Statistics for Anesthesia Using OnlineStatBook. Franklin Dexter, M. D., Ph. D. Director, Division of Management Consulting Professor, Department of Anesthesia. Normal distribution Wikipedia. In probability theory, the normal or Gaussian distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate. Normal Probability Curve Pdf' title='Normal Probability Curve Pdf' />The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions which include finite variance, it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes such as measurement errors often have distributions that are nearly normal. Moreover, many results and methods such as propagation of uncertainty and least squares parameter fitting can be derived analytically in explicit form when the relevant variables are normally distributed. The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell shaped such as the Cauchy, Students t, and logistic distributions. Even the term Gaussian bell curve is ambiguous because it may be used to refer to some function defined in terms of the Gaussian function which is not a probability distribution because it is not normalized does not integrate to 1. Normal Probability Curve Pdf' title='Normal Probability Curve Pdf' />The probability density of the normal distribution is fx,21. Where DefinitioneditStandard normal distributioneditThe simplest case of a normal distribution is known as the standard normal distribution. This is a special case when 0displaystyle mu 0 and 1displaystyle sigma 1, and it is described by this probability density function x1. The factor 12displaystyle 1sqrt 2pi in this expression ensures that the total area under the curve xdisplaystyle varphi x is equal to one. The factor 12displaystyle 12 in the exponent ensures that the distribution has unit variance and therefore also unit standard deviation. This function is symmetric around x0displaystyle x0, where it attains its maximum value 12displaystyle 1sqrt 2pi and has inflection points at x1displaystyle x1 and x1displaystyle x 1. Authors may differ also on which normal distribution should be called the standard one. Gauss defined the standard normal as having variance 212displaystyle sigma 212, that isxex. Surface Texture Contour Measuring Instruments 230 Surface TextureContour Measuring Instruments Explanation of Surface CharacteristicsStandards. Normal+Distributions+on+TI-83.jpg' alt='Normal Probability Curve Pdf' title='Normal Probability Curve Pdf' />Stigler5 goes even further, defining the standard normal with variance 212displaystyle sigma 212pi   xex. General normal distributioneditEvery normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor displaystyle sigma the standard deviation and then translated by displaystyle mu the mean value fx,21x. The probability density must be scaled by 1displaystyle 1sigma so that the integral is still 1. If Zdisplaystyle Z is a standard normal deviate, then XZdisplaystyle Xsigma Zmu will have a normal distribution with expected value displaystyle mu and standard deviation displaystyle sigma. Conversely, if Xdisplaystyle X is a normal deviate with parameters displaystyle mu and 2displaystyle sigma 2, then ZXdisplaystyle ZX mu sigma will have a standard normal distribution. This variate is called the standardized form of Xdisplaystyle XEvery normal distribution is the exponential of a quadratic function fxeax. In this form, the mean value is b2adisplaystyle mu b2a, and the variance is 212adisplaystyle sigma 2 12a. For the standard normal distribution, a12displaystyle a 12, b0displaystyle b0, and cln22displaystyle c ln2pi 2. Matlab Portable 2012 Full Hd. NotationeditThe probability density of the standard Gaussian distribution standard normal distribution with zero mean and unit variance is often denoted with the Greek letter displaystyle phi phi. The alternative form of the Greek letter phi, displaystyle varphi, is also used quite often. The normal distribution is often referred to as N,2displaystyle Nmu ,sigma 2 or N,2displaystyle mathcal Nmu ,sigma 2. Thus when a random variable Xdisplaystyle X is distributed normally with mean displaystyle mu and variance 2displaystyle sigma 2, one may write. X  N,2. displaystyle X sim mathcal Nmu ,sigma 2. Alternative parameterizationseditSome authors advocate using the precisiondisplaystyle tau as the parameter defining the width of the distribution, instead of the deviation displaystyle sigma or the variance 2displaystyle sigma 2. The precision is normally defined as the reciprocal of the variance, 12displaystyle 1sigma 2. The formula for the distribution then becomesfx2ex22. This choice is claimed to have advantages in numerical computations when displaystyle sigma is very close to zero and simplify formulas in some contexts, such as in the Bayesian inference of variables with multivariate normal distribution. Also the reciprocal of the standard deviation 1displaystyle tau prime 1sigma might be defined as the precision and the expression of the normal distribution becomesfx2e2x22. According to Stigler, this formulation is advantageous because of a much simpler and easier to remember formula, and simple approximate formulas for the quantiles of the distribution. PropertieseditThe normal distribution is the only absolutely continuous distribution whose cumulants beyond the first two i. It is also the continuous distribution with the maximum entropy for a specified mean and variance. Geary has shown, assuming that the mean and variance were finite, that the normal distribution is the only distribution where the mean and variance are independent. This theorem was later reproved by Lukas who only assumed that the variance was finite. The normal distribution is a subclass of the elliptical distributions. The normal distribution is symmetric about its mean, and is non zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the weight of a person or the price of a share. Such variables may be better described by other distributions, such as the log normal distribution or the Pareto distribution. The value of the normal distribution is practically zero when the value xdisplaystyle x lies more than a few standard deviations away from the mean.