Middle value, distribution

The median of an even number of results is nothing but the average of the two middle values provided the results are listed in order whereas for an odd number of results the median is the middle value itself. However, the mean and the median are exactly identical in the case of a truly symmetrical distribution. In short, median is an useful measure specifically when dealing with very small samples. 

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. 

Median The middle value in a population distribution, above and below which he an equal number of individual values midpoint (CARB, 2000). 

When a distribution of data is asymmetrieal, it is sometimes desirable to compute a different measure of eentral value. This seeond measure, known as the median, is simply the middle value of a distribution, or the quantity above which half the data lie and below which the other half lie. If n data points are listed in their order of magnitude, the median is the [ n + l)/2]th value. 

This distribution of molecular masses and sizes can now be routinely quantified for all soluble polymers by gel permeation chromatography (1). While this distribution is useful both in practice and in theory, many properties of the polymer sample depend on a single middle value of the distribution. There are, however, several ways to reduce the distribution to a middle value. Each of these reductions is important because they correlate with or predict a certain subset of physical or chemical properties of the polymer. The common averages of a polymer molecular mass distribution are number (m=1), viscosity (m=1 +a ), weight (m=2), and z or zeta average (m=3). These "averages" are actually ratios of the m- moment of the molecular mass distribution to the preceding moment in the above list. The moments of a distribution are fundamental properties of any distributed variable and are covered in detail in reference 2. 

Median me-de-9n n. The value in an arrayed set of repeated measurements that divides the set into two equal-numbered groups. If the sample size is odd, the medium is the middle value. The median is a useful measure of the center when the distribution is strongly skewed toward low or high values. Compare arithmetic mean. 

The median is defined as the middle value (or arithmetic mean of the two middle values) of a set of the numbers. Thus, the median of 4, 5, 9, 10, and 15 is 9. It is also occasionally defined as the distribution midpoint. Further, the median of a continuous probability distribution function f(x) is that value of c so that... 

Figure 18. Long-term extreme value distributions obtained by two different approaches (a) All load peaks (b) Only the extreme peaks observed during a voyage of m nautical miles. Three different initial distributions are considered in each case A. 1-parameter exponential (leftmost curves) B. Weibull distribution (middle curve) C. 3-parameter exponential distribution (rightmost curve).

Fig. 2 Numeric simulations of the EPR spectra of nitroxide radicals ((A)-(Q) and Gd(iii) complexes ((D)-(F)). For both species the spectra were computed for the X-band detection frequency (9.5 GFIz, (A) and (D)) as well as for Q band (35 GFIz, (B) and (E)) and W band (95 GFIz, (C) and (F)). Spectra were simulated with EasySpin software (www. easyspin.org). Spectroscopic parameters for nitroxide radicals g-tensor eigenvalues -[2.0085 2.0061 2.0022], hyperfine tensor eigenvalues - [13 13 100] MFIz, FWFIM -[0.3 0.3] mT (mixed Lorentzian/Gaussian line shape). For nitroxide radicals subspectra corresponding to the spin projection of -Fl (left subspectrum), 0 (middle subspectrum), and -1 (right subspectrum) are plotted as dashed lines. Spectroscopic parameters for Gd(iii) centres isotropic g-value of 1.991 D-values normally distributed with =1500 MHz and a D)= /5 D/ values distributed, according to P(x)=x/3-2x /9 (see ref. 29 and 65).

The interlaminar shear stress, t, has a distribution through half the cross-section thickness shown as several profiles at various distances from the middle of the laminate in Figure 4-54. Stress values that have been extrapolated from the numerical data at material points are shown with dashed lines. The value of is zero at the upper surface of the laminate and at the middle surface. The maximum value for any profile always occurs at the interface between the top two layers. The largest value of occurs, of course, at the intersection of the free edge with the interface between layers and appears to be a singularity, although such a contention cannot be proved by use of a numerical technique. 

Figure 1.21. Monte Carlo simulation of six groups of eight normally distributed measurements each raw data are depicted as x,- vs. i (top) the mean (gaps) and its upper and lower confidence limits (full lines, middle) the confidence limits CL(s ) of the standard deviation converge toward a = 1 (bottom, Eq. 1.42). The vertical divisions are in units of 1 a. The CL are clipped to +5a resp. 0. .. 5ct for better overview. Case A shows the expected behavior, that is for every increase in n the CL(x,nean) bracket /r = 0 and the CL(i t) bracket a - 1. Cases B, C, and D illustrate the rather frequent occurrence of the CL not bracketing either ii and/or ff, cf. Case B n = 5. In Case C the low initial value (arrow ) makes Xmean low and Sx high from the beginning. In Case D the 7 measurement makes both Cl n = 7 widen relative to the n 6 situation. Case F depicts what happens when the same measurements as in Case E are clipped by the DVM.

Ouchiyama and Tanaka (Ol) further established that the ratio of Dn/Dx is uniquely related to the exponent y hence, it is possible to compute the value of the exponent y from experimental size distributions. From the data on limestone and sand systems, these authors found that y = 1,2, and 3 respectively, in the initial, middle, and later stages. Hence, from Eq. (91), in the initial region,... 

Fig. 2 0---0 intermolecular/interionic interactions between COOH and COOH/COO( 1 groups in protonated or partially deprotonated polycarboxylic acids. Top histogram the distribution of 0---0 interactions obtained without neutral/charge discrimination (mean value 2.632 A). When the presence of an ionic charge is taken into account the distributions of 0---0 distances are those in the middle [(n)0(H)-"OCOOH(n) and (")0(H)-"Ocoon( mean value 2.650 A] and in the bottom [( )0(H) Ocoo( ), mean value 2.5333 A] histograms... 

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