Root Mean Squared Error (RMSE)

A type of metric. It is a value that tells us, on average, how close a set of predicted values is from the actual values. The smaller the RMSE, the better.

It is calculated by:

  1. First taking the difference between each respective predicted and actual value.
  2. Then the squaring all obtained values.
  3. Taking the average.
  4. And finally taking the square root.

Let’s say we have a set of predicted values \(1, 2, 3, 4\). The set of actual values is \(1, 4, 3, 3\)

  1. \(1-1, 2-4, 3-3, 4-3, = 0, -2, 0, 1\)
  2. \((0)^2, (-2)^2, (0)^2, (1)^2 = 0, 4, 0, 1\)
  3. \(\frac{0 + 4 + 0 + 1}{4} = \frac{5}{4} = 1.25\)
  4. \(\sqrt{1.25} \approx 1.12\)

This tells us, that on average, our set of predicted values is \(1.12\) units off from the actual values.

In a nutshell, you take the root of the mean of the square of the differences between the predicted and actual values.

The main difference between MSE and RMSE is that RMSE undoes the squaring step by taking the square root.

The reason the square value is taken is due to the averaging step. Let’s say the first predicted value is off from the first actual value by \(-3\) units. And let’s say that the second predicted value is off from the second actual value by \(3\) units.

If we didn’t take the square, the average would be zero \(\left( \frac{-3 + 3}{2} = \frac{0}{2} = 0 \right)\). This is incorrect as both values are off from the actual value.

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