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- Harmonic Mean
- 3.1: Statistics of Central Tendency
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- The Arithmetic, Geometric and Harmonic Means
All of the tests in the first part of this handbook have analyzed nominal variables. You summarize data from a nominal variable as a percentage or a proportion. The rest of the tests in this handbook analyze measurement variables. Summarizing data from a measurement variable is more complicated, and requires a number that represents the "middle" of a set of numbers known as a "statistic of central tendency" or "statistic of location" , along with a measure of the "spread" of the numbers known as a "statistic of dispersion". The arithmetic mean is the most common statistic of central tendency, while the variance or standard deviation are usually used to describe the dispersion.
In mathematics , the harmonic mean is one of several kinds of average , and in particular, one of the Pythagorean means. Typically, it is appropriate for situations when the average of rates is desired. The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations. As a simple example, the harmonic mean of 1, 4, and 4 is. The third formula in the above equation expresses the harmonic mean as the reciprocal of the arithmetic mean of the reciprocals. It is the reciprocal dual of the arithmetic mean for positive inputs:. Thus, the harmonic mean cannot be made arbitrarily large by changing some values to bigger ones while having at least one value unchanged.
PDF version. It is a useful tool for problems solving and building relationships with other mathematics. It should find more use in school mathematics than currently. I will rely heavily on a collection of problems and essays on my web site:. The link poses the problem of generalizing the proof following the lines of argument advanced by Courant and Robbins
3.1: Statistics of Central Tendency
In any research, enormous data is collected and, to describe it meaningfully, one needs to summarise the same. The bulkiness of the data can be reduced by organising it into a frequency table or histogram. These measures may also help in the comparison of data. The mean, median and mode are the three commonly used measures of central tendency. Mean is the most commonly used measure of central tendency. There are different types of mean, viz.
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Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I have probability density of function of some data it's triangular.
Handbook of Means and Their Inequalities pp Cite as. This chapter is devoted to the properties and inequalities of the classical arithmetic, geometric and harmonic means. In particular the basic inequality between these means, the Geometric Mean-Arithmetic Mean Inequality, is discussed at length with many proofs being given.
Quantitative Methods 1 Reading 7. Statistical Concepts and Market Returns Subject 4. Measures of Center Tendency. Why should I choose AnalystNotes?
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The Arithmetic, Geometric and Harmonic Means
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It is well known and has been demonstrated by Xia et al. () that for the same sequence, Am ≥ Gm ≥ Hm. The harmonic mean produces an averaged value.
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