# Root Mean Square as Residual and Shortcut (Especially for GATE-Geospatial 2022)

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Root Mean Square Error is the standard deviation of the residuals. Residuals are a measure of how far from the regression line data points are; RMSE is a measure of how spread out these residuals are.

Root mean square error is commonly used in climatology, forecasting, and regression analysis to verify experimental results.

**The formula is**:

Where:

forecasts (expected values or unknown results) ,

observed values (known results)

The bar above the squared differences is the mean (similar to ) . The same formula can be written with the following,

Where:

summation ( “add up” )

differences, squared

sample size

You can find the RMSE by:

1. Squaring the residuals.

2. Finding the average of the residuals.

3. Taking the square root of the result.

A shortcut to finding the root mean square error is:

Where, is the standard deviation of y.

When standardized observations and forecasts are used as RMSE inputs, there is a direct relationship with the correlation coefficient. For example, if the correlation coefficient is 1, the RMSE will be 0, because all the points lie on the regression line (and therefore there are no errors) .

If the RMS errors exceed the established tolerance, then control points need to be adjusted.