Quadratic Mean of a Statistical List

Parameters

 data - statistical list

Description

 • Important: The stats package has been deprecated. Use the superseding package Statistics instead.
 • The function quadraticmean of the subpackage stats[describe, ...] computes the quadratic mean of the given data.
 • The quadratic mean, also known as the root mean square, is the square root of the mean of the squares of the observations. It is a measure of central tendency. For more information about such measures, see the help for the (arithmetic) mean.
 • This type of mean is used in some physical applications. For example, it gives the average'' speed of particles of the same mass, where average is defined in such a way as to give each particle the same kinetic energy.
 • Classes are assumed to be represented by the class mark, for example 10..12 has the value 11. Missing data are ignored.
 • The command with(stats[describe],quadraticmean) allows the use of the abbreviated form of this command.

Examples

Important: The stats package has been deprecated. Use the superseding package Statistics instead.

 > $\mathrm{with}\left(\mathrm{stats}\right):$

If V is the velocity of a particle of mass M, then its kinetic energy is defined to be 1/2*M*V^2. If we have two particles, one with velocity 3 and the other with velocity 4, the total kinematic energy is given by

 > $\mathrm{Tot_Kinetic}≔\frac{1M{3}^{2}}{2}+\frac{1M{4}^{2}}{2}$
 ${\mathrm{Tot_Kinetic}}{:=}\frac{{25}}{{2}}{}{M}$ (1)

The RMS-velocity is

 > $\mathrm{Rv}≔\mathrm{describe}[\mathrm{quadraticmean}]\left(\left[3,4\right]\right)$
 ${\mathrm{Rv}}{:=}\frac{{5}}{{2}}{}\sqrt{{2}}$ (2)

The energy associated with the 2 particles of mass M and velocity Rv is

 > $2\frac{1M{\mathrm{Rv}}^{2}}{2}=\mathrm{Tot_Kinetic}$
 $\frac{{25}}{{2}}{}{M}{=}\frac{{25}}{{2}}{}{M}$ (3)

Note that the arithmetic average is not suitable here:

 > $\mathrm{Av}≔\mathrm{describe}[\mathrm{mean}]\left(\left[3,4\right]\right)$
 ${\mathrm{Av}}{:=}\frac{{7}}{{2}}$ (4)
 > $2\frac{1M{\mathrm{Av}}^{2}}{2}\ne \mathrm{Tot_Kinetic}$
 $\frac{{49}}{{4}}{}{M}{\ne }\frac{{25}}{{2}}{}{M}$ (5)