Geometric mean arithmetic mean or means calculate center frequeny calculation average of numbers bandwidth center frequency Hi-Fi phone telephone two values value

• Calculation of the geometric mean of two numbers •
e.g. calculating the center frequency of a bandwidth BW = f₂ − f₁
Comparison between the arithmetic mean (average) and the geometric mean
or the difference between arithmetic average and geometric average.

Enter two values in the boxes, click on the calculation button and compare the answers.

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The geometric mean between two numbers is:

The arithmetic mean between two numbers is:

Example: The cut-off frequencies of a phone line are f₁ = 300 Hz
and f₂ = 3300 Hz. What is the center frequency?

The center frequency is f₀ = 995 Hz as geometric mean and
not f₀ = 1800 Hz (arithmetic mean). What a difference!

Phone frequencies - logarithmic scale - sengpielaudio

f₁ f₀ f₂

Telephone transmission range as logarithmic scale. The center frequency is 995 Hz.
The interval from 300 Hz to 995 Hz is equal to the interval from 995 Hz to 3300 Hz.
The frequency range 300 Hz to 3.3 kHz is the bandwidth of the transmission of 3 kHz.
Sometimes the phone transmission goes even up to 3.4 kHz.

The HiFi range goes from f₁ = 20 Hz to f₂ = 20000 Hz. The correct
center frequency is f₀ = 632.5 Hz (!) as geometric mean and
not the value 10.010 kHz of the arithmetic mean calculation. Look here:

Audible frequencies logarithmic scale - sengpielaudio

f₁ f₀ f₂

Audible frequency range with a logarithmic scale. The distance from 20 Hz to 632 Hz is
equal to the distance from 632 Hz to 20 kHz. Look at the marked points in the figure.

By defining the center frequency, the ratios of the cut-off frequencies to the
center frequency are the same:
Formula center frequency

The geometric mean of two numbers is the square root of their product.
The geometric mean of three numbers is the cubic root of their product.

The arithmetic mean is the sum of the numbers, divided by the quantity of the numbers.
Other names for arithmetic mean: average, mean, arithmetic average.

In general, you can only take the geometric mean of positive numbers.

The geometric mean, by definition, is the nth root of the product of the n units in a data
set. For example, the geometric mean of 5, 7, 2, 1 is (5 × 7 × 2 × 1)^1/4 = 2.893.
Alternatively, if you log transform each of the individual units the geometric will be the
exponential of the arithmetic mean of these log-transformed values. So, reusing the
example above, exp [ ( ln(5) + ln(7) + ln(2) + ln(1) ) / 4 ] = 2.893.

Geometric Mean
Formeula geometric mean - sengpielaudio

Arithmetic Mean
Formula arithmetic mean - sengpielaudio

Calculating the -3 dB cut-off frequencies f1 and f2 when center frequency and Q factor is given.

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