question: what is the geometric mean and why is it good as a type of average for comparing things that are indexes of lots of other things?
comment: The geometric mean of n values is the nth root of the product of the n values. By contrast the usual mean, the arithmetic mean, is the sum of the n values divided by n. The geometric mean is interesting when you are interested in tracking proportional changes between different values. Say, you have two values: 1 and 100. The arithmetic mean is 50.5 and the geometric mean is 10. Now, say you double one of the values. * If you double the first value -- (1, 100) - > (2, 100) -- the arithmetic mean becomes 51 and the geometric mean is 14.1. * If you double the second value -- (1, 100) - > (1, 200) -- the arithmetic mean jumps to 100.5 while the geometric mean is the same 14.1. With the geometric mean, the same proportional change to any of the values has the same effect on the mean. With the arithmetic mean, the larger values dominate. For this reason, some like to use it for component indexes. For example, a 25% gain in a 10 dollar stock has the same effect as a 25% gain in a 100 dollar stock in the same index. There are other ways of handling this problem (such as pre-scaling the data) but the geometric mean is one tool in a statisticians toolbox.
passages: For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product, that is, formula_2. As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2, that is, formula_3.
   The geometric mean applies only to positive numbers. It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment.
   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 formula_50. This makes the choice of the geometric mean less obvious than one would expect from the "Properties" section above.
  
  Section::::Applications.:Aspect ratios.
   This makes the geometric mean the only correct mean when averaging "normalized" results; that is, results that are presented as ratios to reference values. This is the case when presenting computer performance with respect to a reference computer, or when computing a single average index from several heterogeneous sources (for example, life expectancy, education years, and infant mortality). In this scenario, using the arithmetic or harmonic mean would change the ranking of the results depending on what is used as a reference. For example, take the following comparison of execution time of computer programs:
   In cases where the geometric mean is being used to determine the average growth rate of some quantity, and the initial and final values formula_40 and formula_41 of that quantity are known, the product of the measured growth rate at every step need not be taken: If the values are formula_42, then the growth rate between measurement formula_43 and formula_44 is formula_45. The geometric mean of these growth rates is then just
  
  where formula_47 is the number of steps from the initial to final state.
  
  Section::::Properties.
   A geometric mean is often used when comparing different items—finding a single "figure of merit" for these items—when each item has multiple properties that have different numeric ranges. For example, the geometric mean can give a meaningful value to compare two companies which are each rated at 0 to 5 for their environmental sustainability, and are rated at 0 to 100 for their financial viability. If an arithmetic mean were used instead of a geometric mean, the financial viability would have greater weight because its numeric range is larger. That is, a small percentage change in the financial rating (e.g. going from 80 to 90) makes a much larger difference in the arithmetic mean than a large percentage change in environmental sustainability (e.g. going from 2 to 5). The use of a geometric mean normalizes the differently-ranged values, meaning a given percentage change in any of the properties has the same effect on the geometric mean. So, a 20% change in environmental sustainability from 4 to 4.8 has the same effect on the geometric mean as a 20% change in financial viability from 60 to 72.
   The geometric mean can be understood in terms of geometry. The geometric mean of two numbers, formula_4 and formula_5, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths formula_4 and formula_5. Similarly, the geometric mean of three numbers, formula_4, formula_5, and formula_10, is the length of one edge of a cube whose volume is the same as that of a cuboid with sides whose lengths are equal to the three given numbers.
   Geometric mean
  
  In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the th root of the product of numbers, i.e., for a set of numbers , the geometric mean is defined as
   The geometric mean of a data set is less than the data set's arithmetic mean unless all members of the data set are equal, in which case the geometric and arithmetic means are equal. This allows the definition of the arithmetic-geometric mean, an intersection of the two which always lies in between.
  
  The geometric mean is also the arithmetic-harmonic mean in the sense that if two sequences (formula_18) and (formula_19) are defined:
  
  and
   Section::::Applications.
  
  Section::::Applications.:Proportional growth.
  
  The geometric mean is more appropriate than the arithmetic mean for describing proportional growth, both exponential growth (constant proportional growth) and varying growth; in business the geometric mean of growth rates is known as the compound annual growth rate (CAGR). The geometric mean of growth over periods yields the equivalent constant growth rate that would yield the same final amount.
   Both in the approximation of squaring the circle according to S.A. Ramanujan (1914) and in the construction of the Heptadecagon according to ""sent by T. P. Stowell, credited to Leybourn's Math. Repository, 1818"", the geometric mean is employed.
  
  Section::::Applications.:Financial.
  
  The geometric mean has from time to time been used to calculate financial indices (the averaging is over the components of the index). For example, in the past the FT 30 index used a geometric mean. It is also used in the recently introduced "RPIJ" measure of inflation in the United Kingdom and elsewhere in the European Union.
   In the case of a right triangle, its altitude is the length of a line extending perpendicularly from the hypotenuse to its 90° vertex. Imagining that this line splits the hypotenuse into two segments, the geometric mean of these segment lengths is the length of the altitude.
   Geometric process
  
  In probability, statistics and related fields, the geometric process is a counting process, introduced by Lam in 1988. It is defined as
  
  The geometric process. Given a sequence of non-negative random variables :formula_1, if they are independent and the cdf of formula_2 is given by formula_3 for formula_4, where formula_5 is a positive constant, then formula_6 is called a geometric process (GP).
  
  The GP has been widely applied in reliability engineering 
  
  Below are some of its extensions.
   Weighted geometric mean
  
  In statistics, given a set of data, 
  
  and corresponding weights, 
  
  the weighted geometric mean is calculated as
  
  Note that if all the weights are equal, the weighted geometric mean is the same as the geometric mean.
  
  Weighted versions of other means can also be calculated. Probably the best known weighted mean is the weighted arithmetic mean, usually simply called the weighted mean. Another example of a weighted mean is the weighted harmonic mean.
   Section::::Measures of spatial central tendency.
  
  The coordinate-wise mean of a point set is the centroid, which solves the same variational problem in the plane (or higher-dimensional Euclidean space) that the familiar average solves on the real line — that is, the centroid has the smallest possible average squared distance to all points in the set.
  
  Section::::Measures of spatial dispersion.
   Related to the above, it can be seen that for a given sample of points formula_37, the geometric mean is the minimizer of formula_38, whereas the arithmetic mean is the minimizer of formula_39. Thus, the geometric mean provides a summary of the samples whose exponent best matches the exponents of the samples (in the least squares sense).
   The fundamental property of the geometric mean, which can be proven to be false for any other mean, is
   The log form of the geometric mean is generally the preferred alternative for implementation in computer languages because calculating the product of many numbers can lead to an arithmetic overflow or arithmetic underflow. This is less likely to occur with the sum of the logarithms for each number.
  
  Section::::Calculation.:Relationship with arithmetic mean and mean-preserving spread.
  
  If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are "spread apart" from each other while leaving the arithmetic mean unchanged — then the geometric mean always decreases.
  
  Section::::Calculation.:Computation in constant time.
   The geometric median is an important estimator of location in statistics, where it is also known as the "L" estimator. It is also a standard problem in facility location, where it models the problem of locating a facility to minimize the cost of transportation.
   The above figure uses capital pi notation to show a series of multiplications. Each side of the equal sign shows that a set of values is multiplied in succession (the number of values is represented by "n") to give a total product of the set, and then the "n"th root of the total product is taken to give the geometric mean of the original set. For example, in a set of four numbers formula_13, the product of formula_14 is formula_15, and the geometric mean is the fourth root of 24, or ~ 2.213. The exponent formula_16 on the left side is equivalent to the taking "n"th root. For example, formula_17.
   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 aspect ratio their geometric mean.
   Moment distance index
  
  The moment distance index (MDI) is a shape-based metric or shape index that can be used to analyze spectral reflectance curves and waveform LiDAR, proposed by Salas and Henebry in 2014. In the case of spectral data, the shape of the reflectance curve should unmask fine points of the spectra usually not considered by existing band-specific indices. It has been used to identify spectral regions for chlorophyll and carotenoids, detect greenhouses using WorldView-2 and Landsat satellite data, identify greenhouse crops, compute canopy heights, and estimate green vegetation fraction.
   Suppose an orange tree yields 100 oranges one year and then 180, 210 and 300 the following years, so the growth is 80%, 16.6666% and 42.8571% for each year respectively. Using the arithmetic mean calculates a (linear) average growth of 46.5079% (80% + 16.6666% + 42.8571%, that sum then divided by 3). However, if we start with 100 oranges and let it grow 46.5079% each year, the result is 314 oranges, not 300, so the linear average "over"-states the year-on-year growth.
   The geometric mean is also one of the three classical Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean. For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between (see Inequality of arithmetic and geometric means.)
  
  Section::::Calculation.
  
  The geometric mean of a data set formula_11 is given by:
   Instead, we can use the geometric mean. Growing with 80% corresponds to multiplying with 1.80, so we take the geometric mean of 1.80, 1.166666 and 1.428571, i.e. formula_49; thus the "average" growth per year is 44.2249%. If we start with 100 oranges and let the number grow with 44.2249% each year, the result is 300 oranges.
  
  Section::::Applications.:Applications in the social sciences.
  