Las distribuciones normales son importantes en estadística y se utilizan a menudo en las ciencias naturales y sociales para representar variables aleatorias de valor real cuyas distribuciones no se conocen. [2] [3] Su importancia se debe en parte al teorema del límite central . Este afirma que, en determinadas condiciones, el promedio de muchas muestras (observaciones) de una variable aleatoria con media y varianza finitas es en sí mismo una variable aleatoria, cuya distribución converge a una distribución normal a medida que aumenta el número de muestras. Por lo tanto, las cantidades físicas que se espera que sean la suma de muchos procesos independientes, como los errores de medición , a menudo tienen distribuciones que son casi normales. [4]
Además, las distribuciones gaussianas tienen algunas propiedades únicas que son valiosas en los estudios analíticos. Por ejemplo, cualquier combinación lineal de un conjunto fijo de desviaciones normales independientes es una desviación normal. Muchos resultados y métodos, como la propagación de la incertidumbre y el ajuste de parámetros por mínimos cuadrados [5] , se pueden derivar analíticamente en forma explícita cuando las variables relevantes se distribuyen normalmente.
A veces, a una distribución normal se la denomina informalmente curva de campana . [6] Sin embargo, muchas otras distribuciones tienen forma de campana (como la distribución de Cauchy , la t de Student y las distribuciones logísticas ). Para conocer otros nombres, consulte Nomenclatura .
El caso más simple de una distribución normal se conoce como distribución normal estándar o distribución normal unitaria . Este es un caso especial cuando y , y se describe mediante esta función de densidad de probabilidad (o densidad):
La variable tiene una media de 0 y una varianza y desviación estándar de 1. La densidad tiene su pico en y puntos de inflexión en y .
Aunque la densidad anterior se conoce más comúnmente como la normal estándar, algunos autores han utilizado ese término para describir otras versiones de la distribución normal. Carl Friedrich Gauss , por ejemplo, una vez definió la normal estándar como
que tiene una varianza de , y Stephen Stigler [7] una vez definió la normal estándar como
que tiene una forma funcional simple y una varianza de
Distribución normal general
Toda distribución normal es una versión de la distribución normal estándar, cuyo dominio ha sido ampliado por un factor (la desviación estándar) y luego traducido por (el valor medio):
La densidad de probabilidad debe escalarse de modo que la integral siga siendo 1.
Si es una desviación normal estándar , entonces tendrá una distribución normal con valor esperado y desviación estándar . Esto es equivalente a decir que la distribución normal estándar se puede escalar/estirar por un factor de y desplazar por para producir una distribución normal diferente, llamada . Por el contrario, si es una desviación normal con parámetros y , entonces esta distribución se puede volver a escalar y desplazar mediante la fórmula para convertirla en la distribución normal estándar. Esta variante también se denomina forma estandarizada de .
Notación
La densidad de probabilidad de la distribución gaussiana estándar (distribución normal estándar, con media cero y varianza unitaria) se denota a menudo con la letra griega ( phi ). [8] La forma alternativa de la letra griega phi, , también se utiliza con bastante frecuencia.
La distribución normal a menudo se denomina o . [9] Por lo tanto, cuando una variable aleatoria se distribuye normalmente con media y desviación estándar , se puede escribir
Parametrizaciones alternativas
Algunos autores recomiendan utilizar la precisión como parámetro que define el ancho de la distribución, en lugar de la desviación estándar o la varianza . La precisión se define normalmente como el recíproco de la varianza, . [10] La fórmula para la distribución se convierte entonces en
Se afirma que esta elección tiene ventajas en los cálculos numéricos cuando está muy cerca de cero y simplifica las fórmulas en algunos contextos, como en la inferencia bayesiana de variables con distribución normal multivariada .
Alternativamente, el recíproco de la desviación estándar podría definirse como la precisión , en cuyo caso la expresión de la distribución normal se convierte en
Según Stigler, esta formulación es ventajosa debido a una fórmula mucho más simple y fácil de recordar, y a fórmulas aproximadas simples para los cuantiles de la distribución.
Las distribuciones normales forman una familia exponencial con parámetros naturales y , y estadísticas naturales x y x 2 . Los parámetros de expectativa dual para la distribución normal son η 1 = μ y η 2 = μ 2 + σ 2 .
Función de distribución acumulativa
La función de distribución acumulativa (CDF) de la distribución normal estándar, usualmente denotada con la letra griega mayúscula ( phi ), es la integral
Función de error
La función de error relacionada da la probabilidad de una variable aleatoria, con distribución normal de media 0 y varianza 1/2 dentro del rango . Es decir:
Estas integrales no se pueden expresar en términos de funciones elementales y, a menudo, se dice que son funciones especiales . Sin embargo, se conocen muchas aproximaciones numéricas; consulte más información a continuación.
Las dos funciones están estrechamente relacionadas, a saber:
Para una distribución normal genérica con densidad , media y varianza , la función de distribución acumulativa es
El complemento de la función de distribución acumulativa normal estándar, , se suele denominar función Q , especialmente en textos de ingeniería. [11] [12] Da la probabilidad de que el valor de una variable aleatoria normal estándar supere a : . Otras definiciones de la función -, todas las cuales son transformaciones simples de , también se utilizan ocasionalmente. [13]
La gráfica de la función de distribución acumulativa normal estándar tiene simetría rotacional doble alrededor del punto (0,1/2); es decir, . Su antiderivada (integral indefinida) se puede expresar de la siguiente manera:
La función de distribución acumulativa de la distribución normal estándar se puede ampliar mediante la integración por partes en una serie:
También se puede derivar una expansión asintótica de la función de distribución acumulativa para valores grandes de x mediante la integración por partes. Para obtener más información, consulte Función de error#Expansión asintótica . [14]
Se puede encontrar una aproximación rápida a la función de distribución acumulativa de la distribución normal estándar utilizando una aproximación de la serie de Taylor:
Cálculo recursivo con desarrollo de la serie de Taylor
La naturaleza recursiva de la familia de derivadas se puede utilizar para construir fácilmente una expansión de la serie de Taylor rápidamente convergente utilizando entradas recursivas sobre cualquier punto de valor conocido de la distribución :
dónde:
Utilizando la serie de Taylor y el método de Newton para la función inversa
Una aplicación de la expansión de la serie de Taylor anterior es utilizar el método de Newton para invertir el cálculo. Es decir, si tenemos un valor para la función de distribución acumulativa , , pero no conocemos la x necesaria para obtener , podemos utilizar el método de Newton para encontrar x, y utilizar la expansión de la serie de Taylor anterior para minimizar el número de cálculos. El método de Newton es ideal para resolver este problema porque la primera derivada de , que es una integral de la distribución estándar normal, es la distribución estándar normal y está disponible para su uso en la solución del método de Newton.
Para resolver, seleccione una solución aproximada conocida, , para el . deseado puede ser un valor de una tabla de distribución, o una estimación inteligente seguida de un cálculo de utilizando cualquier medio deseado para calcular. Utilice este valor de y la expansión de la serie de Taylor anterior para minimizar los cálculos.
Repita el siguiente proceso hasta que la diferencia entre el calculado y el deseado , que llamaremos , esté por debajo de un error aceptablemente pequeño, como 10 −5 , 10 −15 , etc.:
dónde
es la solución de una serie de Taylor usando y
Cuando los cálculos repetidos convergen a un error inferior al valor aceptablemente pequeño elegido, x será el valor necesario para obtener a del valor deseado, .
Desviación estándar y cobertura
Alrededor del 68% de los valores extraídos de una distribución normal están dentro de una desviación estándar σ de la media; alrededor del 95% de los valores se encuentran dentro de dos desviaciones estándar; y alrededor del 99,7% están dentro de tres desviaciones estándar. [6] Este hecho se conoce como la regla 68-95-99,7 (empírica) o la regla de 3 sigma .
Más precisamente, la probabilidad de que una desviación normal se encuentre en el rango entre y está dada por
Para 12 dígitos significativos, los valores para son:
Para valores grandes , se puede utilizar la aproximación .
Función cuantil
La función cuantil de una distribución es la inversa de la función de distribución acumulativa. La función cuantil de la distribución normal estándar se denomina función probit y se puede expresar en términos de la función de error inversa :
Para una variable aleatoria normal con media y varianza , la función cuantil es
El cuantil de la distribución normal estándar se denota comúnmente como . Estos valores se utilizan en pruebas de hipótesis , construcción de intervalos de confianza y gráficos Q-Q . Una variable aleatoria normal excederá con probabilidad , y quedará fuera del intervalo con probabilidad . En particular, el cuantil es 1,96 ; por lo tanto, una variable aleatoria normal quedará fuera del intervalo solo en el 5% de los casos.
La siguiente tabla muestra el cuartil que se encontrará en el rango con una probabilidad especificada . Estos valores son útiles para determinar el intervalo de tolerancia para promedios de muestra y otros estimadores estadísticos con distribuciones normales (o asintóticamente normales). [15] La siguiente tabla muestra , no como se definió anteriormente.
La distribución normal es la única distribución cuyos cumulantes más allá de los dos primeros (es decir, distintos de la media y la varianza ) son cero. También es la distribución continua con la entropía máxima para una media y varianza especificadas. [16] [17] Geary ha demostrado, suponiendo que la media y la varianza son finitas, que la distribución normal es la única distribución donde la media y la varianza calculadas a partir de un conjunto de valores extraídos independientes son independientes entre sí. [18] [19]
La distribución normal es una subclase de las distribuciones elípticas . La distribución normal es simétrica respecto de su media y no es cero en toda la línea real. Como tal, puede no ser un modelo adecuado para variables que son inherentemente positivas o fuertemente sesgadas, como el peso de una persona o el precio de una acción . Dichas variables pueden describirse mejor mediante otras distribuciones, como la distribución log-normal o la distribución de Pareto .
El valor de la distribución normal es prácticamente cero cuando el valor se encuentra a más de unas pocas desviaciones estándar de la media (por ejemplo, una dispersión de tres desviaciones estándar cubre toda la distribución total, excepto el 0,27%). Por lo tanto, puede no ser un modelo apropiado cuando se espera una fracción significativa de valores atípicos (valores que se encuentran a muchas desviaciones estándar de la media) y los mínimos cuadrados y otros métodos de inferencia estadística que son óptimos para las variables distribuidas normalmente suelen volverse muy poco confiables cuando se aplican a esos datos. En esos casos, se debe suponer una distribución de colas más pesadas y aplicar los métodos de inferencia estadística robustos adecuados .
La distribución gaussiana pertenece a la familia de distribuciones estables que son las que atraen las sumas de distribuciones independientes idénticamente distribuidas, independientemente de que la media o la varianza sean finitas o no. A excepción de la gaussiana, que es un caso límite, todas las distribuciones estables tienen colas pesadas y varianza infinita. Es una de las pocas distribuciones que son estables y que tienen funciones de densidad de probabilidad que se pueden expresar analíticamente, las otras son la distribución de Cauchy y la distribución de Lévy .
Simetrías y derivadas
La distribución normal con densidad (media y varianza ) tiene las siguientes propiedades:
Es simétrica alrededor del punto que es al mismo tiempo la moda , la mediana y la media de la distribución. [20]
Es unimodal : su primera derivada es positiva para negativa para y cero sólo en
El área delimitada por la curva y el eje es la unidad (es decir, igual a uno).
Su primera derivada es
Su segunda derivada es
Su densidad tiene dos puntos de inflexión (donde la segunda derivada de es cero y cambia de signo), ubicados a una desviación estándar de la media, es decir en y [20]
Además, la densidad de la distribución normal estándar (es decir, y ) también tiene las siguientes propiedades:
Su primera derivada es
Su segunda derivada es
De manera más general, su derivada n- ésima es donde es el n -ésimo polinomio de Hermite (probabilista) . [22]
La probabilidad de que una variable distribuida normalmente con y conocidos esté en un conjunto particular, se puede calcular utilizando el hecho de que la fracción tiene una distribución normal estándar.
Momentos
Los momentos simples y absolutos de una variable son los valores esperados de y , respectivamente. Si el valor esperado de es cero, estos parámetros se denominan momentos centrales; de lo contrario, estos parámetros se denominan momentos no centrales. Por lo general, solo nos interesan los momentos con orden entero .
Si tiene una distribución normal, los momentos no centrales existen y son finitos para cualquier número cuya parte real sea mayor que −1. Para cualquier entero no negativo , los momentos centrales simples son: [23]
Aquí denota el factorial doble , es decir, el producto de todos los números de a 1 que tienen la misma paridad que
Los momentos absolutos centrales coinciden con los momentos simples para todos los órdenes pares, pero son distintos de cero para los órdenes impares. Para cualquier entero no negativo
La última fórmula es válida también para cualquier número no entero. Cuando la media de los momentos simples y absolutos se pueden expresar en términos de funciones hipergeométricas confluentes y [24]
La esperanza de condicionada al evento que se encuentra en un intervalo está dada por
donde y respectivamente son la densidad y la función de distribución acumulada de . Esto se conoce como el coeficiente de Mills inverso . Tenga en cuenta que anteriormente, se utiliza la densidad de en lugar de la densidad normal estándar como en el coeficiente de Mills inverso, por lo que aquí tenemos en lugar de .
donde es la unidad imaginaria . Si la media , el primer factor es 1, y la transformada de Fourier es, además de un factor constante, una densidad normal en el dominio de la frecuencia , con media 0 y varianza . En particular, la distribución normal estándar es una función propia de la transformada de Fourier.
En teoría de la probabilidad, la transformada de Fourier de la distribución de probabilidad de una variable aleatoria de valor real está estrechamente relacionada con la función característica de esa variable, que se define como el valor esperado de , en función de la variable real (el parámetro de frecuencia de la transformada de Fourier). Esta definición se puede extender analíticamente a una variable de valor complejo . [26] La relación entre ambas es:
Funciones generadoras de momentos y cumulantes
La función generadora de momentos de una variable aleatoria real es el valor esperado de , en función del parámetro real . Para una distribución normal con densidad , media y varianza , la función generadora de momentos existe y es igual a
Para cualquier , el coeficiente de en la función generadora de momentos (expresada como una serie de potencia exponencial en ) es el valor esperado de la distribución normal .
Los coeficientes de esta serie de potencias exponenciales definen los cumulantes, pero como se trata de un polinomio cuadrático en , solo los dos primeros cumulantes son distintos de cero, es decir, la media y la varianza .
Algunos autores prefieren trabajar con E[ e itX ] = e iμt − σ 2 t 2 /2 y ln E[ e itX ] = iμt − 1/2 σ2t2 .
Operador y clase Stein
Dentro del método de Stein, el operador de Stein y la clase de una variable aleatoria son y la clase de todas las funciones absolutamente continuas .
Límite de varianza cero
En el límite cuando tiende a cero, la densidad de probabilidad tiende eventualmente a cero en cualquier , pero crece sin límite si , mientras que su integral permanece igual a 1. Por lo tanto, la distribución normal no puede definirse como una función ordinaria cuando .
Sin embargo, se puede definir la distribución normal con varianza cero como una función generalizada ; específicamente, como una función delta de Dirac traducida por la media , es decir
Su función de distribución acumulativa es entonces la función escalón de Heaviside traducida por la media , es decir
Entropía máxima
De todas las distribuciones de probabilidad sobre los números reales con una media finita especificada y una varianza finita , la distribución normal es la que tiene una entropía máxima . [27] Para ver esto, sea una variable aleatoria continua con densidad de probabilidad . La entropía de se define como [28] [29] [30]
donde se entiende que es cero siempre que . Esta función se puede maximizar, sujeta a las restricciones de que la distribución esté correctamente normalizada y tenga una media y varianza especificadas, mediante el uso del cálculo variacional . Se define una función con tres multiplicadores de Lagrange :
Con máxima entropía, una pequeña variación de producirá una variación de que es igual a 0:
Dado que esto debe ser válido para cualquier α pequeño , el factor que se multiplica debe ser cero, y al resolver se obtiene:
Las restricciones de Lagrange que están correctamente normalizadas y tienen la media y varianza especificadas se satisfacen si y solo si , , y se eligen de modo que
La entropía de una distribución normal es igual a
que es independiente de la media .
Otras propiedades
Si la función característica de alguna variable aleatoria es de la forma en un entorno de cero, donde es un polinomio , entonces el teorema de Marcinkiewicz (llamado así por Józef Marcinkiewicz ) afirma que puede ser como máximo un polinomio cuadrático y, por lo tanto, es una variable aleatoria normal. [31] La consecuencia de este resultado es que la distribución normal es la única distribución con un número finito (dos) de cumulantes distintos de cero .
Si y son conjuntamente normales y no correlacionados , entonces son independientes . El requisito de que y sean conjuntamente normales es esencial; sin él la propiedad no se cumple. [32] [33] [prueba] Para las variables aleatorias no normales, la no correlación no implica independencia.
La distribución conjugada anterior de la media de una distribución normal es otra distribución normal. [35] Específicamente, si son iid y la distribución anterior es , entonces la distribución posterior para el estimador de será
La familia de distribuciones normales no solo forma una familia exponencial (EF), sino que de hecho forma una familia exponencial natural (NEF) con función de varianza cuadrática ( NEF-QVF ). Muchas propiedades de las distribuciones normales se generalizan a propiedades de las distribuciones NEF-QVF, distribuciones NEF o distribuciones EF en general. Las distribuciones NEF-QVF comprenden 6 familias, incluidas las distribuciones Poisson, Gamma, binomial y binomial negativa, mientras que muchas de las familias comunes estudiadas en probabilidad y estadística son NEF o EF.
Si se distribuyen según , entonces . Nótese que no hay ningún supuesto de independencia. [37]
Distribuciones relacionadas
Teorema del límite central
El teorema del límite central establece que, en determinadas condiciones (bastante comunes), la suma de muchas variables aleatorias tendrá una distribución aproximadamente normal. Más específicamente, donde son variables aleatorias independientes e idénticamente distribuidas con la misma distribución arbitraria, media y varianza cero y su media está escalada por
Entonces, a medida que aumenta, la distribución de probabilidad de tenderá a la distribución normal con media y varianza cero .
El teorema puede extenderse a variables que no son independientes y/o no están distribuidas idénticamente si se imponen ciertas restricciones al grado de dependencia y a los momentos de las distribuciones.
Muchas estadísticas de prueba , puntuaciones y estimadores que se encuentran en la práctica contienen sumas de ciertas variables aleatorias, e incluso más estimadores pueden representarse como sumas de variables aleatorias mediante el uso de funciones de influencia . El teorema del límite central implica que esos parámetros estadísticos tendrán distribuciones asintóticamente normales.
El teorema del límite central también implica que ciertas distribuciones pueden aproximarse mediante la distribución normal, por ejemplo:
La precisión de estas aproximaciones depende del propósito para el que se necesitan y de la tasa de convergencia a la distribución normal. Por lo general, estas aproximaciones son menos precisas en los extremos de la distribución.
El teorema de Berry-Esseen proporciona un límite superior general para el error de aproximación en el teorema del límite central y las expansiones de Edgeworth proporcionan mejoras en la aproximación .
Este teorema también se puede utilizar para justificar la modelización de la suma de muchas fuentes de ruido uniformes como ruido gaussiano . Véase AWGN .
Si se distribuye normalmente con media y varianza , entonces
, para cualquier número real y , también se distribuye normalmente, con media y varianza . Es decir, la familia de distribuciones normales está cerrada ante transformaciones lineales .
Operaciones sobre dos variables normales independientes
Si y son dos variables aleatorias normales independientes , con medias , y varianzas , , entonces su suma también estará distribuida normalmente, [prueba] con media y varianza .
En particular, si y son desviaciones normales independientes con media y varianza cero , entonces y también son independientes y se distribuyen normalmente, con media y varianza cero . Este es un caso especial de la identidad de polarización . [40]
Si , son dos desviaciones normales independientes con media y varianza , y , son números reales arbitrarios, entonces la variable también se distribuye normalmente con media y varianza . De ello se deduce que la distribución normal es estable (con exponente ).
Si , son distribuciones normales, entonces su media geométrica normalizada es una distribución normal con y (ver aquí para una visualización).
Operaciones sobre dos variables normales estándar independientes
Si y son dos variables aleatorias normales estándar independientes con media 0 y varianza 1, entonces
Su suma y diferencia se distribuyen normalmente con media cero y varianza dos: .
Operaciones sobre múltiples variables normales independientes
Cualquier combinación lineal de desviaciones normales independientes es una desviación normal.
Si son variables aleatorias normales estándar independientes, entonces la suma de sus cuadrados tiene la distribución de chi-cuadrado con grados de libertad.
Si , son variables aleatorias normales estándar independientes, entonces la relación de sus sumas de cuadrados normalizadas tendrá la distribución F con ( n , m ) grados de libertad: [44]
Operaciones sobre múltiples variables normales correlacionadas
Una forma cuadrática de un vector normal, es decir, una función cuadrática de múltiples variables normales independientes o correlacionadas, es una variable de chi-cuadrado generalizada .
Operaciones sobre la función de densidad
La distribución normal dividida se define de manera más directa en términos de unir secciones escaladas de las funciones de densidad de diferentes distribuciones normales y reescalar la densidad para integrarla en una sola. La distribución normal truncada resulta de reescalar una sección de una sola función de densidad.
Divisibilidad infinita y teorema de Cramér
Para cualquier entero positivo , cualquier distribución normal con media y varianza es la distribución de la suma de desviaciones normales independientes, cada una con media y varianza . Esta propiedad se llama divisibilidad infinita . [45]
Por el contrario, si y son variables aleatorias independientes y su suma tiene una distribución normal, entonces tanto como deben ser desviaciones normales. [46]
Este resultado se conoce como teorema de descomposición de Cramér y equivale a decir que la convolución de dos distribuciones es normal si y solo si ambas son normales. El teorema de Cramér implica que una combinación lineal de variables independientes no gaussianas nunca tendrá una distribución exactamente normal, aunque puede aproximarse a ella de forma arbitraria. [31]
El teorema de Kac-Bernstein
The Kac–Bernstein theorem states that if and are independent and and are also independent, then both X and Y must necessarily have normal distributions.[47][48]
More generally, if are independent random variables, then two distinct linear combinations and will be independent if and only if all are normal and , where denotes the variance of .[47]
Extensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists.
The multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector X ∈ Rk is multivariate-normally distributed if any linear combination of its components Σk j=1aj Xj has a (univariate) normal distribution. The variance of X is a k×k symmetric positive-definite matrix V. The multivariate normal distribution is a special case of the elliptical distributions. As such, its iso-density loci in the k = 2 case are ellipses and in the case of arbitrary k are ellipsoids.
Complex normal distribution deals with the complex normal vectors. A complex vector X ∈ Ck is said to be normal if both its real and imaginary components jointly possess a 2k-dimensional multivariate normal distribution. The variance-covariance structure of X is described by two matrices: the variance matrix Γ, and the relation matrix C.
Gaussian processes are the normally distributed stochastic processes. These can be viewed as elements of some infinite-dimensional Hilbert spaceH, and thus are the analogues of multivariate normal vectors for the case k = ∞. A random element h ∈ H is said to be normal if for any constant a ∈ H the scalar product(a, h) has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear covariance operator K: H → H. Several Gaussian processes became popular enough to have their own names:
A random variable X has a two-piece normal distribution if it has a distribution
where μ is the mean and σ12 and σ22 are the variances of the distribution to the left and right of the mean respectively.
The mean, variance and third central moment of this distribution have been determined[49]
where E(X), V(X) and T(X) are the mean, variance, and third central moment respectively.
One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
Pearson distribution — a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
The generalized normal distribution, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.
Statistical inference
Estimation of parameters
It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate them. That is, having a sample from a normal population we would like to learn the approximate values of parameters and . The standard approach to this problem is the maximum likelihood method, which requires maximization of the log-likelihood function:Taking derivatives with respect to and and solving the resulting system of first order conditions yields the maximum likelihood estimates:
Then is as follows:
Sample mean
Estimator is called the sample mean, since it is the arithmetic mean of all observations. The statistic is complete and sufficient for , and therefore by the Lehmann–Scheffé theorem, is the uniformly minimum variance unbiased (UMVU) estimator.[50] In finite samples it is distributed normally:The variance of this estimator is equal to the μμ-element of the inverse Fisher information matrix. This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error of is proportional to , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulations.
The estimator is called the sample variance, since it is the variance of the sample (). In practice, another estimator is often used instead of the . This other estimator is denoted , and is also called the sample variance, which represents a certain ambiguity in terminology; its square root is called the sample standard deviation. The estimator differs from by having (n − 1) instead of n in the denominator (the so-called Bessel's correction):The difference between and becomes negligibly small for large n's. In finite samples however, the motivation behind the use of is that it is an unbiased estimator of the underlying parameter , whereas is biased. Also, by the Lehmann–Scheffé theorem the estimator is uniformly minimum variance unbiased (UMVU),[50] which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator is better than the in terms of the mean squared error (MSE) criterion. In finite samples both and have scaled chi-squared distribution with (n − 1) degrees of freedom:The first of these expressions shows that the variance of is equal to , which is slightly greater than the σσ-element of the inverse Fisher information matrix . Thus, is not an efficient estimator for , and moreover, since is UMVU, we can conclude that the finite-sample efficient estimator for does not exist.
Applying the asymptotic theory, both estimators and are consistent, that is they converge in probability to as the sample size . The two estimators are also both asymptotically normal:In particular, both estimators are asymptotically efficient for .
Confidence intervals
By Cochran's theorem, for normal distributions the sample mean and the sample variance s2 are independent, which means there can be no gain in considering their joint distribution. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between and s can be employed to construct the so-called t-statistic:This quantity t has the Student's t-distribution with (n − 1) degrees of freedom, and it is an ancillary statistic (independent of the value of the parameters). Inverting the distribution of this t-statistics will allow us to construct the confidence interval for μ;[51] similarly, inverting the χ2 distribution of the statistic s2 will give us the confidence interval for σ2:[52]where tk,p and χ2 k,p are the pth quantiles of the t- and χ2-distributions respectively. These confidence intervals are of the confidence level1 − α, meaning that the true values μ and σ2 fall outside of these intervals with probability (or significance level) α. In practice people usually take α = 5%, resulting in the 95% confidence intervals. The confidence interval for σ can be found by taking the square root of the interval bounds for σ2.
Approximate formulas can be derived from the asymptotic distributions of and s2:The approximate formulas become valid for large values of n, and are more convenient for the manual calculation since the standard normal quantiles zα/2 do not depend on n. In particular, the most popular value of α = 5%, results in |z0.025| = 1.96.
Normality tests
Normality tests assess the likelihood that the given data set {x1, ..., xn} comes from a normal distribution. Typically the null hypothesisH0 is that the observations are distributed normally with unspecified mean μ and variance σ2, versus the alternative Ha that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:
Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
Q–Q plot, also known as normal probability plot or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ−1(pk), x(k)), where plotting points pk are equal to pk = (k − α)/(n + 1 − 2α) and α is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
P–P plot – similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(z(k)), pk), where . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Shapiro–Wilk test: This is based on the fact that the line in the Q–Q plot has the slope of σ. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
Tests based on the empirical distribution function:
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
Either the mean, or the variance, or neither, may be considered a fixed quantity.
When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
Both univariate and multivariate cases need to be considered.
The formulas for the non-linear-regression cases are summarized in the conjugate prior article.
Sum of two quadratics
Scalar form
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.
This equation rewrites the sum of two quadratics in x by expanding the squares, grouping the terms in x, and completing the square. Note the following about the complex constant factors attached to some of the terms:
This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities a and b add directly, so to combine a and b themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean, so it is not surprising that is one-half the harmonic mean of a and b.
Vector form
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length k, and A and B are symmetric, invertible matrices of size , then
where
The form x′ Ax is called a quadratic form and is a scalar:In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since , only the sum matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric. Furthermore, if A is symmetric, then the form
Sum of differences from the mean
Another useful formula is as follows:where
With known variance
For a set of i.i.d. normally distributed data points X of size n where each individual point x follows with known variance σ2, the conjugate prior distribution is also normally distributed.
This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ2. Then if and we proceed as follows.
First, the likelihood function is (using the formula above for the sum of differences from the mean):
Then, we proceed as follows:
In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving μ. The result is the kernel of a normal distribution, with mean and precision , i.e.
This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:
That is, to combine n data points with total precision of nτ (or equivalently, total variance of n/σ2) and mean of values , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a precision-weighted average, i.e. a weighted average of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)
The above formula reveals why it is more convenient to do Bayesian analysis of conjugate priors for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas
With known mean
For a set of i.i.d. normally distributed data points X of size n where each individual point x follows with known mean μ, the conjugate prior of the variance has an inverse gamma distribution or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameterizations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ2 is as follows:
For a set of i.i.d. normally distributed data points X of size n where each individual point x follows with unknown mean μ and unknown variance σ2, a combined (multivariate) conjugate prior is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistics computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
This suggests that we create a conditional prior of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate priors used (an inverse gamma distribution over the variance, and a normal distribution over the mean, conditional on the variance) and with the same four parameters just defined.
The priors are normally defined as follows:
The update equations can be derived, and look as follows:
The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.
Writing it in terms of variance rather than precision, we get:where
Therefore, the posterior is (dropping the hyperparameters as conditioning factors):
In other words, the posterior distribution has the form of a product of a normal distribution over times an inverse gamma distribution over , with parameters that are the same as the update equations above.
Occurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories:
Exactly normal distributions;
Approximately normal laws, for example when such approximation is justified by the central limit theorem; and
Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.
Exact normality
Certain quantities in physics are distributed normally, as was first demonstrated by James Clerk Maxwell. Examples of such quantities are:
The position of a particle that experiences diffusion. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function), then after time t its location is described by a normal distribution with variance t, which satisfies the diffusion equation. If the initial location is given by a certain density function , then the density at time t is the convolution of g and the normal probability density function.
Approximate normality
Approximately normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting additively and independently, its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible and decomposable distributions are involved, such as
Thermal radiation has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.
Assumed normality
I can only recognize the occurrence of the normal curve – the Laplacian curve of errors – as a very abnormal phenomenon. It is roughly approximated to in certain distributions; for this reason, and on account for its beautiful simplicity, we may, perhaps, use it as a first approximation, particularly in theoretical investigations.
— Pearson (1901)
There are statistical methods to empirically test that assumption; see the above Normality tests section.
In biology, the logarithm of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution (after separation on male/female subpopulations), with examples including:
Measures of size of living tissue (length, height, skin area, weight);[53]
The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth; presumably the thickness of tree bark also falls under this category;
Certain physiological measurements, such as blood pressure of adult humans.
In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb in his works.
Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.[54]
In standardized testing, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.
Many scores are derived from the normal distribution, including percentile ranks (percentiles or quantiles), normal curve equivalents, stanines, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
In hydrology the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem.[55] The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.
Methodological problems and peer review
John Ioannidisargued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.[56]
Computational methods
Generating values from normal distribution
In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a N(μ, σ2) can be generated as X = μ + σZ, where Z is standard normal. All these algorithms rely on the availability of a random number generatorU capable of producing uniform random variates.
The most straightforward method is based on the probability integral transform property: if U is distributed uniformly on (0,1), then Φ−1(U) will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ−1, which cannot be done analytically. Some approximate methods are described in Hart (1968) and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places,[57] which is used by R to compute random variates of the normal distribution.
An easy-to-program approximate approach that relies on the central limit theorem is as follows: generate 12 uniform U(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6).[58] Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
The Box–Muller method uses two independent random numbers U and V distributed uniformly on (0,1). Then the two random variables X and Y will both have the standard normal distribution, and will be independent. This formulation arises because for a bivariate normal random vector (X, Y) the squared norm X2 + Y2 will have the chi-squared distribution with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(U) in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable V.
The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, U and V are drawn from the uniform (−1,1) distribution, and then S = U2 + V2 is computed. If S is greater or equal to 1, then the method starts over, otherwise the two quantities are returned. Again, X and Y are independent, standard normal random variables.
The Ratio method[59] is a rejection method. The algorithm proceeds as follows:
Generate two independent uniform deviates U and V;
Compute X = √8/e (V − 0.5)/U;
Optional: if X2 ≤ 5 − 4e1/4U then accept X and terminate algorithm;
Optional: if X2 ≥ 4e−1.35/U + 1.4 then reject X and start over from step 1;
If X2 ≤ −4 lnU then accept X, otherwise start over the algorithm.
The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved[60] so that the logarithm is rarely evaluated.
The ziggurat algorithm[61] is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
Integer arithmetic can be used to sample from the standard normal distribution.[62] This method is exact in the sense that it satisfies the conditions of ideal approximation;[63] i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
There is also some investigation[64] into the connection between the fast Hadamard transform and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.
Numerical approximations for the normal cumulative distribution function and normal quantile function
Zelen & Severo (1964) give the approximation for Φ(x) for x > 0 with the absolute error |ε(x)| < 7.5·10−8 (algorithm 26.2.17): where ϕ(x) is the standard normal probability density function, and b0 = 0.2316419, b1 = 0.319381530, b2 = −0.356563782, b3 = 1.781477937, b4 = −1.821255978, b5 = 1.330274429.
Hart (1968) lists some dozens of approximations – by means of rational functions, with or without exponentials – for the erfc() function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by West (2009) combines Hart's algorithm 5666 with a continued fraction approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
Cody (1969) after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
Marsaglia (2004) suggested a simple algorithm[note 1] based on the Taylor series expansion for calculating Φ(x) with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when x = 10).
Dia (2023) proposes the following approximation of with a maximum relative error less than in absolute value: for and for ,
Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting p = Φ(z), the simplest approximation for the quantile function is:
This approximation delivers for z a maximum absolute error of 0.026 (for 0.5 ≤ p ≤ 0.9999, corresponding to 0 ≤ z ≤ 3.719). For p < 1/2 replace p by 1 − p and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
This approximation is particularly accurate for the right far-tail (maximum error of 10−3 for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).
Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small relative error on the whole domain for the cumulative distribution function and the quantile function as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.
History
Development
Some authors[65][66] attribute the credit for the discovery of the normal distribution to de Moivre, who in 1738[note 2] published in the second edition of his The Doctrine of Chances the study of the coefficients in the binomial expansion of (a + b)n. De Moivre proved that the middle term in this expansion has the approximate magnitude of , and that "If m or 1/2n be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ℓ, has to the middle Term, is ."[67] Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.[68]
In 1823 Gauss published his monograph "Theoria combinationis observationum erroribus minimis obnoxiae" where among other things he introduces several important statistical concepts, such as the method of least squares, the method of maximum likelihood, and the normal distribution. Gauss used M, M′, M′′, ... to denote the measurements of some unknown quantity V, and sought the most probable estimator of that quantity: the one that maximizes the probability φ(M − V) · φ(M′ − V) · φ(M′′ − V) · ... of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function φ is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values.[note 3] Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:[69]
where h is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linearweighted least squares method.[70]
Although Gauss was the first to suggest the normal distribution law, Laplace made significant contributions.[note 4] It was Laplace who first posed the problem of aggregating several observations in 1774,[71] although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral ∫ e−t2dt = √π in 1782, providing the normalization constant for the normal distribution.[72] Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem, which emphasized the theoretical importance of the normal distribution.[73]
It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss.[74] His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.[75]
In the middle of the 19th century Maxwell demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena:[76] The number of particles whose velocity, resolved in a certain direction, lies between x and x + dx is
Naming
Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.
Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual.[77] However, by the end of the 19th century some authors[note 5] had started using the name normal distribution, where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what would, in the long run, occur under certain circumstances."[78] Around the turn of the 20th century Pearson popularized the term normal as a designation for this distribution.[79]
Many years ago I called the Laplace–Gaussian curve the normal curve, which name, while it avoids an international question of priority, has the disadvantage of leading people to believe that all other distributions of frequency are in one sense or another 'abnormal'.
— Pearson (1920)
Also, it was Pearson who first wrote the distribution in terms of the standard deviation σ as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
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 (1947) Introduction to Mathematical Statistics and A. M. Mood (1950) Introduction to the Theory of Statistics.[80]
^De Moivre first published his findings in 1733, in a pamphlet Approximatio ad Summam Terminorum Binomii (a + b)n in Seriem Expansi that was designated for private circulation only. But it was not until the year 1738 that he made his results publicly available. The original pamphlet was reprinted several times, see for example Walker (1985).
^"It has been customary certainly to regard as an axiom the hypothesis that if any quantity has been determined by several direct observations, made under the same circumstances and with equal care, the arithmetical mean of the observed values affords the most probable value, if not rigorously, yet very nearly at least, so that it is always most safe to adhere to it." — Gauss (1809, section 177)
^"My custom of terming the curve the Gauss–Laplacian or normal curve saves us from proportioning the merit of discovery between the two great astronomer mathematicians." quote from Pearson (1905, p. 189)
^Besides those specifically referenced here, such use is encountered in the works of Peirce, Galton (Galton (1889, chapter V)) and Lexis (Lexis (1878), Rohrbasser & Véron (2003)) c. 1875.[citation needed]
References
Citations
^Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019). "Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation" (PDF). Annals of Operations Research. 299 (1–2). Springer: 1281–1315. arXiv:1811.11301. doi:10.1007/s10479-019-03373-1. S2CID 254231768. Retrieved February 27, 2023.
^Normal Distribution, Gale Encyclopedia of Psychology
^Casella & Berger (2001, p. 102)
^Lyon, A. (2014). Why are Normal Distributions Normal?, The British Journal for the Philosophy of Science.
^Jorge, Nocedal; Stephan, J. Wright (2006). Numerical Optimization (2nd ed.). Springer. p. 249. ISBN 978-0387-30303-1.
^ a b"Normal Distribution". www.mathsisfun.com. Retrieved August 15, 2020.
^Stigler (1982)
^Halperin, Hartley & Hoel (1965, item 7)
^McPherson (1990, p. 110)
^Bernardo & Smith (2000, p. 121)
^Scott, Clayton; Nowak, Robert (August 7, 2003). "The Q-function". Connexions.
^Barak, Ohad (April 6, 2006). "Q Function and Error Function" (PDF). Tel Aviv University. Archived from the original (PDF) on March 25, 2009.
^Vaart, A. W. van der (October 13, 1998). Asymptotic Statistics. Cambridge University Press. doi:10.1017/cbo9780511802256. ISBN 978-0-511-80225-6.
^Cover, Thomas M.; Thomas, Joy A. (2006). Elements of Information Theory. John Wiley and Sons. p. 254. ISBN 9780471748816.
^Park, Sung Y.; Bera, Anil K. (2009). "Maximum Entropy Autoregressive Conditional Heteroskedasticity Model" (PDF). Journal of Econometrics. 150 (2): 219–230. CiteSeerX10.1.1.511.9750. doi:10.1016/j.jeconom.2008.12.014. Archived from the original (PDF) on March 7, 2016. Retrieved June 2, 2011.
^Geary RC(1936) The distribution of the "Student's ratio for the non-normal samples". Supplement to the Journal of the Royal Statistical Society 3 (2): 178–184
^Papoulis, Athanasios. Probability, Random Variables and Stochastic Processes (4th ed.). p. 148.
^Winkelbauer, Andreas (2012). "Moments and Absolute Moments of the Normal Distribution". arXiv:1209.4340 [math.ST].
^Bryc (1995, p. 23)
^Bryc (1995, p. 24)
^Cover & Thomas (2006, p. 254)
^Williams, David (2001). Weighing the odds : a course in probability and statistics (Reprinted. ed.). Cambridge [u.a.]: Cambridge Univ. Press. pp. 197–199. ISBN 978-0-521-00618-7.
^Smith, José M. Bernardo; Adrian F. M. (2000). Bayesian theory (Reprint ed.). Chichester [u.a.]: Wiley. pp. 209, 366. ISBN 978-0-471-49464-5.{{cite book}}: CS1 maint: multiple names: authors list (link)
^O'Hagan, A. (1994) Kendall's Advanced Theory of statistics, Vol 2B, Bayesian Inference, Edward Arnold. ISBN 0-340-52922-9 (Section 5.40)
^ a bBryc (1995, p. 35)
^UIUC, Lecture 21. The Multivariate Normal Distribution, 21.6:"Individually Gaussian Versus Jointly Gaussian".
^Edward L. Melnick and Aaron Tenenbein, "Misspecifications of the Normal Distribution", The American Statistician, volume 36, number 4 November 1982, pages 372–373
^"Kullback Leibler (KL) Distance of Two Normal (Gaussian) Probability Distributions". Allisons.org. December 5, 2007. Retrieved March 3, 2017.
^Jordan, Michael I. (February 8, 2010). "Stat260: Bayesian Modeling and Inference: The Conjugate Prior for the Normal Distribution" (PDF).
^Amari & Nagaoka (2000)
^"Expectation of the maximum of gaussian random variables". Mathematics Stack Exchange. Retrieved April 7, 2024.
^"Normal Approximation to Poisson Distribution". Stat.ucla.edu. Retrieved March 3, 2017.
^ a bDas, Abhranil (2021). "A method to integrate and classify normal distributions". Journal of Vision. 21 (10): 1. arXiv:2012.14331. doi:10.1167/jov.21.10.1. PMC 8419883. PMID 34468706.
^Bryc (1995, p. 27)
^Weisstein, Eric W. "Normal Product Distribution". MathWorld. wolfram.com.
^Lukacs, Eugene (1942). "A Characterization of the Normal Distribution". The Annals of Mathematical Statistics. 13 (1): 91–3. doi:10.1214/aoms/1177731647. ISSN 0003-4851. JSTOR 2236166.
^Basu, D.; Laha, R. G. (1954). "On Some Characterizations of the Normal Distribution". Sankhyā. 13 (4): 359–62. ISSN 0036-4452. JSTOR 25048183.
^Lehmann, E. L. (1997). Testing Statistical Hypotheses (2nd ed.). Springer. p. 199. ISBN 978-0-387-94919-2.
^Patel & Read (1996, [2.3.6])
^Galambos & Simonelli (2004, Theorem 3.5)
^ a bLukacs & King (1954)
^Quine, M.P. (1993). "On three characterisations of the normal distribution". Probability and Mathematical Statistics. 14 (2): 257–263.
^John, S (1982). "The three parameter two-piece normal family of distributions and its fitting". Communications in Statistics – Theory and Methods. 11 (8): 879–885. doi:10.1080/03610928208828279.
^ a bKrishnamoorthy (2006, p. 127)
^Krishnamoorthy (2006, p. 130)
^Krishnamoorthy (2006, p. 133)
^Huxley (1932)
^Jaynes, Edwin T. (2003). Probability Theory: The Logic of Science. Cambridge University Press. pp. 592–593. ISBN 9780521592710.
^Oosterbaan, Roland J. (1994). "Chapter 6: Frequency and Regression Analysis of Hydrologic Data" (PDF). In Ritzema, Henk P. (ed.). Drainage Principles and Applications, Publication 16 (second revised ed.). Wageningen, The Netherlands: International Institute for Land Reclamation and Improvement (ILRI). pp. 175–224. ISBN 978-90-70754-33-4.
^Why Most Published Research Findings Are False, John P. A. Ioannidis, 2005
^Wichura, Michael J. (1988). "Algorithm AS241: The Percentage Points of the Normal Distribution". Applied Statistics. 37 (3): 477–84. doi:10.2307/2347330. JSTOR 2347330.
^De Moivre, Abraham (1733), Corollary I – see Walker (1985, p. 77)
^Stigler (1986, p. 76)
^Gauss (1809, section 177)
^Gauss (1809, section 179)
^Laplace (1774, Problem III)
^Pearson (1905, p. 189)
^Stigler (1986, p. 144)
^Stigler (1978, p. 243)
^Stigler (1978, p. 244)
^Maxwell (1860, p. 23)
^Jaynes, Edwin J.; Probability Theory: The Logic of Science, Ch. 7.
^Peirce, Charles S. (c. 1909 MS), Collected Papers v. 6, paragraph 327.
^Kruskal & Stigler (1997).
^"Earliest Uses... (Entry Standard Normal Curve)".
^Sun, Jingchao; Kong, Maiying; Pal, Subhadip (June 22, 2021). "The Modified-Half-Normal distribution: Properties and an efficient sampling scheme". Communications in Statistics – Theory and Methods. 52 (5): 1591–1613. doi:10.1080/03610926.2021.1934700. ISSN 0361-0926. S2CID 237919587.
Sources
Aldrich, John; Miller, Jeff. "Earliest Uses of Symbols in Probability and Statistics".
Aldrich, John; Miller, Jeff. "Earliest Known Uses of Some of the Words of Mathematics". In particular, the entries for "bell-shaped and bell curve", "normal (distribution)", "Gaussian", and "Error, law of error, theory of errors, etc.".
Amari, Shun-ichi; Nagaoka, Hiroshi (2000). Methods of Information Geometry. Oxford University Press. ISBN 978-0-8218-0531-2.
Bernardo, José M.; Smith, Adrian F. M. (2000). Bayesian Theory. Wiley. ISBN 978-0-471-49464-5.
Bryc, Wlodzimierz (1995). The Normal Distribution: Characterizations with Applications. Springer-Verlag. ISBN 978-0-387-97990-8.
Casella, George; Berger, Roger L. (2001). Statistical Inference (2nd ed.). Duxbury. ISBN 978-0-534-24312-8.
Fan, Jianqing (1991). "On the optimal rates of convergence for nonparametric deconvolution problems". The Annals of Statistics. 19 (3): 1257–1272. doi:10.1214/aos/1176348248. JSTOR 2241949.
Galton, Francis (1889). Natural Inheritance (PDF). London, UK: Richard Clay and Sons.
Galambos, Janos; Simonelli, Italo (2004). Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions. Marcel Dekker, Inc. ISBN 978-0-8247-5402-0.
Gauss, Carolo Friderico (1809). Theoria motvs corporvm coelestivm in sectionibvs conicis Solem ambientivm [Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections] (in Latin). Hambvrgi, Svmtibvs F. Perthes et I. H. Besser. English translation.
Halperin, Max; Hartley, Herman O.; Hoel, Paul G. (1965). "Recommended Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation". The American Statistician. 19 (3): 12–14. doi:10.2307/2681417. JSTOR 2681417.
Hart, John F.; et al. (1968). Computer Approximations. New York, NY: John Wiley & Sons, Inc. ISBN 978-0-88275-642-4.
Huxley, Julian S. (1932). Problems of Relative Growth. London. ISBN 978-0-486-61114-3. OCLC 476909537.
Johnson, Norman L.; Kotz, Samuel; Balakrishnan, Narayanaswamy (1994). Continuous Univariate Distributions, Volume 1. Wiley. ISBN 978-0-471-58495-7.
Johnson, Norman L.; Kotz, Samuel; Balakrishnan, Narayanaswamy (1995). Continuous Univariate Distributions, Volume 2. Wiley. ISBN 978-0-471-58494-0.
Karney, C. F. F. (2016). "Sampling exactly from the normal distribution". ACM Transactions on Mathematical Software. 42 (1): 3:1–14. arXiv:1303.6257. doi:10.1145/2710016. S2CID 14252035.
Kinderman, Albert J.; Monahan, John F. (1977). "Computer Generation of Random Variables Using the Ratio of Uniform Deviates". ACM Transactions on Mathematical Software. 3 (3): 257–260. doi:10.1145/355744.355750. S2CID 12884505.
Krishnamoorthy, Kalimuthu (2006). Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC. ISBN 978-1-58488-635-8.
Kruskal, William H.; Stigler, Stephen M. (1997). Spencer, Bruce D. (ed.). Normative Terminology: 'Normal' in Statistics and Elsewhere. Statistics and Public Policy. Oxford University Press. ISBN 978-0-19-852341-3.
Laplace, Pierre-Simon de (1774). "Mémoire sur la probabilité des causes par les événements". Mémoires de l'Académie Royale des Sciences de Paris (Savants étrangers), Tome 6: 621–656. Translated by Stephen M. Stigler in Statistical Science1 (3), 1986: JSTOR 2245476.
Le Cam, Lucien; Lo Yang, Grace (2000). Asymptotics in Statistics: Some Basic Concepts (second ed.). Springer. ISBN 978-0-387-95036-5.
Leva, Joseph L. (1992). "A fast normal random number generator" (PDF). ACM Transactions on Mathematical Software. 18 (4): 449–453. CiteSeerX10.1.1.544.5806. doi:10.1145/138351.138364. S2CID 15802663. Archived from the original (PDF) on July 16, 2010.
Lexis, Wilhelm (1878). "Sur la durée normale de la vie humaine et sur la théorie de la stabilité des rapports statistiques". Annales de Démographie Internationale. II. Paris: 447–462.
Lukacs, Eugene; King, Edgar P. (1954). "A Property of Normal Distribution". The Annals of Mathematical Statistics. 25 (2): 389–394. doi:10.1214/aoms/1177728796. JSTOR 2236741.
McPherson, Glen (1990). Statistics in Scientific Investigation: Its Basis, Application and Interpretation. Springer-Verlag. ISBN 978-0-387-97137-7.
Marsaglia, George; Tsang, Wai Wan (2000). "The Ziggurat Method for Generating Random Variables". Journal of Statistical Software. 5 (8). doi:10.18637/jss.v005.i08.
Marsaglia, George (2004). "Evaluating the Normal Distribution". Journal of Statistical Software. 11 (4). doi:10.18637/jss.v011.i04.
Maxwell, James Clerk (1860). "V. Illustrations of the dynamical theory of gases. — Part I: On the motions and collisions of perfectly elastic spheres". Philosophical Magazine. Series 4. 19 (124): 19–32. doi:10.1080/14786446008642818.
Monahan, J. F. (1985). "Accuracy in random number generation". Mathematics of Computation. 45 (172): 559–568. doi:10.1090/S0025-5718-1985-0804945-X.
Patel, Jagdish K.; Read, Campbell B. (1996). Handbook of the Normal Distribution (2nd ed.). CRC Press. ISBN 978-0-8247-9342-5.
Pearson, Karl (1901). "On Lines and Planes of Closest Fit to Systems of Points in Space" (PDF). Philosophical Magazine. 6. 2 (11): 559–572. doi:10.1080/14786440109462720. S2CID 125037489.
Pearson, Karl (1905). "'Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und Pearson'. A rejoinder". Biometrika. 4 (1): 169–212. doi:10.2307/2331536. JSTOR 2331536.
Pearson, Karl (1920). "Notes on the History of Correlation". Biometrika. 13 (1): 25–45. doi:10.1093/biomet/13.1.25. JSTOR 2331722.
Rohrbasser, Jean-Marc; Véron, Jacques (2003). "Wilhelm Lexis: The Normal Length of Life as an Expression of the "Nature of Things"". Population. 58 (3): 303–322. doi:10.3917/pope.303.0303.
Shore, H (1982). "Simple Approximations for the Inverse Cumulative Function, the Density Function and the Loss Integral of the Normal Distribution". Journal of the Royal Statistical Society. Series C (Applied Statistics). 31 (2): 108–114. doi:10.2307/2347972. JSTOR 2347972.
Shore, H (2005). "Accurate RMM-Based Approximations for the CDF of the Normal Distribution". Communications in Statistics – Theory and Methods. 34 (3): 507–513. doi:10.1081/sta-200052102. S2CID 122148043.
Stigler, Stephen M. (1978). "Mathematical Statistics in the Early States". The Annals of Statistics. 6 (2): 239–265. doi:10.1214/aos/1176344123. JSTOR 2958876.
Stigler, Stephen M. (1982). "A Modest Proposal: A New Standard for the Normal". The American Statistician. 36 (2): 137–138. doi:10.2307/2684031. JSTOR 2684031.
Stigler, Stephen M. (1986). The History of Statistics: The Measurement of Uncertainty before 1900. Harvard University Press. ISBN 978-0-674-40340-6.
Stigler, Stephen M. (1999). Statistics on the Table. Harvard University Press. ISBN 978-0-674-83601-3.
Walker, Helen M. (1985). "De Moivre on the Law of Normal Probability" (PDF). In Smith, David Eugene (ed.). A Source Book in Mathematics. Dover. ISBN 978-0-486-64690-9.
Wallace, C. S. (1996). "Fast pseudo-random generators for normal and exponential variates". ACM Transactions on Mathematical Software. 22 (1): 119–127. doi:10.1145/225545.225554. S2CID 18514848.
West, Graeme (2009). "Better Approximations to Cumulative Normal Functions" (PDF). Wilmott Magazine: 70–76. Archived from the original (PDF) on February 29, 2012.