What is defined as the AM of various powers of deviations of the variable taken from zero?

Arithmetic versus Geometric Mean

The arithmetic mean of percentage changes or log returns is the simple average. Because of the potentially powerful effects of compounding returns over time, however, the geometric average can provide a more representative statistical characterisation of price change in a typical year. For example, suppose a house price produces returns of r1 = r2 = 0, r3 = 0.50, and r4 = r5 = 1.00, the arithmetic average return is (0.00 + 0.00 + 0.50 + 1.00 + 1.00)/5 = 0.50. If one simply adds up the percentage changes of 50 and 100, and then 100 percentage points again, one might mistakenly surmise that every dollar invested would at the end of these 5 years be worth $2.50. But because of compounding, the 5-year gross return would be (1)(1)(1.50)(2)(2) = 6. In other words, every dollar invested at time t = 0 would be worth $6.00 at the end of these 5 years.

If the arithmetic average return is used to estimate the 5-year gross return, a distortion in the opposite direction occurs: 1.505 = 7.5938. Every dollar invested at the outset did not in fact turn into $7.59. To correct this distortion, the geometric average provides a characterisation that properly accounts for compounding returns over time: ((1)(1)(1.50)(2)(2))1/5−1 = 0.4310. In other words, each dollar invested grew by around 43% each year. Compounding annual returns of 43% over 5 years produces the exact 5-year gross return, because 1.43105 = (1)(1)(1.50)(2)(2).

Table 1 presents 5-year arithmetic and annualised geometric mean returns for each of the three houses. The geometric mean is generally smaller than the arithmetic mean return. Note, too, that the standard deviation risk measure is based on average distance from the arithmetic rather than geometric mean.

5.2.1.1 Mean

The arithmetic mean, or simply called the mean is calculated by dividing the sum of all observations in the dataset by the total number of observations. The mean is significantly affected by the outliers, that is, extremely large or small values. The mean is also called as mathematical expectation, or average. The sample mean is denoted by x¯ (pronounced as “x-bar”) and is calculated using the following equation.

(5.1)x¯=1n∑i=1nxi

where n is the total number of observations in the sample and x1,x2,….xn are individual observations. As the sample mean changes from one sample to another, it is considered as a random variable. If the whole population is used, then x¯ is replaced by the Greek symbol, μ and is given by

(5.2)μ=1N ∑i=1Nxi

where N is the total number of observations in the population. The population mean is always fixed and is a nonrandom variable.

1.3 Statistical Indexes

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Arithmetic mean x¯: when observing a quantity x for n times under the same conditions, the mean x¯ is computed as follows:

(1.3)x¯=∑i=1n xinn→∞

Variance σ2: the theoretical mean of the squared residual errors that can be computed as follows:

(1.4) σ2=∑i=1nxi−x¯2n−1=∑i=1nvi 2n−1

The denominator n − 1 is called the redundancy r or the degree of freedom. The minimum observations we need to determine the variance is only 1 value, which is termed as no; accordingly, the rest of the observations are redundant, and r can be formulated as:

(1.5)r=n−no

Standard deviation σ: the square root of the variance that expresses by how much the observations differ from the MPV or the mean value. Further, it is a measure of how spread out the observations are. Therefore, standard deviation is an expression of precision computed as:

(1.6)σ=∑i=1nxi−x¯2n−1=∑i=1nvi2n−1

When the reference values are available, the standard deviation is termed as the Root Mean Squared Error (RMSE), which indicates by how much the observed or derived quantities deviate from the reference (true) values.

(1.7)RMSE=∑i=1nxi−x¯REFRENCE2n

Standard error σx¯ : represents the standard deviation of the mean, which is computed as:

(1.8)σx¯=σn=∑i=1nxi−x¯2n n−1=∑i=1nvi2nn−1

Example 1.1

Other Simple Means or Combinations of Rates

The simple arithmetic mean, i.e., half the sum of the male and female rates, is an obvious possibility. However, it has received little attention because it yields marriages even when there are no persons of one sex.

The geometric mean, i.e., the square root of the product of male and female rates, is a simple average that avoids the problem of marriages when there are no persons of one sex. Goodman (1967) discussed a geometric mean variant where the two sexes are given unequal weight, that is where the male rate would be raised to the power α (0 ≤ α ≤ 1) and the female rate would be raised to the power (1 − α). Parameter α would indicate the ‘dominance’ of one sex.

Rather than averaging male and female rates, one could simply take the minimum of the two rates. That ‘minimum’ approach avoids the problem of marriages in the absence of one sex, and has been incorporated into more sophisticated models.

1.10.1.1 Mean Center, Median Center, and Central Feature

Similar to the arithmetic mean in regular statistics that represents the average value of observations, the mean center may be interpreted as the average location of a set of points. If one wishes to assess the mean center of lines or areas, they may be represented by the coordinates of the center points of the lines or the centroids of the areas (Mitchell, 2005, p. 33). As a location can be represented by its x and y coordinates, the mean center’s coordinates (X¯, Y¯) are the average x coordinate and average y coordinate of all points (Xi, Yi) for i = 1, 2, …, n such that

X¯=∑iXi/n;Y¯ =∑iYi/n.

If the points are weighted by a variable (e.g., population), denoted by wi, the weighted mean center has coordinates

X¯=∑iwiXi/∑iwi;Y¯= ∑iwiYi/∑i wi.

Two other measures of central tendency are used less often: median center is the location that has the shortest total distance to all points, and central feature is the point feature (among all input points) that has the shortest total distance to all other points. The median center or central feature can be viewed as the most accessible location to the point set as the total distance from all data points is the minimum. The difference is that the central feature refers to an existing point feature and the median center does not. With this restriction, the total distance from the central feature to all other point features is always greater than or equal to the total distance from the median center to all point features. Computing the central feature is straightforward: select the point with the shortest total distance from the others. The median center is approximated by an iterative algorithm (e.g., Kuhn and Kuenne, 1962). Beginning with the mean center, a candidate median center is found and then refined until it approximately represents the location with the minimum total distance to all features (Wong and Lee, 2005, Chapter 5).

Measures of Variability Based on Deviations

The sum of the deviations from the arithmetic mean is always zero:

∑i=1N(Xi−μ) =∑(Xi−∑XiN)=∑Xi−N(∑XiN)=0.

Because the positive and negative deviations cancel out, measures of variability must dispense with the signs of the deviations; after all, a large negative deviation from the mean is as much of an indication of variability as a large positive deviation.

In practice, there are two methods to eradicate the negative signs: either taking the absolute value of the deviations or squaring the deviations. The mean absolute deviation is one measure of deviation, but it is seldom used. The primary measure of variability is, in effect, the mean squared deviation. For a population, the variance parameter of the variable X, denoted by σX2, is defined as:

σX2=∑i=1N(Xi−μ)2N .

However, the units of the variance are different than the units of the mean or the data themselves. For example, the variance of wages is in the units of dollars squared, an odd concept. For this reason, it is more common for researchers to report the standard deviation, denoted by σX, which for a population is defined as:

σX= ∑i=1N(Xi−μ)2N.

Unlike the formulas for the population and sample means, there is an important computational difference between the formulas for the variance and standard deviation depending on whether one is dealing with a population or a sample. The formulas for the sample variance and sample standard deviation are as follows:

SX2 =∑i=1n(Xi−X¯) 2n−1,SX=∑i=1n (Xi−X¯)2n−1.

Note that the divisor in these calculations is the sample size, n, minus 1. The reduction is necessary because the calculation of the sample mean used up some of the information that was contained in the data. Each time an estimate is calculated from a fixed number of observations, 1 degree of freedom is used up. For example, from a sample of 1 person, an estimate (probably a very bad one) of the mean income of the population could be obtained, but there would be no information left to calculate the population variance. One cannot extract two estimates from one data point. The correction for degrees of freedom enforces this restriction; in the example, the denominator would be zero and the sample standard deviation would be undefined.

The variance and the standard deviation, whether for populations or samples, should not be thought of as two different measures. They are two different ways of presenting the same information.

A final consideration is how the variability of two different samples or populations should be compared. A standard deviation of 50 points on a test means something different if the maximum score is 100 or 800. For this reason, it is often useful to consider the coefficient of relative variation, usually indicated by CV, which is equal to the standard deviation divided by the mean. The CV facilitates comparisons among standard deviations of heterogeneous groups by normalizing each by the appropriate mean.

AVERAGE AGE AT INFECTION (A)

Average age at infection (A) is a useful summary measure indicating the arithmetic mean age at infection of all cases over some (often unspecified) period of time. For a childhood infection such as measles, it also provides an approximate indication of how narrow is the age window for vaccination which public health programs should aim for between the loss of passive immunity from maternal antibodies and the age at which the majority of children will already have been infected. The average age at infection will also provide some indication of the likely disease burden from an infection in which processes of morbidity or mortality are age-related. For infections provoking long-lasting immunity the average age at infection will change according to the evolution of the age profile of herd immunity in the population, so that it is likely to be higher when an infection is introduced into the population for the first time and lower as the infection becomes endemic in the population and many older individuals are immune following recovery from infection; the infectious agent is then primarily reliant on births to supply new susceptible individuals to the population. If an infection is characterized by epidemics repeated at longer intervals of time, the average age at infection will rise and fall accordingly. Note that the introduction of a vaccination program, through its impact on the age distribution of immunity, will itself change the average age at infection, and as infection becomes more rare as a result of a successful vaccination program, those who have not been vaccinated will themselves be less at risk of infection (herd immunity effect). If these individuals do eventually become infected it is likely to be at an older age than would otherwise have been the case in the absence of vaccination. Should morbidity increase with age it is therefore theoretically possible, dependent on circumstances, for vaccination to result in an increased burden of disease (Williams and Manfredi, 2004). Rubella infection constitutes a prime example of such a risk, being a mild infection in childhood, but should the average age at infection be delayed there is likely to be an increase in risk of infection for women in their fertile years, with a resulting risk of congenital rubella if infection occurs in the first trimester of pregnancy (Edmunds et al., 2000). Processes of demographic change whether a result of increase or reduction in the population growth rate or the onset of population decline, by changing the population age structure (and indirectly therefore the age profile of susceptibility), may also change the average age at infection (Williams and Manfredi, 2004). The arithmetic mean, of course, tells us little about the variance in ages at infection which may itself be important—if there is substantial variance in ages at infection, a substantial proportion of cases may occur in younger or older (perhaps much older) ages, so that when considering how changes in the age distribution of infection may affect age-dependent morbidity or mortality, median and percentile measures also become important.

4.4.5 Pricing Options with Microsoft Excel

In this section, we show how the Visual Basic within Excel can be used to create powerful derivative pricing applications based on the Black–Scholes formula. We will explain how Excel’s Visual Basic can be used to create an application that prices a selection of simple European put and call options at the press of a button.

In Section  4.4.3, we derived the Black–Scholes formula

c(S,E,τ)=SN1(d1)− e−rτEN1(d2)andp(S,E,τ)=−SN1(−d1)+e−rτEN 1(−d2),

where

d2=log (S∕E)+(r−σ2∕2)τστ=d1−στ,

where S is the current value of the asset and σ is the volatility of the asset, and N1(x)=(1∕2π)∫−∞xe−x2∕2dx.

The univariate cumulative standard normal distribution, N1(x), can be evaluated in Excel by using its built-in function NORMDIST. The definition of this function is as follows:

NORMDIST(x,mean,standard_dev,cumulative)

This function returns the normal cumulative distribution for the specified mean and standard deviation.

Function parameters:

x: the value for which you want the distribution.

mean: the arithmetic mean of the distribution.

standard_dev: the standard deviation of the distribution.

cumulative: a logical value that determines the form of the function. If cumulative is TRUE, NORMDIST returns the cumulative distribution function; if FALSE, it returns the probability density function.

If mean = 0 and standard_dev = 1, NORMDIST returns the standard normal distribution.

This function can be used to create a Visual Basic function to calculate European option values within Excel, see Code excerpt 4.5.

Code excerpt 4.5

Visual basic code to price European options using the Black–Scholes formula.

Once the function has been defined, it can be accessed interactively using the Paste Function facility within Excel as shown in Fig. 4.1.

The function bs_opt can also be incorporated into other Visual Basic code within Excel. To illustrate, if the following Visual Basic subroutine is defined in Code excerpt 4.6.

Code excerpt 4.6

Visual basic code that uses the function bs_opt.

When the button labelled ”CALCULATE OPTIONS” is clicked, the values of 22 European options will be calculated using the data in columns 1–3 on worksheet 1, see Figs 4.2 and 4.3.

The cumulative standard normal distribution can also be used to provide analytic solutions for a range of other exotic options such as Barrier options, Exchange options, Lookback options, Binary options, etc.

Surface Finish on Stainless Steel

The surface finish on stainless steel is best expressed as the Ra value, which is the arithmetic mean deviation of the readings taken by a surface profile meter measured in microns. Expressed mathematically

R a=1lm∫x=0x=lm|y|dx

An Ra value of less than 0.8 μm is regarded as adequate for most purposes. In some instances where high turbulence in the fluid product occurs, Ra values higher than 0.8 μm may be acceptable. A sheet that has been cold rolled will usually have an acceptable Ra. The requirement can also be achieved by polishing.