Mathematics
Geometric Mean
The geometric mean is a type of average that is calculated by taking the nth root of the product of n numbers. It is commonly used to find the average growth rate or to compare different quantities that have different units. In finance, it is used to calculate the average return on investment.
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10 Key excerpts on "Geometric Mean"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
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 cuboid 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. Calculation The Geometric Mean of a data set is given by: The Geometric Mean of a data set is less than or equal to the data set's arithmetic mean (the two means are equal if and only if all members of the data set are equal). This allows ________________________ WORLD TECHNOLOGIES ________________________ the definition of the arithmetic-Geometric Mean, a mixture of the two which always lies in between. The Geometric Mean is also the arithmetic-harmonic mean in the sense that if two sequences ( a n ) and ( h n ) are defined: and then a n and h n will converge to the Geometric Mean of x and y . This can be seen easily from the fact that the sequences do converge to a common limit (which can be shown by Bolzano-Weierstrass theorem) and the fact that Geometric Mean is preserved: Replacing arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
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 cuboid 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. Calculation The Geometric Mean of a data set is given by: The Geometric Mean of a data set is less than or equal to the data set's arithmetic mean (the two means are equal if and only if all members of the data set are equal). This allows ________________________ WORLD TECHNOLOGIES ________________________ the definition of the arithmetic-Geometric Mean, a mixture of the two which always lies in between. The Geometric Mean is also the arithmetic-harmonic mean in the sense that if two sequences ( a n ) and ( h n ) are defined: and then a n and h n will converge to the Geometric Mean of x and y . This can be seen easily from the fact that the sequences do converge to a common limit (which can be shown by Bolzano-Weierstrass theorem) and the fact that Geometric Mean is preserved: Replacing arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
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 ________________________ WORLD TECHNOLOGIES ________________________ 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 cuboid 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 invest-tment. The Geometric Mean is also one of the three classic Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean. Calculation The Geometric Mean of a data set is given by: The Geometric Mean of a data set is less than or equal to the data set's arithmetic mean (the two means are equal if and only if all members of the data set are equal). This allows the definition of the arithmetic-Geometric Mean, a mixture of the two which always lies in between. The Geometric Mean is also the arithmetic-harmonic mean in the sense that if two sequences ( a n ) and ( h n ) are defined: and then a n and h n will converge to the Geometric Mean of x and y . This can be seen easily from the fact that the sequences do converge to a common limit (which can be shown by Bolzano-Weierstrass theorem) and the fact that Geometric Mean is preserved: ________________________ WORLD TECHNOLOGIES ________________________ Replacing arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.
eBook - ePubThat Doesn't Work Anymore
Retooling Investment Economics in the Age of Disruption
- Robert S. Kricheff(Author)
- 2018(Publication Date)
- De Gruyter(Publisher)
While the mean is the most common to use there are instances where the mode and the median can be more helpful. For example, if you want to find out the most common salary in a group of workers you would use the mode, as the foreman’s salary might skew the mean. The median can be good when most numbers in the data set are clustered close together but there are one or two extreme outliers that really skew the mean.Some other averages are quite common when running investment analysis. These include the Geometric Mean, a weighted average and a moving average.Geometric Means are good when each figure in the data set is dependent on the others. An example of this would be if you were tracking a company’s revenue growth from one year to another. The Geometric Mean multiplies all of the numbers in the data set and then takes that product to the “n” root, “n” standing for the number of figures in the data set. This type of mean is often used to examine investment returns. Typically, when using a Geometric Mean all of the numbers in the data set need to be positive, but you can manage around this by adding one to each figure in the data set. If the data set included annual investment returns (0.2, 0.1, 0.2, -0.1, 0.1), you could add one to each data point (1.2, 1.1, 1.2, 0.9, 1.1) then take the product of these figures to the 1/5 power and subtract one to get the Geometric Mean (the other option is to just never have any negative returns).Weighted averages are commonly used in indexes, portfolio, analysis and probability theory to develop expected values. Investors often use indexes that use weighted averages. For example, an index may track the stock prices of the 100 largest public companies. The index could weight each stock the same and calculate the average price of the index. Getting this figure at the end of the day on a percentage basis would show you how much the average stock in the index moved. However, it would not show you how the value of the overall index moved. A common way of showing you the move in the value of the index is to “weight” each stock in the index based on what the equity market capitalization is (equity market capitalization = (number of shares outstanding * stock price).
eBook - ePub- Shahdad Naghshpour(Author)
- 2015(Publication Date)
- Business Expert Press(Publisher)
outliers and their exclusions provide more meaningful and representative statistics. The sample data are sorted and a given percentage, say 5%, of the top and the bottom of the data are discarded, and the regular mean is calculated for the remaining data. The trimmed mean is less susceptible to extreme values.Geometric MeanThe Geometric Mean is useful when the values change in geometric progression instead of arithmetic progression, as is the case with growth rates or interest rates. The Geometric Mean is calculated using the following formula.The formula can be expressed using logarithm to avoid taking the nth root, as shown in the next section. This was more important before the advent of powerful calculators. The logarithmic formula is a linear sum of its elements.Example 2.4 Assume that a new company grew at 28% the first year, 15% the second year, and 13% the third year. What is the rate of the growth of company?Solution 2.4Note that at the beginning of each year, the following amounts are available. Therefore, $100 will grow to $166.336 over 3 years at the above growth rates for each year. The Geometric Mean for the growth rate over 3 years is The growth rate is (1.184846 – 1) × 100 = 18.4846%. Therefore, $100 will grow to $166.336 over 3 years at this rate as well, which is the case as seen below: In the case of the average growth rate, the same result is obtained if the rate is raised to power 3:= 100 × (1 + 0.184846)3 = 166.336The expression for calculating the Geometric Mean in Excel is given below: = geomean(range)where the range is any valid Excel range. Make sure the above data are entered as 128, 115, and 113 but not as 28, 15, and 13. In Stata, the following command will display arithmetic, geometric, and harmonic means. In addition, it will display their 95% confidence intervals. The level of confidence can be modified.
eBook - ePubQuantitative Techniques in Business, Management and Finance
A Case-Study Approach
- Umeshkumar Dubey, D P Kothari, G K Awari(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
Table 3.20 for a comparison of the arithmetic mean and median.3.10 Geometric Mean
Managers often come across quantities that change over a period of time, and may need to know the average rate of change over this period. Arithmetic mean is inaccurate in tracing such a change. Hence, a new measure of central tendency is needed to calculate the change rate – the ‘Geometric Mean’.TABLE 3.20 Comparison of Arithmetic Mean and MedianSerial No. Arithmetic Mean Median 1. It is calculated value, and not based on position in the series. It is especially useful in the case of open-ended classes, since only the position, and not values of items, must be known. The median is also recommended if the distribution has unequal classes, since it is much easier to compute than the mean.2. It is affected by the value of every item in the series. It is not influenced by the magnitude of extreme deviations from it. 3. It cannot be graphically ascertained. It can be determined graphically. 4. Being determined by a rigid formula, it lends itself to subsequent algebraic treatment better than the median. It is not capable of further algebraic treatment. TABLE 3.21 Growth Rate of Textile Unitswhere n is the number of values.GM=product of all values n=nx 1,x 2, ....... ,x n=(x 1,x 2,x 3...x n)1 / nGeometric Mean is applicable in many cases. Its use in calculating the growth rates of a textile unit in the southern region for the last 5 years is given in Table 3.21 .The Geometric Mean =nx 1,x 2, ....... ,x nwhere x1 , x2 , x3 ,..., xn are the terms of the growth factor and are equal to 1 + (rate/100).GM == 1.10931.07 × 1.08 × 1.10 × 1.12 × 1.1851.1093 is the average growth factor. The growth rate is calculated as 1.1093 – 1 = 0.1093. Then, 0.1093 × 100 = 10.93. So, the growth rate is 10.93% per year.The Geometric Mean, like the arithmetic mean, is a calculated average. The Geometric Mean (GM) of a series of numbers, X1 , X2 ,..., Xn
- Alan R. Jones(Author)
- 2018(Publication Date)
- Routledge(Publisher)
40 | Measures of Central Tendency pass through the Arithmetic Mean of those Logarithmic values, which is equivalent to us deriving a best fit curve which passes through the Geometric Mean of the untrans- formed (raw) data, We will have to wait until Volume III Chapters 5 and 6 to explore these dizzy delights any further. ( I know I can hardly wait myself.) For the Formula-philes: Log of the Geometric Mean = Arithmetic Mean of the Log Values Consider n terms in a series x 1 , x 2 , . . . x n with corresponding Logarithmic values L 1 , L 2 , . . . L n The Geometric Mean (GMean) is the nth root of the n terms: GMean x n i n ⎛ ⎝ ⎜ ⎛ ⎛ ⎝ ⎝ ⎞ ⎠ ⎟ ⎞ ⎞ ⎠ ⎠ ∏ x 1 1/ (1) Expanding the Product in (1): GMean n = ( ) … 1/ (2) Taking Logarithmic values of (2): log log ( ) l og ( ) /n ( ) … (3) Expanding the power: log log n ( ) l og ( ) /n ( ) … 1 (4) Expanding the Log of the Product in (4) as a Sum of its constituent values and substituting for L i : log n ( ) GMean = ( ) +… + 1 + + (5) . . . which is the Arithmetic Mean of the Log Values: 2.4.6 Average of averages: Can we take the Geometric Mean of a Geometric Mean? In short, the answer is, ‘ Yes . . . so long as we follow some basic rules’, equivalent to those for Arithmetic Means: GM Rule 1: We should not take the Geometric Mean of a number of other Geo- metric Means if the other sub-sample Means are based on different sample quantities. GM Rule 2: We should not take the Geometric Mean of a number of other Geo- metric Means if one or more known factors from the entire sample are not included in the sub-sample Means. GM Rule 3: We should not take the Geometric Mean of a number of other Geo- metric Means if the sub-sample Means are based on overlapping factors.
eBook - ePub- Dr. Russell A. Langley(Author)
- 2013(Publication Date)
- Dover Publications(Publisher)
Averages are also needed for making comparisons between different groups. One group of measurements might tell us that ‘Prince Hairylegs’ galloped a mile in 1 minute 40 seconds on Monday, 1 minute 53 seconds on Tuesday, and 1 minute 44 seconds on Wednesday; to compare such performances with those of another racehorse, we would need to work out the averages.In fact, averages are so useful that they are used on practically everything that can be measured. In view of the great variety of applications, then, it has been found necessary to devise a number of different averages, each with its own particular use.The Arithmetic MeanThe ordinary average which we learnt at school is properly termed the arithmetic mean. It is used so commonly that it is often simply called ‘the mean’ or ‘the average’, but it is generally best to give it its full title so as to avoid confusion with other averages.The arithmetic mean of any set of numbers can be calculated by applying this formula –where x1 , x2 , x3 , etc., are the values of all the individual measurements forming the set, and n = the total number of measurements in the set.Thus the arithmetic mean of 20, 23, 25, and 26 is –A shortcut (which is perfectly accurate) is to first subtract a fixed amount from each value, then find the arithmetic mean of the reduced values, and finally replace the fixed amount by adding it to the reduced mean. Thus in the above example we could subtract, say 20 from each value; the calculation would then be –Whenever possible, we shall be using shortcuts like this, because they save time and also, by keeping the numbers small, they reduce our liability to arithmetical errors.One point should be mentioned here. The arithmetic mean is only accurate as an indicator of the centre of data when that data is in units of an equi-intervalled scale
eBook - PDF- Michel Grabisch, Jean-Luc Marichal, Radko Mesiar, Endre Pap(Authors)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
4 Means and averages 4.1 Introduction and definitions It would be very unnatural to propose a book on aggregation functions without dealing somehow with means and averaging functions. Already discovered and studied by the ancient Greeks, 1 the concept of mean has given rise today to a very wide field of investigation with a huge variety of applications. Actually, a tremendous amount of literature on the properties of several means (such as the arithmetic mean, the Geometric Mean, etc.) has already been produced, especially since the nineteenth century, and is still developing today. For a good overview, see the expository paper by Frosini [124] and the remarkable monograph by Bullen [47]. The first modern definition of mean was probably due to Cauchy [56], who con- sidered in 1821 a mean as an internal (see Definition 2.52) function. We adopt this approach and assume further that a mean should be a nondecreasing function. As usual, I represents a nonempty real interval, bounded or not. The more general cases where I includes −∞ and/or ∞ will always be mentioned explicitly. Definition 4.1. An n-ary mean in I n is an internal aggregation function M : I n → I. An extended mean in ∪ n∈N I n is an extended function M : ∪ n∈N I n → I whose restriction to each I n is a mean. It follows immediately from Proposition 2.54 that a mean is nothing other than an idempotent aggregation function. Moreover, by Corollary 1.8, if M : I n → I is a mean in I n , then it is also a mean in J n , for any subinterval J ⊆ I. The concept of mean as an average or numerical equalizer is usually ascribed to Chisini [60, p. 108], who gave in 1929 the following definition: Let y = F(x 1 , . . . , x n ) be a function of n independent variables x 1 , . . . , x n . A mean of x 1 , . . . , x n with respect to the function F is a number M such that, if each of x 1 , . . . , x n is replaced by M , the function value is unchanged, that is, F(M , . . . , M ) = F(x 1 , . . . , x n ).
eBook - PDF- Valery A. Slaev, Anna G. Chunovkina, Leonid A. Mironovsky(Authors)
- 2013(Publication Date)
- De Gruyter(Publisher)
The measurement of the diameter of a tree, which is conventionally done in forestries, is a quadratic mean diameter, rather than an arithmetic mean diameter [133]. Example 4.2.17. Given three square lots with sides x 1 D 100 m; x 2 D 200 m; x 3 D 300 m. We want to replace the different values of the side lengths by a mean reasoning in order to have the same general area for all the lots. The arithmetic mean of the side length . 100 C 200 C 300 / : 3 D 200 m does not satisfy this condition, since the joint area of the three lots with the side of 200 m would be equal to 3 . 200 m / 2 D 120 000 m 2 . 220 Chapter 4 Algorithms for evaluating the result of two or three measurements At the same time, the area of the given three lots is equal to . 100 m / 2 C . 200 m / 2 C . 300 m / 2 D 140 000 m 2 . The quadratic mean will be the correct answer: O x D r . 100 / 2 C . 200 / 2 C . 300 / 2 3 D 216 m. For two measurements x 1 D a , x 2 D b the quadratic mean can be expressed through an arithmetic, geometric, and contraharmonic means with the help of the formula Q.a , b/ D G.A.a , b/ , C.a , b// , or for short, Q D G.A , C/ . Its correctness follows from the chain of equalities: G.A.a , b/ , C.a , b// D G a C b 2 , a 2 C b 2 a C b D s a C b 2 a 2 C b 2 a C b D r a 2 C b 2 2 D Q.a , b/ . 4.2.4 Geometrical interpretation of the means The arithmetic, geometric, and harmonic means are known as the Pythagorean means. One of the geometric presentations which allow the Pythagorean means of two num-bers to be visually compared is shown in Figure 4.12. They are shown in the form of the segments a and b , forming a diameter of a semicircumference. Its radius is equal to the arithmetic mean of the numbers a and b , and the length of the perpendicular G is equal to their Geometric Mean. The length of the leg H of a right-angle triangle with the hypotenuse G is equal to the harmonic mean.
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