x̄ - > Standard normal distribution

The term "standard normal", which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel "Introduction to mathematical statistics" and A. M. Mood "Introduction to the theory of statistics". In popular culture, The concept of a normal distribution is widespread in popular culture and its application can be observed in a number of common situations. For example, asking a group of people to vote over a range of orders, numerical values will typically result in a normal distribution. See also Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have the same means; Bhattacharyya distance – the method used to separate mixtures of normal distributions Erdős–Kac theorem – on the occurrence of the normal distribution in number theory Full width at half maximum Gaussian blur – convolution, which uses the normal distribution as a kernel Modified half-normal distribution Normally distributed and uncorrelated does not imply independent Ratio normal distribution Reciprocal normal distribution Standard normal table Stein's lemma Sub-Gaussian distribution Sum of normally distributed random variables Tweedie distribution – The normal distribution is a member of the family of Tweedie exponential dispersion models. Wrapped normal distribution – the Normal distribution applied to a circular domain Z-distribution – a special case of the normal distribution with a mean of 0 and standard deviation of 1 Z-test – using the normal distribution

In mathematics, the logarithm is the inverse function of exponentiation. That means the logarithm of a given number is the exponent to which another fixed number, the base must be raised, to produce that number. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g. since the "logarithm base 10" of 1000 is 3, or. The logarithm of to base is denoted as or without parentheses, or even without the explicit base, when no confusion is possible, or when the base does not matter such as in big O notation. The logarithm base is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number as its base; its use is widespread in mathematics and physics, because of its simpler integral and derivative. The binary logarithm uses base and is frequently used in computer science. Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors, and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors: provided that and are all positive. The slide rule, also based on logarithms, allows quick calculations without tables, but at a lower precision. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century, and who also introduced the letter as the base of natural logarithms. Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, a decibel is a unit used to express ratios as logarithms, mostly for signal power and amplitude. In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and geometric objects called fractals. They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting. The concept of the logarithm is the inverse of exponentiation and extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete logarithm is the multi-valued inverse of the exponential function infinite groups; it has uses in public-key cryptography. Motivation Addition, multiplication, and exponentiation are three of the most fundamental arithmetic operations. The inverse of addition is subtraction, and the inverse of multiplication is division. Similarly, a logarithm is the inverse operation of exponentiation. Exponentiation is when a number the base is raised to a certain power the exponent for giving a value this denoted, For example, raising to the power of gives: 2^3 8 The logarithm of the base is the inverse operation, that provides the output from the input. That is, y \log_b x is equivalent to x b^y if is a positive real number. One of the main historical motivations for introducing logarithms is the formula