Some times geometric mean and harmonic mean should be used instead of arithmetic mean in medical data analysis.
The use of a geometric mean "normalizes" the ranges being averaged, so that no range dominates the weighting, and 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 is related to logarithmic transformation of the data. If there are no zero values, then the geometric mean equals the exponential of the mean of the log-values. Any exponent can be used, but it must equal the logarithm base, e.g. 10 to the power of the mean of the log10-values.
Compared to the arithmetic mean, the geometric mean is said to be ‘not overly influenced by the very large values in a skewed distribution’.
The choice of measure of location may not always be so straightforward. If we consider malaria parasite density, then we could imagine either the arithmetic or the geometric mean having more interest, depending on the purpose of the study.
Part of this came from an interesting article that could be useful for you (attached).
Finally, please if you consider this answer appropriate, please upvote it using the green up arrow click. Thanks.
The following link has a flowchart that helps to choose on of these formulas.
http://economistatlarge.com/r-guide/arithmetic-harmonic-geometric-means-r
The use of a geometric mean "normalizes" the ranges being averaged, so that no range dominates the weighting, and 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 is related to logarithmic transformation of the data. If there are no zero values, then the geometric mean equals the exponential of the mean of the log-values. Any exponent can be used, but it must equal the logarithm base, e.g. 10 to the power of the mean of the log10-values.
Compared to the arithmetic mean, the geometric mean is said to be ‘not overly influenced by the very large values in a skewed distribution’.
The choice of measure of location may not always be so straightforward. If we consider malaria parasite density, then we could imagine either the arithmetic or the geometric mean having more interest, depending on the purpose of the study.
Part of this came from an interesting article that could be useful for you (attached).
Finally, please if you consider this answer appropriate, please upvote it using the green up arrow click. Thanks.
The geometric mean is essentially the ordinary mean when the data are log transformed. I've used the log transform with activity data that can have large positive outliers. The harmonic mean is the average of data transformed by computing the reciprocal as is done when heart inter-beat intervals are transformed to heart rate.
Use geometric mean when you expect a log-normally distributed distribution (standard in pharmacokinetics and in several other areas; in general, when null or negative values are impossible to get, despite distinction with Gaussian may sometimes be difficult).
[Corrected: I was too quick for this last point...] Use harmonic mean when you expect an exponentially distributed distribution (basic model for times between two events and similar quantities) and using rates instead of times.
Here are some good resources on advantages and disadvantages of different types of mean as a measure of central tendency!
http://influentialpoints.com/Training/Measures_of_location_Use_and_misuse.htm
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3127352/
The geometric mean is a summary statistic which is useful when the measurement scale is not linear. It is used in the case of quantitative data measured on a proportion scale. Are the values of variables relative changes, e.g. rates of growth the geometric mean is used for calculating the average rate of changes because the summary of changes is not described by a sum, but by a product. The geometric mean is also connected to logarithmic data or logarithmic transformed data (s. first answer). And it is used in the case of count data resulting on multiplicative effects, if their values have very different magnitudes (range over several powers of ten) and a strong asymmetric distribution, e.g. in the case of microbial or baterial counts in liquid media.
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