Geometric mean

⁠ of each number, finding the arithmetic mean of the logarithms, and then returning the result to linear scale using the exponential function ⁠

Suppose for example a person invests $1000 and achieves annual returns of +10%, −12%, +90%, −30% and +25%, giving a final value of $1609.

The arithmetic mean of these annual returns – 16.6% per annum – is not a meaningful average because growth rates do not combine additively.

⁠ can be calculated using logarithms base 2: Related to the above, it can be seen that for a given sample of points

In computer implementations, naïvely multiplying many numbers together can cause arithmetic overflow or underflow.

Calculating the geometric mean using logarithms is one way to avoid this problem.

over the unit interval shows that the geometric mean of the positive numbers between 0 and 1 is equal to

Instead, using the geometric mean, the average yearly growth is approximately 44.2% (calculated by

of equal length, This makes the geometric mean the only correct mean when averaging normalized results; that is, results that are presented as ratios to reference values.

In this scenario, using the arithmetic or harmonic mean would change the ranking of the results depending on what is used as a reference.

However, by presenting appropriately normalized values and using the arithmetic mean, we can show either of the other two computers to be the fastest.

In general, it is more rigorous to assign weights to each of the programs, calculate the average weighted execution time (using the arithmetic mean), and then normalize that result to one of the computers.

Metrics that are inversely proportional to time (speedup, IPC) should be averaged using the harmonic mean.

[8] It is also used in the CPI calculation[9] and recently introduced "RPIJ" measure of inflation in the United Kingdom and in the European Union.

This has the effect of understating movements in the index compared to using the arithmetic mean.

[8] Although the geometric mean has been relatively rare in computing social statistics, starting from 2010 the United Nations Human Development Index did switch to this mode of calculation, on the grounds that it better reflected the non-substitutable nature of the statistics being compiled and compared: Not all values used to compute the HDI (Human Development Index) are normalized; some of them instead have the form

This makes the choice of the geometric mean less obvious than one would expect from the "Properties" section above.

The equally distributed welfare equivalent income associated with an Atkinson Index with an inequality aversion parameter of 1.0 is simply the geometric mean of incomes.

In the case of a right triangle, its altitude is the length of a line extending perpendicularly from the hypotenuse to its 90° vertex.

In an ellipse, the semi-minor axis is the geometric mean of the maximum and minimum distances of the ellipse from a focus; it is also the geometric mean of the semi-major axis and the semi-latus rectum.

Now take two diametrically opposite points on the circle and apply pressure from both ends to deform it into an ellipse with semi-major and semi-minor axes of lengths

Distance to the horizon of a sphere (ignoring the effect of atmospheric refraction when atmosphere is present) is equal to the geometric mean of the distance to the closest point of the sphere and the distance to the farthest point of the sphere.

[12] The geometric mean has been used in choosing a compromise aspect ratio in film and video: given two aspect ratios, the geometric mean of them provides a compromise between them, distorting or cropping both in some sense equally.

Concretely, two equal area rectangles (with the same center and parallel sides) of different aspect ratios intersect in a rectangle whose aspect ratio is the geometric mean, and their hull (smallest rectangle which contains both of them) likewise has the aspect ratio of their geometric mean.

In the choice of 16:9 aspect ratio by the SMPTE, balancing 2.35 and 4:3, the geometric mean is

This was discovered empirically by Kerns Powers, who cut out rectangles with equal areas and shaped them to match each of the popular aspect ratios.

When overlapped with their center points aligned, he found that all of those aspect ratio rectangles fit within an outer rectangle with an aspect ratio of 1.77:1 and all of them also covered a smaller common inner rectangle with the same aspect ratio 1.77:1.

[13] The value found by Powers is exactly the geometric mean of the extreme aspect ratios, 4:3 (1.33:1) and CinemaScope (2.35:1), which is coincidentally close to

Applying the same geometric mean technique to 16:9 and 4:3 approximately yields the 14:9 (

The geometric mean is also used to calculate B and C series paper formats.

Example of the geometric mean: (red) is the geometric mean of and , [ 1 ] [ 2 ] is an example in which the line segment is given as a perpendicular to . is the diameter of a circle and .
Proof without words of the AM–GM inequality :
PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b . Using the geometric mean theorem , triangle PGR's altitude GQ is the geometric mean . For any ratio a : b , AO ≥ GQ.
Geometric proof without words that max ( a , b ) > root mean square ( RMS ) or quadratic mean ( QM ) > arithmetic mean ( AM ) > geometric mean ( GM ) > harmonic mean ( HM ) > min ( a , b ) of two distinct positive numbers a and b [ note 1 ]
The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. Using Pythagoras' theorem on the 3 triangles of sides ( p + q , r , s ) , ( r , p , h ) and ( s , h , q ) ,
Equal area comparison of the aspect ratios used by Kerns Powers to derive the SMPTE 16:9 standard. [ 13 ] TV 4:3/1.33 in red, 1.66 in orange, 16:9/1.7 7 in blue , 1.85 in yellow, Panavision /2.2 in mauve and CinemaScope /2.35 in purple.