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.
6 Key excerpts on "Geometric Mean"
eBook - ePub- Perumal Mariappan(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...When the average is weighted, the importance of all the items is taken into account. The ordinary average becomes a special case of a weighted average. If we take the weight uniformly as one, the weighted mean becomes ordinary mean. 4.9.1 Advantages of the Weighted Mean The weighted mean is used in the following instances. • It is used in constructing index numbers. The relative weights of the expenditure like food, clothing, housing, and such are obtained by surveys and the cost-of-living index is calculated with those weights. • In educational institutions, it is used to assess the real merit of the student. • It is used in evaluating standardized death rates. 4.10 Geometric Mean The Geometric Mean (GM) is the n th root of the product of n items. G M = (X 1 × X 2 × … … ….. × X n) 1 / n It is a mathematical average and not a positional average. It takes all the given values in its evaluation. It gives less weight to the end values than the arithmetic mean. Usually this value will be less than the mean. If any one of the element takes the value 0, GM = 0, and if any element is negative, the value of GM is imaginary. So, it is useful only in certain special situations. NOTE: GM can be used in the following situations: • to calculate the rates of change. • certain cases of averaging ratios and percentages. • problems involving rates of interest of invested money. • can be used to interpolate between items that have a uniform rate of change. • Used in the evaluation of index numbers. Example : A sum of money was invested for 5 years. The average rates of return for the investment for the 5 successive years were as follows: 5 %, 4 %, 5 %, 6 %, 3 % What was the average rate of interest for these 5 years? If we assume that the amount invested as $100. The total amount earned in the 5 years is 105, 104, 105, 106, and 103. (i.e., 5% = 100 + 5 = 105)...
eBook - ePub- Rafael Perera, Carl Heneghan, Douglas Badenoch(Authors)
- 2011(Publication Date)
- BMJ Books(Publisher)
...If you’re going to interpret what your data are telling you, and communicate it to others, you will need to summarize your data in a meaningful way. Typical mathematical summaries include percentages, risks and the mean. The benefit of mathematical summaries is that they can convey information with just a few numbers; these summaries are known as descriptive statistics. Summaries that capture the average are known as measures of central tendency, whereas summaries that indicate the spread of the data usually around the average are known as measures of dispersion. The arithmetic mean (numeric data) The arithmetic mean is the sum of the data divided by the number of measurements. It is the most common measure of central tendency and represents the average value in a sample. To calculate the mean, add up all the measurements in a group and then divide by the total number of measurements. The Geometric Mean If the data we have sampled are skewed to the right (see p. 7) then we transform the data using a natural logarithm (base e = 2.72) of each value in the sample. The arithmetic mean of these transformed values provides a more stable measure of location because the influence of extreme values is smaller. To obtain the average in the same units as the original data – called the Geometric Mean – we need to back transform the arithmetic mean of the transformed data: The weighted mean The weighted mean is used when certain values are more important than others: they supply more information. If all weights are equal then the weighted mean is the same as the arithmetic mean (see p. 54 for more). We attach a weight (w i) to each of our observations (x i): The median and mode The easiest way to find the median and the mode is to sort each score in order, from the smallest to the largest: The median is the value at the midpoint, such that half the values are smaller than the median and half are greater than the median...
eBook - ePubSensory Evaluation of Food
Statistical Methods and Procedures
- Michael O'Mahony(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...The mode is easily computed and interpreted (except where there are several possible modes) but is rarely employed; no common inferential procedure makes use of the mode. The median is a useful descriptive statistic which fits the requirements of clear and effective communication, but it is of only limited use where inferences about population parameters are to be made from a sample. It is less affected by a preponderance of extreme values on one side of a distribution (skew) than the mean, and is to be preferred to the mean in cases where there are extreme values or values of indeterminable magnitude (e.g., when values are too big to be measured on the scale being used). The mean uses more information than either of the other measures, shows less fluctuation over successive samples, and is employed in all classical (parametric: see Section 2.7) statistical procedures. The mean that we have been discussing is the arithmetic mean; there are other means. Instead of adding all the N values in a sample and dividing the total by N, we could multiply all the numbers together and take the Mh root of the value obtained; this is called the Geometric Mean. It is equivalent to calculating a mean by converting all the numbers to logarithms, taking their arithmetic mean, and then converting the answer back to an antilogarithm. The Geometric Mean is the appropriate mean to use when plotting power functions of perceived versus physical intensities, obtained by the magnitude estimation intensity scaling procedure. Another type of mean is the harmonic mean which is obtained from reciprocal values. This is discussed in Section 9.6, dealing with Duncan’s multiple range test. There is one more important point. An average or mean of a sample of scores is a middle value or a measure of central tendency. Obviously, not all the scores are equal to the mean; it is the middle value where the high scores cancel out the low scores...
eBook - ePub- Ram C. Bhujel(Author)
- 2011(Publication Date)
- Wiley-Blackwell(Publisher)
...The difference in importance (weight) may be due to a different number of observations for particular figures or the values may carry different weights/ranks. In such a case, AM is often referred to as weighted mean/average: which is expressed as: The example in Table 5.1 shows the scores of two students for aquaculture entrance using weights for different subjects based on their relevance. The score in biology gets 3 weights because it is the most important subject for aquaculture, and other subjects get lower scores because they are of less importance to the study of aquaculture. In this case, although Student 2 has a higher total score (210 vs. 215), Student 1 has higher weighted scores (total and weighted mean) because of the higher weight given to biology, which is the most important background knowledge required for aquaculture. Because Student 1 received a higher score in biology, s/he will be preferred over Student 2. Table 5.1 Comparison of two students based on the simple score and weighted scores. 5.2.1.2 Geometric Mean If data are in geometric series (multiplication of a constant value), e.g. 2, 4, 8, 16…(X × 2), then Geometric Mean (GM) represents the central location. It is calculated by multiplying the values of all the observations and nth root, such as: If data are in a geometric series, log transformation is required before computing the GM. After computing the GM, it is then transformed back to the actual GM. 5.2.1.3 Harmonic mean Harmonic series is the reciprocal of arithmetic series; for example, if the arithmetic series is 2, 4, 6, 8 then the harmonic series is 1/2, 1/4, 1/6, 1/8, which is expressed as 1/(x + 2)...
eBook - ePub- Derek L. Waller(Author)
- 2016(Publication Date)
- Routledge(Publisher)
...For this a questionnaire is used to determine a customer’s subjective (qualitative) opinion. These opinions are then given a quantitative score as shown in Table 1.18 (The same as the fourth row of Table 1.7). Table 1.18 The results from the questionnaire are then analyzed using weighted averages. In this case we look for a central value that considers the importance of each opinion. An illustration is shown in Table 1.19 that is the opinion of 15 clients on the hotel breakfast service. The weighted average of the response is calculated by: Weighted average = ∑ Number of responses*score Total responses From Table 1.19 : Weighted average = 2 * 1 + 1 * 2 + 1 * 3 + 5 * 4 + 6 * 5 15 = 3.80 This translates into saying the breakfast service is considered between acceptable and good, and closer to being good. [ In Excel for a matrix such as this the function, SUMPRODUCT ] Table 1.19 Geometric Mean The Geometric Mean is a measure of central tendency for data, such as revenues, costs, interest rates, prices, productivity etc., that are changing over time. Consider Table 1.20 that gives food sales over six years. Table 1.20 The information from this table has been expanded to calculate the percentage annual change and the growth factor each year as shown in Table 1.21. For example, the annual percentage change for Year 2 is (912,140 − 876,000)/876,000 or 0.0413 or 4.13%. The corresponding growth factor for Year 2 is 1.0413 (1 + 0.0413). The growth factor is the amount by which the value of the previous period has to be multiplied in order to give the current value. If the percentage change is negative, then the growth factor is less than 1 as for example in Year 4. Table 1.21 The compounded average growth rate, or Geometric Mean for a dataset with “n” time periods, is: (product of growth rates) n In this case the Geometric Mean growth rate for the data in Table 1.21 is: 1.0413 * 1.1221 * 0.9769 * 1.0238 * 1.0897 5 = 1.0495 The Geometric Mean growth is 4.95% (1.0495 – 1)...
eBook - ePubThe Handbook of Traditional and Alternative Investment Vehicles
Investment Characteristics and Strategies
- Mark J. P. Anson, Frank J. Fabozzi, Frank J. Jones(Authors)
- 2010(Publication Date)
- Wiley(Publisher)
...The average return on an asset, observed over a long period of time, is often used as the expected return for future years. Although the future return is unknown and cannot be predicted with great accuracy, the historical average return is as good a guess as any of what the return will be in the future. The Geometric Mean is a bit more complicated. It uses compounding to determine the mean return. For a set of observations related to an asset return stream, the Geometric Mean is equal to where R (G) = the return for the Geometric Mean R (1), R (2), R (T) = the returns to asset X in periods 1, 2, all the way to period T T = the number of periods over which we calculate the Geometric Mean means that we take the T th root of our compound return stream to determine R (G) Using the data in Exhibit A.1 and the previous equation, we calculate the Geometric Mean as: We note that the Geometric Mean is slightly lower than the arithmetic mean. This is the case when the returns are changing through time. In fact, there is an approximate relationship between the two: where Var (R) is the variance of the rate of return on asset X. It can be seen that if the annual rate of return on the asset has no volatility, then the Geometric Mean and arithmetic mean will be equal. One more observation is appropriate. The Geometric Mean is also used in the calculation for the cumulative average growth rate (CAGR). CAGR assumes that an asset, cash flow, or some other random variable grows at a constant rate of return compounded over a sample period of time. For example, assume that we purchase a stock of company A at $50 and hold it for three years until it reaches the value of $100. What is our CAGR? The calculation is simple: We combine our holding period return calculation with the equation for the Geometric Mean. First, determine the holding period return as $100/$50 = 2 Then our CAGR is equal to...
