Root mean square

Quadratic mean ( quadratic ) ^[1] - number $s$ ${\ displaystyle s}$ $s$ equal to the square root of the arithmetic mean of the squares of the given numbers $a_{1},a_{2},...,a_{n}$ ${\ displaystyle a_ {1}, a_ {2}, ..., a_ {n}}$ $a_ {1}, a_ {2}, ..., a_ {n}$ :

s={\sqrt {\frac {a_{1}^{2}+a_{2}^{2}+\ldots +a_{n}^{2}}{n}}}

{\ displaystyle s = {\ sqrt {\ frac {a_ {1} ^ {2} + a_ {2} ^ {2} + \ ldots + a_ {n} ^ {2}} {n}}}}

s = {\ sqrt {{\ frac {a_ {1} ^ {2} + a_ {2} ^ {2} + \ ldots + a_ {n} ^ {2}} {n}}}}

The quadratic mean is a special case of the average power and therefore obeys the inequality about means . In particular, for any numbers it is not less than the arithmetic mean :

{\frac {a_{1}+a_{2}+\ldots +a_{n}}{n}}\leqslant {\sqrt {\frac {a_{1}^{2}+a_{2}^{2}+\ldots +a_{n}^{2}}{n}}}

{\ displaystyle {\ frac {a_ {1} + a_ {2} + \ ldots + a_ {n}} {n}} \ leqslant {\ sqrt {\ frac {a_ {1} ^ {2} + a_ {2 } ^ {2} + \ ldots + a_ {n} ^ {2}} {n}}}}

{\ frac {a_ {1} + a_ {2} + \ ldots + a_ {n}} {n}} \ leqslant {\ sqrt {{\ frac {a_ {1} ^ {2} + a_ {2} ^ {2} + \ ldots + a_ {n} ^ {2}} {n}}}}

The quadratic mean is widely used in many sciences. In particular, through it the basic concept of probability theory and mathematical statistics is determined - variance (the square root of which is called the standard deviation ). The least squares method , which is of general scientific significance, is also closely connected with this concept.

Notes

↑ Great Encyclopedic Dictionary. 2000.

Properties

The quadratic mean of a set of non-negative numbers lies between the minimum and maximum numbers from this set.

[1] Great Encyclopedic Dictionary. 2000.

Root mean square

Notes

Properties

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