula:

$RMSE = {\sqrt {\frac{1} {N}{\sum\limits_{i = 1}^N {(x_{i} - \hat{x}_{i} } })^{2} } }$

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

2. If you have 10 observations, place observed elevation values in A2 to A11. Place predicted values in B2 to B11.

3. In column C2, subtract observed value and predicted value: =A2-B2. Repeat for all rows below where predicted and observed values exist.

4. 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.

### What’s Next?

RMSE quantifies how different a set of values are.

The smaller an RMSE value, the closer predicted and observed values are.

Give this quick RMSE guide a try to master one of the most widely used statistics in GIS…

Root Mean Square Error

#### 7 Comments

1. Sedigheh says:

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

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

3. G. H. Sambo says:

Hello,
How do you interprete the result of RMSE?

4. kevin says:

can you calculate within arcmap ?

5. Ritabrata Roy says:

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))

6. Tasya says:

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

7. GISGeography says:

Yes, that is a good example of using RMSE