# Statistics/Summary/Averages/Geometric Mean

Statistics
* Introduction
** What Is Statistics?
** Subjects in Modern Statistics
** Why Should I Learn Statistics?
** What Do I Need to Know to Learn Statistics?
* Different Types of Data
** Primary and Secondary Data
** Quantitative and Qualitative Data
* Methods of Data Collection
** Experiments
** Sample Surveys
** Observational Studies
* Data Analysis
** Data Cleaning
** Moving Average
* Summary Statistics
** Measures of center
*** Mean, Median, and Mode
*** Geometric Mean
*** Harmonic Mean
*** Relationships among Arithmetic, Geometric, and Harmonic Mean
*** Geometric Median
** Measures of dispersion
*** Range of the Data
*** Variance and Standard Deviation
*** Quartiles and Quartile Range
*** Quantiles
* Displaying Data
** Bar Charts
** Comparative Bar Charts
** Histograms
** Scatter Plots
** Box Plots
** Pie Charts
** Comparative Pie Charts
** Pictograms
** Line Graphs
** Frequency Polygon
* Probability
** Combinatorics
** Bernoulli Trials
** Introductory Bayesian Analysis
* Distributions
** Discrete Distributions
*** Uniform Distribution
*** Bernoulli Distribution
*** Binomial Distribution
*** Poisson Distribution
*** Geometric Distribution
*** Negative Binomial Distribution
*** Hypergeometric Distribution
** Continuous Distributions
*** Uniform Distribution
*** Exponential Distribution
*** Gamma Distribution
*** Normal Distribution
*** Chi-Square Distribution
*** Student-t Distribution
*** F Distribution
*** Beta Distribution
*** Weibull Distribution
* Testing Statistical Hypothesis
** Purpose of Statistical Tests
** Formalism Used
** Different Types of Tests
** z Test for a Single Mean
** z Test for Two Means
** t Test for a single mean
** t Test for Two Means
** paired t Test for comparing Means
** One-Way ANOVA F Test
** z Test for a Single Proportion
** z Test for Two Proportions
** Testing whether Proportion A Is Greater than Proportion B in Microsoft Excel
** Spearman's Rank Coefficient
** Pearson's Product Moment Correlation Coefficient
** Chi-Squared Tests
*** Chi-Squared Test for Multiple Proportions
*** Chi-Squared Test for Contingency
** Approximations of distributions
* Point Estimates (12:07, 28 March 2007 (UTC))
** Unbiasedness
** Measures of goodness
** UMVUE
** Completeness
** Sufficiency and Minimal Sufficiency
** Ancillarity
* Practice Problems
** Summary Statistics Problems
** Data-Display Problems
** Distributions Problems
** Data-Testing Problems
* Numerical Methods
** Basic Linear Algebra and Gram-Schmidt Orthogonalization
** Unconstrained Optimization
** Quantile Regression
** Numerical Comparison of Statistical Software
** Numerics in Excel
** Statistics/Numerical_Methods/Random Number Generation
* Time Series Analysis
* Multivariate Data Analysis
** Principal Component Analysis
** Factor Analysis for metrical data
** Factor Analysis for ordinal data
** Canonical Correlation Analysis
** Discriminant Analysis
* Analysis of Specific Datasets
** Analysis of Tuberculosis
* Appendix
** Authors
** Glossary
** Index
** Links
Statistics | Mean

### Geometric Mean

The Geometric Mean is calculated by taking the nth root of the product of a set of data.
${\tilde {x}}={\sqrt[{n\;}]{\prod _{i=1}^{n}x_{i}}}$
For example, if the set of data was:
1,2,3,4,5
The geometric mean would be calculated:
${\sqrt[{5}]{1\times 2\times 3\times 4\times 5}}={\sqrt[{5}]{120}}=2.61$
Of course, with large n this can be difficult to calculate. Taking advantage of two properties of the logarithm:
$\log(a\cdot b)=\log(a)+\log(b)$
$\log(a^{n})=n\cdot \log(a)$
We find that by taking the logarithmic transformation of the geometric mean, we get:
$\log \left({\sqrt[{n}]{x_{1}\times x_{2}\times x_{3}\cdots x_{n}}}\right)={\frac {1}{n}}\sum _{i=1}^{n}\log(x_{i})$
Which leads us to the equation for the geometric mean:
${\tilde {x}}=\exp \left({\frac {1}{n}}\sum _{i=1}^{n}\log(x_{i})\right)$

### When to use the geometric mean

The arithmetic mean is relevant at any time several quantities add together to produce a total. The arithmetic mean answers the question, "if all the quantities had the same value, what would that value have to be in order to achieve the same total?"
In the same way, the geometric mean is relevant any time several quantities multiply together to produce a product. The geometric mean answers the question, "if all the quantities had the same value, what would that value have to be in order to achieve the same product?"
For example, suppose you have an investment which returns 10% the first year, 50% the second year, and 30% the third year. What is its average rate of return? It is not the arithmetic mean, because what these numbers mean is that on the first year your investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.50, and the third year it was multiplied by 1.30. The relevant quantity is the geometric mean of these three numbers. Written by Hafiz G m
It is known that the geometric mean is always less than or equal to the arithmetic mean (equality holding only when A=B). The proof of this is quite short and follows from the fact that $({\sqrt {A}}-{\sqrt {B}})^{2}$ is always a non-negative number. This inequality can be surprisingly powerful though and comes up from time to time in the proofs of theorems in calculus. Source.
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