Square mean

The root mean square (or the root mean square RMS , English: root mean square RMS ) is that average , which is calculated as the square root of the quotient from the sum of the squares of the noticed numbers and their number.

The two numbers 1 and 2 have e.g. B. the root mean square value ( arithmetic mean  = 1.5; the larger number 2 is rated more strongly in the square mean). ${\displaystyle {\sqrt {\frac {1^{2}+2^{2}}{2}}}\approx 1{,}58}$

Because of the squaring, the root mean square is also called the second (absolute) moment . The "third moment" would be the averaging to the third  power (also called the cubic mean ) etc.

calculation

To calculate the QMW of a series of numbers, the squares of all numerical values ​​are first added and divided by their number  n . The square root of this gives the QMW: ${\displaystyle x_{i}}$

${\displaystyle \mathrm {QMW} ={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}{x_{i}^{2}}}}={\sqrt {\frac {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}{n}}}}$.

From a geometrical point of view, one determines from the series of numbers squares and from them a square of average area or medium size (the radicand under the root). The root or side length of this square is the square mean of the row of numbers or the side lengths of all squares. ${\displaystyle x_{i}}$

For continuously available sizes, the following must be integrated over the area under consideration:

${\displaystyle \mathrm {QMW} ={\sqrt {{\frac {1}{t_{2}-t_{1}}}\int _{t_{1}}^{t_{2}}{f(t)^{2}\,\mathrm {d} t}}}}$ ;

in the case of periodic quantities, for example the sinusoidal alternating current, one integrates over a number of periods.

application

In technology, the root mean square is of great importance for periodically changing quantities such as alternating current , whose power conversion at an ohmic resistor ( Joule heat ) increases with the square of the current strength . One speaks here of the effective value of the current. The same relationship applies to electrical voltages that vary over time .

With an alternating variable with a sinusoidal shape, the QMW is times the peak value , i.e. approx. 70.7%. ${\displaystyle (1/{\sqrt {2}})}$

If one does not know anything about the temporal course of the fluctuations that occur, it should be known from the context in which the mean value is to be calculated whether the equivalent value (e.g. for electrolysis ) or the effective value (e.g. for light and Heat) is meaningful.

See also

• Measurement technology , scatter , variance
• Least squares method , adjustment calculation
• Mean inequality
• Mean square deviation , median
• Control quality