Shouldn't it equal, at least for positive numbers, the mean error?

The formula for RMS is √∑((θ̂̂̂̈̂-θ)²)

However, shouldn't it be ∑(√(θ̂̂̂̈̂-θ)²) ?

i.e, that you should take the root of the squared individual estimates, and THEN sum them, rather summing those squared estimates, and then taking the root?

Eg, you have an observed value of 10, and four estimates, 8, 9, 11, 12

The errors on those estimates would be -2, -1, 1, 2.

Taking the square of these errors would convert them to 4, 1, 1, 4.

Then summing them would give 10.

The mean would then be 10/4 = 2.5

Then taking the square root of this gives 1.58113883008.

Surely that figure should be 1.5?

Eg, if, using those same estimates, and square them, so you again get 4, 1, 1, 4.

Then take the square root individually, the become 2, 1, 1, 2, i.e, you keep the original figures, while removing the signs.

Then summing them would give 6.

The mean would then be 6/4 = 1.5.

Which is accurate.

I've tested a few online RMS calculators, and they all return the same figure, 1.58113883008,. but to me, the correct answer should be 1.5.

Perhaps I am misunderstanding the meaning of RMS, but to me, the point of squaring the differences of the estimates to the observed value was purely to remove the signs, and then the root was to reverse that back to the original, but positive, figures.

Therefore, the figure should be 1.5, i.e. the figure you'd get if you simply calculated the mean of the errors whilst ignoring the signs.

Any illumination you can give would be appreciated.