estadistica para administracion y economia _ paoola ramiirez - academia.pdf
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Estadistica para Administracion y Economia
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Determinación dela varianza y ladesviaciónestándar delporcentaje deimpureza de loscompuestos
Tabla 3-19 Media Desviación Desviación Observación
Observación ( x ) 2.49/15 ( x ) al cuadrado ( x )2 al cuadrado ( x 2)
(1) (2) (3) (1) (2) (4) [(1) (2)]2 (5) (1)2
0.04 0.166 0.126 0.016 0.0016
0.06 0.166 0.106 0.011 0.0036
0.12 0.166 0.046 0.002 0.0144
0.14 0.166 0.026 0.001 0.0196
0.14 0.166 0.026 0.001 0.0196
0.15 0.166 0.016 0.000 0.0225
0.17 0.166 0.004 0.000 0.0289
0.17 0.166 0.004 0.000 0.0289
0.18 0.166 0.014 0.000 0.0324
0.19 0.166 0.024 0.001 0.0361
0.21 0.166 0.044 0.002 0.0441
0.21 0.166 0.044 0.002 0.0441
0.22 0.166 0.054 0.003 0.0484
0.24 0.166 0.074 0.005 0.0576
0.25 0.166 0.084 0.007 0.06252.49← x 0.051← ( x )2 0.4643 ← x
2
2
[3-12] ← O→ 2
[3-12]
(0.166)20.4643
15
0.051
15
x 2
N 2
( x )2
N
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98 Capítulo 3 Medidas de tendencia central y dispersión en distribuciones de frecuencias
FIGURA 3-12
Localización delas observacionesalrededor de lamedia para unadistribución defrecuencias conforma de campana m - 3s m - 2s m - s m m + s m + 2s m + 3s
99%
95%
68%
0.0034 al cuadrado 0.0034 al cuadrado
2 [3-13]
0.0034
0.058%
de los valores caen dentro de 2 desviaciones estándar a partir de la media de la distribución, y al
menos 89% de los valores caen dentro de 3 desviaciones estándar a partir de la media.Podemos medir aún con más precisión el porcentaje de observaciones que caen dentro de un ran-
go específico de una curva simétrica con forma de campana, como la mostrada en la figura 3-12. Enestos casos, podemos decir que:
1. Aproximadamente 68% de los valores de la población cae dentro de ±1 desviación estándara partir de la media.
2. Aproximadamente 95% de los valores estará dentro de ±2 desviaciones estándar a partir de lamedia.
3. Aproximadamente 99% de los valores estará en el intervalo que va desde 3 desviaciones es-tándar a la izquierda de la media hasta 3 desviaciones estándar a la derecha de la media.
A la luz del teorema de Chebyshev, analicemos los datos de la tabla 3-19. En ellos, la impurezamedia de los 15 frascos de compuesto es 0.166% y la desviación estándar es 0.058%. El teorema deChebyshev nos dice que al menos 75% de los valores (11 de nuestros 15 frascos) están entre 0.166 2(0.058) 0.050 y 0.166 2(0.058) 0.282. De hecho, 93% de las observaciones (14 de los 15valores) están realmente en el intervalo. Note que la distribución es razonablemente simétrica y que93% es muy cercano al 95% teórico para un intervalo de ±2 desviaciones estándar a partir de la me-dia de una curva con forma de campana.
La desviación estándar es útil también para describir cuánto se apartan las observaciones individua-les de una distribución de la media de la misma. Una medida que se conoce como resultado están-
dar nos da el número de desviaciones estándar que una observación en particular ocupa por debajoo por encima de la media. Si x simboliza la observación, entonces el resultado estándar calculado apartir de los datos de la población es:
Concepto de
resultado estándar
Uso del teorema
de Chebyshev
Resultado estándar
Resultado estándar de la población x
[3-14]
donde,
• x observación tomada de la población
• media de la población
• desviación estándar de la población
Suponga que observamos un frasco de compuesto que tiene 0.108% de impureza. Como nuestra
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3.9 Dispersión: medidas de desviación promedio 99
Suponga que observamos un frasco de compuesto que tiene 0.108% de impureza. Como nuestrapoblación tiene una media de 0.166 y una desviación estándar de 0.058, una observación de 0.108tendría un resultado estándar de 1:
Resultado estándar [3-14]
1
Una impureza observada del 0.282% tendría un resultado estándar de 2:
Resultado estándar [3-14]
2
El resultado estándar indica que una impureza del 0.282% se desvía de la media en 2(0.058) 0.116unidades, que es igual a 2, en términos del número de desviaciones estándar alejado de la media.
Interpretación del
resultado estándar
0.116
0.058
0.282 0.166
0.058
x
0.058
0.058
0.108 0.1660.058
x
Cálculo del
resultado estándar
Cálculo de la varianza y la desviación estándar
utilizando datos agrupados
En el ejemplo de la página 58, los datos con respecto en las ventas en 100 restaurantes de comida
rápida se encuentran agrupados en una distribución de frecuencias. Con esos datos, podemos utilizarlas siguientes fórmulas para calcular la varianza y la desviación estándar:
Cálculo de la
varianza y de ladesviación estándar
de datos agrupados
Varianza de datos agrupados
2
2 [3-15]
f x 2
N
f ( x )2
N
Desviación estándar de datos agrupados
2 2 [3-16] f x 2
N
f ( x )2
N
donde,
• 2 varianza de la población• desviación estándar de la población
• f frecuencia de cada una de las clases
• x punto medio de cada clase
• media de la población
• N tamaño de la población
La tabla 3-20 muestra cómo aplicar estas ecuaciones para encontrar la varianza y la desviación es-tándar de las ventas en 100 restaurantes de comida rápida.
Dejamos como ejercicio para el lector curioso verificar que la segunda mitad de la ecuación 3-15,
2 da como resultado el mismo valor de
2. f x 2
N
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100 Capítulo 3 Medidas de tendencia central y dispersión en distribuciones de frecuencias
Varianza de una muestra
s2 [3-17]nx
2
n 1 x 2
n 1( x x )2
n 1
Desviación estándar de una muestra
s s2 [3-18]nx
2
n 1 x 2
n 1( x x )2
n 1
Ahora estamos listos para calcular las estadísticas de muestra análogas a la varianza de población 2
y la desviación estándar de la población, . Se trata de la varianza de la muestra s2 y la desviación están-dar de la muestra, s. En la sección siguiente, observará que cambiamos la notación con letras griegas (quedenotan parámetros de población) a las latinas correspondientes a las estadísticas de muestras.
Desviación estándar de una muestra
Para calcular la varianza y la desviación estándar de una muestra, utilizamos las mismas fórmulas delas ecuaciones 3-12 y 3-13, sustituyendo con x y N con n 1. Las fórmulas tienen el siguiente as-pecto:
Cálculo de la
desviación estándar
de una muestra
Cambio a la varianza
y la desviación
estándar de unamuestra
N
F r e c u
e n c i a
M e d
i a
f
f
x
x
( x
) 2
f ( x
) 2
( 2 )
( 3 )
( 2 )
( 1 )
( 4 )
( 1 )
( 4 )
[ ( 1 )
( 4 ) ] 2
( 2
)
[ ( 1 )
( 4 ) ] 2
4
3 , 0
0 0
1 , 2 5
0
5 0 0
2 5 0 , 0
0 0
1 , 0
0 0 , 0
0 0
7
5 , 9
5 0
1 , 2 5
0
4 0 0
1 6 0 , 0
0 0
1 , 1
2 0 , 0
0 0
8
7 , 6
0 0
1 , 2 5
0
3 0 0
9 0 , 0
0 0
7 2 0 , 0
0 0
1 0
1 0 , 5
0 0
1 , 2 5
0
2 0 0
4 0 , 0
0 0
4 0 0 , 0
0 0
1 2
1 3 , 8
0 0
1 , 2 5
0
1 0 0
1 0 , 0
0 0
1 2 0 , 0
0 0
1 7
2 1 , 2
5 0
1 , 2 5
0
0
0
0
1 3
1 7 , 5
5 0
1 , 2 5
0
1 0 0
1 0 , 0
0 0
1 3 0 , 0
0 0
1 0
1 4 , 5
0 0
1 , 2 5
0
2 0 0
4 0 , 0
0 0
4 0 0 , 0
0 0
9
1 3 , 9
5 0
1 , 2 5
0
3 0 0
9 0 , 0
0 0
8 1 0 , 0
0 0
7
1 1 , 5
5 0
1 , 2 5
0
4 0 0
1 6 0 , 0
0 0
1 , 1
2 0 , 0
0 0
2
3 , 5
0 0
1 , 2 5
0
5 0 0
2 5 0 , 0
0 0
5 0 0 , 0
0 0
0 0 1
0 0 1 , 8
5 0
1 , 2 5
0
6 0 0
3 6 0 , 0
0 0
0 0 3 6 0 , 0
0 0
1 0 0
1 2 5
, 0 0 0
6 , 6
8 0
, 0 0 0
x
[ 3 - 3
]
1 , 2
5 0 ( m i l e s d e d ó l a r e s ) ←
M e d i a
2
[ 3 - 1 5 ]
6 6 , 8
0 0 ( o 6 6 , 8
0 0 [ m i l e s d
e d ó l a r e s ] 2 ) ← V
a r i a n z a
2
[ 3 - 1 6 ]
6 6 , 8
0 0
2 5 8 . 5 ← D
e s v i a c i ó n e s t á n d a r
$ 2 5 8 , 5
0 0
6 , 6
8 0 , 0
0 0
1 0 0
f ( x –
) 2
N
1 2 5 , 0
0 0
1 0 0
( f
x )
n
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3.9 Dispersión: medidas de desviación promedio 101
D e t e r m i n a c i ó n d e
l a v
a r i a n z a
y l a
d e s v
i a c i ó n
e s t á n d a r
d e l a s v
e n t
a s e n 1 0 0
r e s t a u
r a n t e s d e
c o m i d a r á p
i d a
s i t u a d o s e n
e l d i s t r i t o
d e l e s t e ( m
i l e s )
T a b l a 3 - 2 0
P u
n t o m e d i o
x
C l a s e
( 1 )
7 0 0 -
7 9 9
1 , 7
5 0
8 0 0 -
8 9 9
1 , 8
5 0
9 0 0 -
9 9 9
1 , 9
5 0
1 , 0
0 0 - 1 , 0
9 9
1 , 0
5 0
1 , 1
0 0 - 1 , 1
9 9
1 , 1
5 0
1 , 2
0 0 - 1 , 2
9 9
1 , 2
5 0
1 , 3
0 0 - 1 , 3
9 9
1 , 3
5 0
1 , 4
0 0 - 1 , 4
9 9
1 , 4
5 0
1 , 5
0 0 - 1 , 5
9 9
1 , 5
5 0
1 , 6
0 0 - 1 , 6
9 9
1 , 6
5 0
1 , 7
0 0 - 1 , 7
9 9
1 , 7
5 0
1 , 8
0 0 - 1 , 8
9 9
1 , 8
5 0
donde,
• s2 Varianza de la muestra
• s Desviación estándar de la muestra
Determinación de lavarianza y la desviaciónestándar de la muestrade los donativosanuales de Blue Cross-Blue Shield al Hospitalde Cumberland (miles)
Tabla 3-21 Observación Media
( x ) ( x ) x x ( x x )2 x 2
(1) (2) (1) (2) [(1) (2)]2 (1)2
863 1,351 488 238,144 744,769
903 1,351 448 200,704 815,409
957 1,351 394 155,236 915,8491,041 1,351 310 96,100 1,083,681
1,138 1,351 213 45,369 1,295,044
1,204 1,351 147 21,609 1,449,616
1,354 1,351 3 9 1,833,316
1,624 1,351 273 74,529 2,637,376
1,698 1,351 347 120,409 2,883,204
1,745 1,351 394 155,236 3,045,025
1,802 1,351 451 203,401 3,247,204
1,883 1,351 532 00283,024 003,545,689
( x x )2→ 1,593,770 23,496,182← x2
s2
[3-17]
144,888 (o $144,888 [miles de dólares]2)← Varianza de la muestra
s s2 [3-18]
14 4, 88 8 O
380.64 (es decir, $380,640)← Desviación estándar de la muestra
s2
[3-17]
144,888
1,593,770
11
12(1,351)2
11
23,496,182
11
n x 2
n 1
x 2
n 1
1,593,770
11
( x x )2
n 1
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102 Capítulo 3 Medidas de tendencia central y dispersión en distribuciones de frecuencias
• s Desviación estándar de la muestra
• x Valor de cada una de las n observaciones
• x Media de la muestra
• n 1 Número de observaciones de la muestra menos 1
¿Por qué utilizamos n 1 como denominador en lugar de n? Los especialistas en estadística pue-den demostrar que si tomamos muchas muestras de una población dada, encontramos la varianza dela muestra (s2) para cada muestra y promediamos los resultados, este promedio no tiende a igualarel valor de la varianza de la población, 2, a menos que usemos n 1 como denominador en nues-tros cálculos. En el capítulo 7, se dará la explicación estadística de por qué esto es cierto.
Las ecuaciones 3-17 y 3-18 nos permiten encontrar la varianza y la desviación estándar de lamuestra de los donativos anuales de Blue Cross-Blue Shield al Hospital de Cumberland que presen-tamos en la tabla 3-21; observe que ambas mitades de la ecuación 3-17 producen el mismo resultado.
Cálculo de la varian-
za y la desviación es-tándar de la muestra
para los datos del
hospital
Uso de n 1 como
denominador
Igual que utilizamos la desviación estándar de la población para derivar los resultados estándarde la misma, podemos usar la desviación estándar de la muestra para calcular los resultados estándar dela muestra. Estos resultados indican a cuántas desviaciones estándar arriba o abajo de la media de lamuestra se encuentra una observación dada. La fórmula adecuada es:
Cálculo de los
resultados estándar
de la muestra
Resultado estándar de una observación de una muestra
Resultado estándar de la muestra [3-19] x x
s
donde:
• x observación tomada de la muestra
• x media de la muestra
• s desviación estándar de la muestra
En el ejemplo anterior, vemos que la observación 863 corresponde a un resultado estándar de 1.28:
Resultado estándar de la muestra [3-19]
1.28
En esta sección hemos demostrado por qué la desviación estándar es la medida de dispersión quemás se utiliza. Podemos usarla para comparar distribuciones y para calcular resultados estándar, que sonun elemento importante de la inferencia estadística que analizaremos más adelante. Al igual que lavarianza, la desviación estándar toma en cuenta cada observación del conjunto de datos. Sin embar-go, la desviación estándar tiene también algunas desventajas. No es fácil calcularla como el rango,y no puede calcularse en distribuciones de extremo abierto. Además, los valores extremos que se en-cuentren en el conjunto de datos distorsionan el valor de la desviación estándar, aunque en menorgrado que en el caso del rango.
488
380.64
863 1,351
380.64
x x
s
Al calcular y usar la desviación estándarse supone que no hay muchos valores de-masiado grandes o demasiado pequeñosen el conjunto de datos porque se sabe
que la desviación estándar usa todos los valores; esos valo-
res extremos distorsionarán la respuesta. Sugerencia: pue-de evitarse la confusión entre usar N o n 1 como deno-minador para las muestras y poblaciones si se asocia el va-lor más pequeño (n 1) con el conjunto más pequeño (lamuestra).
SUGERENCIAS
Y
SUPOSICIONES
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3.9 Dispersión: medidas de desviación promedio 103
Ejercicios 3.9
Ejercicios de autoevaluación
EA 3-13 Talent, Ltd., una compañía en Hollywood de selección de elenco, está en proceso de elegir un grupo deextras para una película. Las edades de los 20 hombres que se entrevistaron primero son:
50 56 55 49 52 57 56 57 56 59
54 55 61 60 51 59 62 52 54 49
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