Measures of dispersion

The most basic step in statistical analysis of a data set is to describe it descriptively, that is, to compute properties associated with the data set and to display the data set in an informative manner. A data set consists of a finite number of samples or data points. In this book, a data set will be denoted using either set notation, that is, jti, X2. x or vector notation, that is, as if = (x],jC2. x ). Set notation is useful for describing and listing the elements of a data set, while vector notation is useful for mathematical manipulation. The size of the data set is equal to n. The most common descriptive statistics include measures of central tendency and dispersion. 

Measures of central tendency provide some information about the most common value in the data set. The basic measures of central tendency include the mean, mode, and median. Since the most common such measure is the mean, which is often colloquially called the average, all of these measures are often referred to as averages. A summary of the basic properties of these measures is provided in Table 1.1. 

The mean is a measure of the central value of the set of numbers. It is often denoted as an overbar (o) over a variable, for example, the mean of x would be written as x. The most common mean is simply the sum of all the values divided by the total number of data points, n, that is. 

Alternatively, a weighted mean can be computed, where for each value a weight w is assigned, that is, 

Mean —r n y. . Xi X= =1 n Easy to compute and interpret Can easily be influenced by extreme values 

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Another measure of dispersion is the coefficient of variation, which is merely the standard deviation expressed as a fraction of the arithmetic mean, viz., s/x. It is useful mainly to show whether the relative or the absolute spread of values is constant as the values are changed. 

Some measure of dispersion of the subgroup data should also be plotted as a parallel control chart. The most reliable measure of scatter is the standard deviation. For small groups, the range becomes increasingly significant as a measure of scatter, and it is usually a simple matter to plot the range as a vertical line and the mean as a point on this line for each group of observations. 

The number of plates, Np, and the height equivalent to a theoretical plate, HETP, are defined as measures of dispersion effects as ... 

These equations are ealled the moment equations, beeause we are effeetively taking moments of the data about a point to measure the dispersion over the whole set of data. Note that in the varianee, the positive and negative deviates when squared do not eaneel eaeh other out but provide a powerful measure of dispersion whieh... 

To express the measure of dispersion in the original scale of measurement, it is usual to take the square root of the variance to give the standard deviation ... 

Tlie probabilities given in Eqs. (20.5.10), (20.5.11), and (20.5.12) are tlie source of the percentages cited in statements 1, 2, and 3 at tlie end of Section 19.10. These can be used to interpret tlie standard deviation S of a sample of observations on a normal random variable, as a measure of dispersion about tlie... 

Rheological methods of measuring the interphase thickness have become very popular in science [50, 62-71]. Usually they use the viscosity versus concentration relationships in the form proposed by Einstein for the purpose [62-66], The factor K0 in Einstein s equation typical of particles of a given shape is evaluated from measurements of dispersion of the filler in question in a low-molecular liquid [61, 62], e.g., in transformer oil [61], Then the viscosity of a suspension of the same filler in a polymer melt or solution is determined, the value of Keff is obtained, and the adsorbed layer thickness is calculated by this formula [61,63,64] ... 

The standard deviation, Sj, is the most commonly used measure of dispersion. Theoretically, the parent population from which the n observations are drawn must meet the criteria set down for the normal distribution (see Section 1.2.1) in practice, the requirements are not as stringent, because the standard deviation is a relatively robust statistic. The almost universal implementation of the standard deviation algorithm in calculators and program packages certainly increases the danger of its misapplication, but this is counterbalanced by the observation that the consistent use of a somewhat inappropriate statistic can also lead to the right conclusions. 

The function that provides a quantitative measure of dispersal is called entropy and is symbolized S. In 1877, the Austrian physicist Ludwig Boltzmann derived Equation, which defines the entropy of a substance in terms of W, the number of ways of describing the system. 

From the variance some measures of dispersion are derived. The most commonly used are... 

For this type of estimation the results of the measurements have to be arranged in numerical order. Together with the median also robust measures of dispersion, such as median absolute deviation, mad yj = med y, — med yi, and interquantile ranges (Sachs [1992]) should be used. 

The open ends boundary conditions apply when the measuring points are some distances from the ends. Such an arrangement is used in making accurate measurements of dispersion coefficients. 

The variance of a population is another useful measure of dispersion and reflects the extent of the differences between the data. Denoted by a2, it is equal to the mean squared deviation of the individual values from the population mean. Usually, the symbols V and s2 are used for the variance deduced from sample data. Thus, for a sample of N data drawn from a population with mean /z, the estimated variance is... 

The use of the mean with either the SD or SEM implies, however, that we have reason to believe that the sample of data being summarized are from a population that is at least approximately normally distributed. If this is not the case, then we should rather use a set of statistical descriptions which do not require a normal distribution. These are the median, for location, and the semiquartile distance, for a measure of dispersion. These somewhat less familiar parameters are characterized as follows. 

Our discussion has led to the measure of dispersion by a dimensionless group D/uL. Let us now see how this affects conversion in reactors. 

Particle size distributions (PSD) measurement of dispersion of metal particles on supports... 

The three terms of the solubility parameter, 5, 5p, and 5h, represent measures of dispersive forces, polar forces, and hydrogen-bonding forces, respectively. Two liquids with similar solubility parameters are soluble. 

A characteristic of biological systems is variability, with most values of a variable clustered around the middle of the range of observed values, and fewer at the extremes of the range. The measure of location or central tendency gives an indication where the distribution is centred, while a measure of dispersion indicates the degree of scatter or spread in the distribution. The most widely used measure of central tendency is the arithmetic mean or average of the observed values, i.e, the sum of all variable values divided by the number of observations. Another measure of central tendency is the median, the middle measurement in the data (if n is odd) or the average of the two middle values (if n is even). The median is the appropriate measure of central tendency for ordinal data. 

The mean summarises only one aspect of a distribution. We also need some measure of spread or dispersion, the tendency for observations to depart from the central tendency. The standard measure of dispersion is the variance ... 

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