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# Error Log Normal Distribution

## Lognormal Distribution Vs Normal Distribution

top Lognormal Standard Error up vote 1 down vote favorite 1 What is the Standard Error of the Lognormal distribution? I am particularly interested in comparing two probabilities from the distribution. I have the two proportions based on experiments, in general these proportions follow a lognormal fit. The standard error of differences from a normal distribution is straightforward (I've seen one other version of this which I think can see on wikipedia): $SE = log uniform distribution \sqrt{\frac{p (1-p)}{n} + \frac{q (1-q)}{m}}$ where $p, q$ are probabilities and $n, m$ sample sizes, expected to be from a normally distributed sample. How would this be extended to the lognormal? Edit: The basic idea is to learn if the difference between two probabilities would be significant or not. I don't think I'm expressing my question correctly as I'm not a strict statistician. standard-error lognormal share|improve this question edited Sep 18 '12 at 21:44 asked Sep 18 '12 at 16:21 Lillian Milagros Carrasquillo 6218 2 When you mention standard error are you talking about the standard deviation of the sample mean from lognormal data or for the difference in means from two samples from possibly different lognormal distirbutions? What you presented above is the standard error for the difference of two independent estimates of proportion for two BINOMIAL distributions and not the difference of sample means for two NORMAL distributions, –Michael Chernick Sep 18 '12 at 16:34 Standard error of the differences between the two independently sampled probabilities. I've used the SE equation above to represent survey data, which is assumed to be normal, I'm noting that I need a Standard Error calculation for a lognormal. If the standard error for the normal distribution is different than above, feel free to share. This is just what was taught

Function A variable X is lognormally distributed if $$Y = \ln(X)$$ is normally distributed with "LN" denoting the natural logarithm. The general formula for the probability density function of the lognormal distribution is $$f(x) = \frac{e^{-((\ln((x-\theta)/m))^{2}/(2\sigma^{2}))}} {(x-\theta)\sigma\sqrt{2\pi}} \hspace{.2in} x > \theta; m, \sigma > 0$$ where σ log poisson distribution is the shape parameter (and is the standard deviation of the log of the distribution),

## Log Exponential Distribution

θ is the location parameter and m is the scale parameter (and is also the median of the distribution). If x =

## Log Normal Curve

θ, then f(x) = 0. The case where θ = 0 and m = 1 is called the standard lognormal distribution. The case where θ equals zero is called the 2-parameter lognormal distribution. The equation for http://stats.stackexchange.com/questions/37502/lognormal-standard-error the standard lognormal distribution is $$f(x) = \frac{e^{-((\ln x)^{2}/2\sigma^{2})}} {x\sigma\sqrt{2\pi}} \hspace{.2in} x > 0; \sigma > 0$$ Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. Note that the lognormal distribution is commonly parameterized with $$\mu = \log(m)$$ The μ parameter is the mean of the http://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm log of the distribution. If the μ parameterization is used, the lognormal pdf is $$f(x) = \frac{e^{-(\ln(x - \theta) - \mu)^2/(2\sigma^2)}} {(x - \theta)\sigma\sqrt{2\pi}} \hspace{.2in} x > 0; \sigma > 0$$ We prefer to use the m parameterization since m is an explicit scale parameter. The following is the plot of the lognormal probability density function for four values of σ. There are several common parameterizations of the lognormal distribution. The form given here is from Evans, Hastings, and Peacock. Cumulative Distribution Function The formula for the cumulative distribution function of the lognormal distribution is $$F(x) = \Phi(\frac{\ln(x)} {\sigma}) \hspace{.2in} x \ge 0; \sigma > 0$$ where $$\Phi$$ is the cumulative distribution function of the normal distribution. The following is the plot of the lognormal cumulative distribution function with the same values of σ as the pdf plots above. Percent Point Function The formula for the percent point function of the lognormal distribution is $$G(p) = \exp(\sigma\Phi^{-1}(p)) \hspace{.2in} 0 \le p < 1; \sigma > 0$$ where $$\Phi^{-1}$$ is the percent point function of the normal distribution. The following is the plot of the lognormal percent point function with the same values of σ as the pdf plots above. Hazard Function The formula for the hazard function of the lo

2005 by Ulf Olsson, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor. Key Words:Generalized confidence interval. Abstract Methods for calculating confidence intervals for the mean are reviewed for the http://www.amstat.org/publications/JSE/v13n1/olsson.html case where the data come from a log-normal distribution. In a simulation study it is found that http://broadleaf.com.au/resource-material/lognormal-distribution-summary/ a variation of the method suggested by Cox works well in practice. An approach based on Generalized confidence intervals also works well. A comparison of our results with those of Zhou and Gao (1997) reveals that it may be preferable to base the interval on t values, rather than on z values. 1. The problem In applied statistics classes we sometimes come across data that log normal need to be transformed prior to analysis. For example, income data can often be considered to be log-normal. One way of analyzing such data is to log-transform the original variable X and to base the inference on the transformed variable Y = log(X). This means that we assume that the distribution from which our data emerges can be approximated with a log-normal distribution. In this paper we will discuss interval estimation of the arithmetic mean value of X in a log-normal log normal distribution distribution. It is true that the median is often used to describe the average of skewed distributions like income distributions. However, there are situations when the arithmetic mean is a parameter of interest. For example, in a sample survey, a confidence interval for the average income can be used to calculate a confidence interval for the total income in the population. Note that if X is log-normal, then the median of Y is equal to the log of the median of X. In this paper we will assume that it is the arithmetic mean of X, and not the median of X, that we want to make inference about. It is a rather straight-forward task to use the log-transformed data Y to calculate a confidence interval for the expected value (mean value) of Y. We will discuss how this result can be used to calculate a confidence interval for the expected value of X. 2. Theory and notation Let X denote the original variable that follows a log-normal distribution. X has expected value E(X)= and variance Var(X)=. We let Y denote the log-transformed, normally distributed variable Y = log(X), that has mean value E(Y)=, and variance Var(Y)=. Denote the sample mean of Y with , and the sample variance of Y with s2. It holds (see e.g. Zhou and Gao, 1997) that (1) This means that the mean value of X is not equal to the antilog of the mean value of Y

under Techniques and special applications Introduction I was stuck in a distant part of Papua New Guinea some years ago without reference sources. I had a lognormal distribution defined in terms of its mean and 95-percentile values, and I needed help in determining its standard deviation. Many people from the RISKANAL list responded to my request (see the list below; many thanks, all of you) with a wealth of specific and general information. I have summarised the main points here. Definition A random variable X is said to follow a lognormal distribution if the random variable Y = log ( X ) is normally distributed, N ( mu, sigma^2 ). A lognormal distribution is defined by a density function of f (y) = EXP( - ((LOG(y) – mu)^2) / (2 * sigma^2) ) / (y * sigma * SQR(2 * pi)), for y > 0 Lognormal distributions are typically specified in one of two ways throughout the literature. One is to specify the mean and standard deviation of the underlying normal distribution (mu and sigma) as described above. The other is to specify the distribution using the mean of the lognormal distribution itself and a term called the ‘error factor’. The error factor for a lognormal distribution is defined as the ratio of the 95th percentile to the median, or, equivalently, the ratio of the median to the 5th percentile. Physically, its square represents the width of a 90% confidence interval with respect to the median. The mathematical relationships between the mean and error factor, and the parameters of the underlying normal distribution (mu and sigma) are shown by the following equations: sigma = LOG(error factor) / 1.645 mu = LOG(mean) – (sigma^2 / 2) When the mean and error factor are used as input for the lognormal distribution, both input parameters must be positive, and the error factor must be greater than one. If mu and sigma are specified, there is no restriction on mu, but sigma must be positive. Formulae Two parameters are generally sufficient to define a lognormal distribution. The majority (but not all) of the formulae listed below are taken from a freeware program called LOGNORM

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