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Discussion Arithmetic mean & Geometric Mean.
A mathematical representation of the typical value of a series of numbers, computed as the sum of all the numbers in the series divided by the count of all numbers in the series.
1. The arithmetic mean is probably the most commonly taught and encountered
Statistic today, appearing in frequent everyday contexts. It is also the one statistic that
is consistently incorporated in the elementary school course in the United States.
Further, the mean is a fundamental concept in statistics courses because of its
Predominant use in most inferential statistics and formulae.
The mean is the most commonly-used type of average and is often referred to simply as the average. The term "mean" or "arithmetic mean" is preferred in mathematics and statistics to distinguish it from other averages such as the median and the mode.
In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set (cardinality). (The word set is used perhaps somewhat loosely; for example, the number 3.8 could occur more than once in such a "set".) If 1 particular number occurs more times than others in the set, it could be called a mode. The arithmetic mean is what pupils are taught very early to call the "average." If the set is a statistical population, then we speak of the population mean.]
In mathematics, there are numerous methods for calculating the average or central tendency of a list of n numbers. The most common method, and the one generally referred to simply as the average, is the arithmetic mean. Please see the table of mathematical symbols for explanations of the symbols used. Average - Arithmetic mean. The arithmetic mean is the standard "average", often simply called the "mean".
The average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio. Technically defined as "the 'n'th root product of 'n' numbers", the formula for calculating geometric mean is most easily written as:
Where 'n' represents the number of returns in the series.
The geometric mean must be used when working with percentages (which are derived from values), whereas the standard arithmetic mean will work with the values themselves.
The main benefit to using the geometric mean is that the actual amounts invested do not need to be known; the calculation focuses entirely on the return figures themselves and presents an "apples-to-apples" comparison when looking at two investment options
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, which is what most people think of with the word "average", except that the numbers are multiplied and then the nth root (where n is the count of numbers in the set) of the resulting product is taken.
For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product which equals 4; that is 2√2 × 8 = 4. As another example, the geometric mean of three numbers 1, ½, ¼ is the cube root of their product (1/8), which is 1/2; that is 3√1 × ½ × ¼ = ½.
The geometric mean can also be understood in terms of geometry. The geometric mean of two numbers, a and b, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths a and b. Similarly, the geometric mean of three numbers, a, b, and c, is the length of one side of a cube whose volume is the same as that of a cuboids with sides whose lengths are equal to the three given numbers.
The geometric mean only applies 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. The geometric mean is also one of the three classic Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean.
The geometric mean is a measure of central tendency, just like a median. It is different than the traditional mean (which we sometimes call the arithmetic mean) because it uses multiplication rather than addition to summarize data values.
The geometric mean is a useful summary when we expect that changes in the data occur in a relative fashion. Here are some examples where we would expect relative changes:
Geometric means are often useful summaries for highly skewed data. They are also natural for summarizing ratios. Don't use a geometric mean, though, if you have any negative or zero values in your data.
3. The geometric mean is useful to determine "average factors". For example, if a stock rose 10% in the first year, 20% in the second year and fell 15% in the third year, then we compute the geometric mean of the factors 1.10, 1.20 and 0.85 as (1.10 × 1.20 × 0.85)1/3 = 1.0391... And we conclude that the stock rose 3.91 percent per year, on average. Put another way... The arithmetic mean is relevant any time several quantities add together to produce a total. The arithmetic mean answers the question, if all the quantities had the same value.
Efficacy calculations in anathematic studies require estimates of the central tendency for the nematode populations. Confusion exists among practitioners regarding which measures of central tendency are most appropriate; although the arithmetic mean is frequently used, there are theoretical reasons for preferring the geometric mean. To investigate this controversy, arithmetic and geometric means were compared for their suitability for use in measuring efficacy. Arithmetic and geometric means were compared as measures of central tendency for skewed distributions. The following criteria were developed to facilitate the comparison: (1) probability around the parameter, (2) influence of extreme values, and (3) proximity to the median. Under log-normality, theoretical results demonstrated the superiority of the geometric mean. Modified-bootstrap simulations using empirical data from cattle were used to confirm theoretical expectations. Simulations on log-normal data supported the geometric mean as the better indicator of the central tendency. Additionally, for data not confirmed as log-normal, the superiority of geometric means was demonstrated. In a comparison of precision, it was shown that mean squared error was always smaller for sample geometric means than for arithmetic means when n> or =2. Simulation results added support to that conclusion.
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