# Statistics/Summary/Averages/Relationships among Arithmetic, Geometric and Harmonic 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

### Relationships among Arithmetic, Geometric and Harmonic Mean

The Means mentioned above are realizations of the generalized mean
    ${\bar {x}}(m)=\left({\frac {1}{n}}\cdot \sum _{i=1}^{n}{|x_{i}|^{m}}\right)^{1/m}$
and ordered this way:
    ${\begin{alignedat}{2}&{\mathit {minimum}}&\;=\;&{\bar {x}}(-\infty )\\{\mathrel {<}}\;&{\mathit {harmonic\ mean}}&\;=\;&{\bar {x}}(-1)\\{\mathrel {<}}\;&{\mathit {geometric\ mean}}&\;=\;&\lim _{m\rightarrow 0}{\bar {x}}(m)\\{\mathrel {<}}\;&{\mathit {arithmetic\ mean}}&\;=\;&{\bar {x}}(1)\\{\mathrel {<}}\;&{\mathit {maximum}}&\;=\;&{\bar {x}}(\infty )\end{alignedat}}$