# RMSE vs Standard deviation in population

RMSE (Root mean square error) and SD (Standard deviation) have similar formulas.

The only difference is that you divide by $n$ and not $n−1$ since you are not subtracting the sample mean here. The RMSE would then correspond to $\sigma$ . Therefore, the population RMSE is $\sigma$ and you want a CI for that.

So I want to know whether RMSE and SD are the same. Also, I want reference about it.

TLDR; While the formulas may be similar, RMSE and standard deviation have different usage.

You are right that both standard deviation and RMSE are similar because they are square roots of squared differences between some values. Nonetheless, they are not the same. Standard deviation is used to measure the spread of data around the mean, while RMSE is used to measure distance between some values and prediction for those values. RMSE is generally used to measure the error of prediction, i.e. how much the predictions you made differ from the predicted data. If you use mean as your prediction for all the cases, then RMSE and SD will be exactly the same.

As a sidenote, you may notice that mean is a value that minimizes the squared distance to all the values in the sample. This is the reason why we use standard deviation along with it -- they are related species!

– Tim
Jun 27, 2017 at 13:40
• @Chill2Macht but then you will be calculating standard deviation.
– Tim
Jun 27, 2017 at 14:12
• @Chill2Macht RMSE is not sd of errors. Sd(errors) = mean((errors - mean(errors))^2) while rmse = mean(errors^2)
– Tim
Jun 27, 2017 at 16:47
• Worth noting that as the mean error approaches 0 and n approaches infinity sd and rmse converge. Aug 15, 2017 at 9:18
• @Tim I think you're missing the square root. It should be Sd(errors) = square root( mean((errors - mean(errors))^2)) Nov 25, 2019 at 12:50

This will make bit clear, RMSE calculated between two sets, eg: set and predicted set, to calculate the error, eg :

price Vs predicted price
10         12
12         10
13         17


$${RMSE}=\sqrt{\frac{\sum_{i=1}^N{(F_i - O_i)^2}}{N}}$$ f = forecasts (expected values or unknown results), o = observed values (known results).

we find the difference of each row, then sum the differences, and square it, divided by N and finally root... (or you can use a single fixed predicted value and subtract from all rows)

RMSD will use a single set to calculate the spread, (not between predicted, but itself)

price
10
12
13


$${RMSD}=\sqrt{\frac{\sum_{i=1}^N{(x_i - \mu_i)^2}}{N}}$$ μ is the average value