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Four Means: Arithmetic, Harmonic, Geometric, RMS

The main concept here is that of finding the value that is most

representative of a group.

Arithmetic Mean: xmean1

N

i

xi

=

N=

The example below shows three resistors in series.

The arithmetic mean of the three resistances is: RR1 R2+ R3+

3=

As seen below, each of the resistors can be substituted by the arithmeticmean and the current will still be the same for the applied voltage because

I V

R1 R2+ R3+=

V

3 R=

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Geometric Mean: xmean

1

N

i

xi

=

1

N

=

The example below shows three voltage amplifiers in series.

Vout A1 A2 A3 Vin=

The geometric mean of the three voltage amplifiers is: A A1 A2 A3( )

1

3=

Below, it is seen that each of the amplifiers can be substituted by the

geometric mean and the output voltage will be the same for the applied

input voltage Vin.

Vout A1 A2 A3 Vin= A3

Vin=

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Root Mean Square (RMS): xmean 1

N

i

xi2

=

N=

Equations characterizing energy or power often involve squared terms. For instance,

Kinetic_Energy 1

2m v

2=

Spring_Energy 1

2k x

2=

Resistive_Power_Dissipation

V2

R=

I

2

R=

The RMS is often used to find the mean (effective) amplitudes of v,x, V, or I that

are representative of the energy or power levels.

For instance, the temperature of a dilute gas is directly proportional to the net kinetic

energy of the molecules of the gas:

where R is the ideal gas constant and mois the

mass of each molecule (assuming only one type

of molecules)

Temperature 2

3

1

R

1

N

i

1

2mo vi

2

=

=

The above equation is typically expressed as:

Temperature 1

3

N

R mo vRMS

2=

where vRMS is the root-mean-square velocity (speed) defined as:

vRMS1

N

i

vi( )

2

=

N=

Thus, if each molecule of a dilute, homogeneous gas was forced to move at the root

mean square velocity (speed), the temperature of the gas would not change.

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The RMS measures the mean amplitude of a group without regard to signs.

For instance the RMS of the measurements 1 and -1 is 1. However, the

arithmetic mean of 1 and -1 is 0.