Estadística Ejemplos
ClassFrequency360-3692370-3793380-3895390-3997400-4095410-4194420-4294430-4391ClassFrequency360−3692370−3793380−3895390−3997400−4095410−4194420−4294430−4391
Paso 1
Paso 1.1
El límite inferior de cada clase es el menor valor de la clase. Por el contrario, el límite superior de cada clase es el mayor valor de la clase.
ClassFrequency(f)LowerLimitsUpperLimits360-3692360369370-3793370379380-3895380389390-3997390399400-4095400409410-4194410419420-4294420429430-4391430439ClassFrequency(f)LowerLimitsUpperLimits360−3692360369370−3793370379380−3895380389390−3997390399400−4095400409410−4194410419420−4294420429430−4391430439
Paso 1.2
El punto medio de la clase es el límite inferior para la clase más el límite superior para la clase, dividido por 22.
ClassFrequency(f)LowerLimitsUpperLimitsMidpoint(M)360-3692360369360+3692370-3793370379370+3792380-3895380389380+3892390-3997390399390+3992400-4095400409400+4092410-4194410419410+4192420-4294420429420+4292430-4391430439430+4392ClassFrequency(f)LowerLimitsUpperLimitsMidpoint(M)360−3692360369360+3692370−3793370379370+3792380−3895380389380+3892390−3997390399390+3992400−4095400409400+4092410−4194410419410+4192420−4294420429420+4292430−4391430439430+4392
Paso 1.3
Simplifica toda la columna de puntos medios.
ClassFrequency(f)LowerLimitsUpperLimitsMidpoint(M)360-3692360369364.5370-3793370379374.5380-3895380389384.5390-3997390399394.5400-4095400409404.5410-4194410419414.5420-4294420429424.5430-4391430439434.5ClassFrequency(f)LowerLimitsUpperLimitsMidpoint(M)360−3692360369364.5370−3793370379374.5380−3895380389384.5390−3997390399394.5400−4095400409404.5410−4194410419414.5420−4294420429424.5430−4391430439434.5
Paso 1.4
Suma la columna de puntos medios a la tabla original.
ClassFrequency(f)Midpoint(M)360-3692364.5370-3793374.5380-3895384.5390-3997394.5400-4095404.5410-4194414.5420-4294424.5430-4391434.5ClassFrequency(f)Midpoint(M)360−3692364.5370−3793374.5380−3895384.5390−3997394.5400−4095404.5410−4194414.5420−4294424.5430−4391434.5
ClassFrequency(f)Midpoint(M)360-3692364.5370-3793374.5380-3895384.5390-3997394.5400-4095404.5410-4194414.5420-4294424.5430-4391434.5ClassFrequency(f)Midpoint(M)360−3692364.5370−3793374.5380−3895384.5390−3997394.5400−4095404.5410−4194414.5420−4294424.5430−4391434.5
Paso 2
Calcula el cuadrado del punto medio de cada grupo M2M2.
ClassFrequency(f)Midpoint(M)M2360-3692364.5364.52370-3793374.5374.52380-3895384.5384.52390-3997394.5394.52400-4095404.5404.52410-4194414.5414.52420-4294424.5424.52430-4391434.5434.52ClassFrequency(f)Midpoint(M)M2360−3692364.5364.52370−3793374.5374.52380−3895384.5384.52390−3997394.5394.52400−4095404.5404.52410−4194414.5414.52420−4294424.5424.52430−4391434.5434.52
Paso 3
Simplifica la columna M2M2.
ClassFrequency(f)Midpoint(M)M2360-3692364.5132860.25370-3793374.5140250.25380-3895384.5147840.25390-3997394.5155630.25400-4095404.5163620.25410-4194414.5171810.25420-4294424.5180200.25430-4391434.5188790.25ClassFrequency(f)Midpoint(M)M2360−3692364.5132860.25370−3793374.5140250.25380−3895384.5147840.25390−3997394.5155630.25400−4095404.5163620.25410−4194414.5171810.25420−4294424.5180200.25430−4391434.5188790.25
Paso 4
Multiplica cada punto medio al cuadrado por su frecuencia ff.
ClassFrequency(f)Midpoint(M)M2f⋅M2360-3692364.5132860.252⋅132860.25370-3793374.5140250.253⋅140250.25380-3895384.5147840.255⋅147840.25390-3997394.5155630.257⋅155630.25400-4095404.5163620.255⋅163620.25410-4194414.5171810.254⋅171810.25420-4294424.5180200.254⋅180200.25430-4391434.5188790.251⋅188790.25ClassFrequency(f)Midpoint(M)M2f⋅M2360−3692364.5132860.252⋅132860.25370−3793374.5140250.253⋅140250.25380−3895384.5147840.255⋅147840.25390−3997394.5155630.257⋅155630.25400−4095404.5163620.255⋅163620.25410−4194414.5171810.254⋅171810.25420−4294424.5180200.254⋅180200.25430−4391434.5188790.251⋅188790.25
Paso 5
Simplifica la columna f⋅M2f⋅M2.
ClassFrequency(f)Midpoint(M)M2f⋅M2360-3692364.5132860.25265720.5370-3793374.5140250.25420750.75380-3895384.5147840.25739201.25390-3997394.5155630.251089411.75400-4095404.5163620.25818101.25410-4194414.5171810.25687241420-4294424.5180200.25720801430-4391434.5188790.25188790.25ClassFrequency(f)Midpoint(M)M2f⋅M2360−3692364.5132860.25265720.5370−3793374.5140250.25420750.75380−3895384.5147840.25739201.25390−3997394.5155630.251089411.75400−4095404.5163620.25818101.25410−4194414.5171810.25687241420−4294424.5180200.25720801430−4391434.5188790.25188790.25
Paso 6
Obtén la suma de todas las frecuencias. En este caso, la suma de todas las frecuencias es n=2,3,5,7,5,4,4,1=31n=2,3,5,7,5,4,4,1=31.
∑f=n=31∑f=n=31
Paso 7
Obtén la suma de la columna f⋅M2f⋅M2. En este caso, 265720.5+420750.75+739201.25+1089411.75+818101.25+687241+720801+188790.25=4930017.75265720.5+420750.75+739201.25+1089411.75+818101.25+687241+720801+188790.25=4930017.75.
∑f⋅M2=4930017.75∑f⋅M2=4930017.75
Paso 8
Paso 8.1
Obtén el punto medio MM para cada clase.
ClassFrequency(f)Midpoint(M)360-3692364.5370-3793374.5380-3895384.5390-3997394.5400-4095404.5410-4194414.5420-4294424.5430-4391434.5ClassFrequency(f)Midpoint(M)360−3692364.5370−3793374.5380−3895384.5390−3997394.5400−4095404.5410−4194414.5420−4294424.5430−4391434.5
Paso 8.2
Multiplica la frecuencia de cada clase por el punto medio de la clase.
ClassFrequency(f)Midpoint(M)f⋅M360-3692364.52⋅364.5370-3793374.53⋅374.5380-3895384.55⋅384.5390-3997394.57⋅394.5400-4095404.55⋅404.5410-4194414.54⋅414.5420-4294424.54⋅424.5430-4391434.51⋅434.5ClassFrequency(f)Midpoint(M)f⋅M360−3692364.52⋅364.5370−3793374.53⋅374.5380−3895384.55⋅384.5390−3997394.57⋅394.5400−4095404.55⋅404.5410−4194414.54⋅414.5420−4294424.54⋅424.5430−4391434.51⋅434.5
Paso 8.3
Simplifica la columna f⋅Mf⋅M.
ClassFrequency(f)Midpoint(M)f⋅M360-3692364.5729370-3793374.51123.5380-3895384.51922.5390-3997394.52761.5400-4095404.52022.5410-4194414.51658420-4294424.51698430-4391434.5434.5ClassFrequency(f)Midpoint(M)f⋅M360−3692364.5729370−3793374.51123.5380−3895384.51922.5390−3997394.52761.5400−4095404.52022.5410−4194414.51658420−4294424.51698430−4391434.5434.5
Paso 8.4
Suma los valores en la columna f⋅Mf⋅M.
729+1123.5+1922.5+2761.5+2022.5+1658+1698+434.5=12349.5729+1123.5+1922.5+2761.5+2022.5+1658+1698+434.5=12349.5
Paso 8.5
Suma los valores en la columna de frecuencia.
n=2+3+5+7+5+4+4+1=31n=2+3+5+7+5+4+4+1=31
Paso 8.6
La media (mu) es la suma de f⋅Mf⋅M dividido por nn, que es la suma de frecuencias.
μ=∑f⋅M∑fμ=∑f⋅M∑f
Paso 8.7
La media es la suma del producto de los puntos medios y las frecuencias, dividida por el total de frecuencias.
μ=12349.531μ=12349.531
Paso 8.8
Simplifica el lado derecho de μ=12349.531μ=12349.531.
398.37096774398.37096774
398.37096774398.37096774
Paso 9
La ecuación para la desviación estándar es S2=∑f⋅M2-n(μ)2n-1S2=∑f⋅M2−n(μ)2n−1.
S2=∑f⋅M2-n(μ)2n-1S2=∑f⋅M2−n(μ)2n−1
Paso 10
Sustituye los valores calculados en S2=∑f⋅M2-n(μ)2n-1S2=∑f⋅M2−n(μ)2n−1.
S2=4930017.75-31(398.37096774)231-1S2=4930017.75−31(398.37096774)231−1
Paso 11
Simplifica el lado derecho de S2=4930017.75-31(398.37096774)231-1S2=4930017.75−31(398.37096774)231−1 para obtener la varianza S2=344.51612903S2=344.51612903.
344.51612903344.51612903
