The center frequency f₀ (resonant frequency) is the geometric mean between the lower cutoff frequency f₁ and the upper cutoff frequency f₂ of a frequency band. See also: Band-pass filter. f2 - f1 is called the bandwidth B.

Only if the bandwidth f₂ - f₁ is very small in comparison to the center frequency it is sometimes possible to use this arithmetic mean for calculations, but this is often calculated in mistake:

At radio stations (medium wave) the bandwidth is often only 9 kHz. A transmitter, which has 1500 kHz, is transmitting from 1495.5 kHz to 1504.5 kHz.
The exact formula gives:

and the short formula gives in this case the very close result of:

The short calculated value is always too large. If the bandwidth is given by B = f₂ - f₁, the difference is:

.

But if for instance we are looking for the center frequency of the telephone audio band from 300 Hz to 3300 Hz, we get (3300 + 300) / 2 = "1800 Hz" for the short arithmetic mean calculation, but the root of 300 x 3300 = "995" Hz with the correct geometric mean formula. What a big difference!

That the geometric mean is not the arithmetic mean can be seen in a calculation program at the bottom in the external link. There one can compare the difference of both values.

External links

• Calculations and comparisons between the geometric mean and the arithmetic mean