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

This link says

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.

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3 Answers 3

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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!

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

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  • $\begingroup$ This answer would be improved if you used math typsetting. More information: math.meta.stackexchange.com/questions/5020/… $\endgroup$
    – Sycorax
    Mar 21, 2022 at 20:04
  • $\begingroup$ "find the difference of each row, then sum the differences, and square it" Not quite, you're summing the squared differences. $\endgroup$
    – ʀᴏʙ
    Sep 13, 2023 at 6:47
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We're in the middle of the Australian Open as I write this, so forgive me if I am stretching analogies.

Suppose we're playing tennis, and our random variable is where the opponent's ball will cross the baseline (the back line of the court).

Some opponents will play it safe. They always hit near the middle of the court. In that case, the Standard Deviation is low. Other opponents will constantly aim for the sidelines. Assuming they go for both the forehand and backhand sidelines, the Standard Deviation will be higher. SD is a property of the population.

The better we are at predicting where any given shot will cross the baseline, the lower our Root Mean Squared Error. RMSE is a property of our estimates. If our forecasts are correct (and we have enough athletic ability), then our RMSE can be much lower than the SD, hopefully close to zero. If we always guess the wrong direction (we're unlikely to play in the Open in that case), our RMSE can be much higher than the SD.

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