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Bioaerosol Particle Statistics
Published in Christopher S. Cox, Christopher M. Wathes, Bioaerosols Handbook, 2020
A straight line on the log-probability graph is convenient for analysis of the distribution. Though they have little physical significance in interpreting experimental results,10 the geometric median diameter and geometric standard deviation describe completely the lognormal distribution on that graph. The geometric median diameter is readily determined from the 50% point of the graph (Figure 5.4). The geometric standard deviation is calculated by dividing the diameter taken from the 84% probability point by the geometric median diameter, or dividing the geometric median diameter by the diameter taken from the 16% probability point. The mode, median, and mean are always unequal in a lognormal distribution, whereas they are equal in a normal distribution. However, the geometric mean diameter (Equation 5.7) is equal to the geometric median diameter in a lognormal distribution.
Fine-Dispersion Aerosols in the Environment of Human Life
Published in Katarzyna Majchrzycka, Nanoaerosols, Air Filtering and Respiratory Protection, 2020
called the geometric mean diameter, and the standard deviation is replaced by the standard deviation of the logarithmic diameters, called the geometric standard deviation, defined as: ln(σg)=∑ni(ln(di)−ln(dg))2N−1.
Overview of air pollution
Published in Abhishek Tiwary, Ian Williams, Air Pollution, 2018
Many air pollutant concentrations are highly variable, with a low average and a large range between near-zero minimum values and maxima during episodes. Furthermore, effects on organisms such as people and plants often depend on short episodes of high concentration. How can we summarise this breadth? The standard method is to examine the frequency distribution of the data, by making a tally of the numbers of hourly or daily means that fall in the ranges 0–4, 5–9, 10–14, and so on; ppb, for example. The number in each range is then converted to a proportion or percentage of the total number of readings – this is the frequency. A common result of such an analysis of air pollutant concentrations is that the data approximate to a log-normal frequency distribution – the log of the frequency is normally distributed with concentration. This fact has certain implications for the way in which the data are summarised, and a new set of statistical values must be used. The geometric mean replaces the arithmetic mean, and the geometric standard deviation replaces the standard deviation.
Trends in passenger exposure to carbon monoxide inside a vehicle on an arterial highway of the San Francisco Peninsula over 30 years: A longitudinal study
Published in Journal of the Air & Waste Management Association, 2019
We applied the Shapiro–Wilk test using Sigma-Plot Version 13 to the logarithm of the observed CO concentrations in each field survey. For Surveys #1 and #4, the results were consistent with the hypothesis at the p = 0.05 level of significance that the observations follow a lognormal distribution. Figure 2 shows that Surveys #2 and #3 had several outliers. When the two lowest CO concentrations were removed from Survey #2 and the lowest and two highest concentrations were removed from Survey #3, the lognormal hypothesis also could not be rejected at the p = 0.05 level of significance for these two surveys. For all four surveys, the results of the Shapiro–Wilk tests, the closeness of the observed data plotted in Figure 2 to the straight lines estimated from the models, and the relatively high R2 values in Table 6 meet the criteria proposed by Larsen (1961) for air quality distributions that are approximately lognormal. In a lognormal distribution, the geometric standard deviation, which determines the slope of the line, is the ratio of the CO concentration at the 84.13th percentile to the CO concentration at the 50th percentile (the median). Despite these fixed slopes, the observed arithmetic mean net exposures showed a major decrease over the 30-yr period from 9.7 ppm to 4.8 ppm to 1.7 ppm to 0.5 ppm (Table 3), and these statistics agreed closely with the model parameters in Table 6.
Effect of borehole location on pile performance
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2022
M. P. Crisp, M. B. Jaksa, Y. L. Kuo
Arguably the second most complex metric is the geometric standard deviation above the geometric mean, termed here as the “geometric product”. Unlike the traditional standard deviation, which is added to the mean, the geometric standard deviation is multiplied by the geometric mean. The other difference from the arithmetic mean and standard deviation is that the geometric statistics are calculated the same way, albeit on the natural logarithm of the values. The remaining metric, the probability of failure, is calculated simply by counting the number of times failure occurs as a ratio of the total number of Monte Carlo realisations.