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viernes, 10 de octubre de 2008

postheadericon La varianza, Parte 2

Sacado de: Mis inicios en el poker

En la anterior entrada sobre la varianza, vimos como se podía calcular y porque la varianza del lanzamiento de un dado era mayor que la del lanzamiento de una moneda.

En esta entrada, sólo me queda comentar un par de propiedades importantes sobre la varianza e introducir el concepto de desviación estándar que es con lo que nos tocará lidiar a diario.

**********************************************************************************

“Este hombre utiliza la Estadística de la misma manera que un borracho utiliza una farola: más para apoyarse que para iluminarse”

Quizás la propiedad más interesante de la varianza es que la varianza de varios intentos es aditiva. ¿Y esto que significa? Volvemos al ejemplo del lanzamiento de moneda que proponía en la entrada anterior. Si salía cara, ganábamos 1$ y si salía cruz, perdíamos un 1$. Vimos que la varianza de esta apuesta era 1. Pero, ¿y si jugamos 10 veces? Pues por la propiedad que acabamos de enunciar, la varianza será 10 veces mayor, es decir 10.

Esto lo podemos escribir como:

VN = N * V

Esta propiedad la usaremos más adelante para tratar con los datos del poker tracker.

El principal inconveniente de usar la varianza para tratar los datos, son las unidades. La varianza vendrá dada por unidades al cuadrado por evento al cuadrado. Esto nos dificulta poder compararla con la media, que vendrá dada en unidades por evento. En el caso que nos ocupa la media será por ejemplo, BB/100, BB/mano, $/100, $/h, etc.

¿Y como solucionamos esto? Pues,¿ para qué complicarnos la vida?, le hacemos la raíz cuadrada a la varianza, de forma que tenga las mismas unidades que la media, y le damos otro nombre, desviación estándar (o desviación típica). La desviación estándar, se designa normalmente como “σ” y por definición, muchas veces nos encontraremos la varianza expresada como σ^2 (sigma al cuadrado).

Por tanto, la relación entre desviación estándar y varianza será:

σ = √V

σ^2 = V

¿Y que sucede con la desviación estándar tras N intentos? Lo más importante es saber que la desviación estándar de varios intentos NO ES ADITIVA como en el caso de la varianza.

Hemos visto que la varianza de N intentos era:

VN = N * V

O lo que es lo mismo:

σ^2 de N intentos = N * σ^2

Operando un poquito:

σN =√( N * σ^2)

Por tanto:


Esto es una propiedad importante, porque vemos como la desviación estándar se va “escalando” a medida que los intentos crecen. Por ejemplo, si consideramos un juego cualquiera que tenga una σ =1. ¿Cuál será la σ de 100, 1000, 10000 intentos?

σ100 = 1 √ 100 = 10
σ1000 = 1 √ 1000 = 31
σ10.000 = 1 √ 10000 = 100

Vemos que en términos absolutos, la desviación estándar crece con los intentos, pero si la comparamos con el numero de intentos, está es cada vez más pequeña. Es decir, 10 sobre 100 es 10 veces mayor que 100 sobre 10.000.

Aquí se empieza a intuir la idea de largo plazo de la que tanto hablamos. Vemos que a medida que crecen los intentos, la desviación estándar decrece en relación a ellos y por tanto, los resultados tienden a estar más cerca de su media.

Bueno, por hoy hasta aquí hemos llegado. Por mi parte, doy ya por introducidos los conceptos de varianza y desviación estándar. En próximas entregas, veremos cómo usar estos datos para sacar conclusiones aunque primero habría que introducir la distribución normal.

A ver si se ven bien las fórmulas porque la verdad es que el editor este de blogger es una mierda. Como lo tengo en word, a ver si encuentro algún lado para colgarlo y añado el enlace de descarga despues de cada link.

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