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Hoja de fórmulas y tablas

Las fórmulas de toda la materia, de la descriptiva a la regresión, y las tablas normal, t, chi-cuadrado y F. Para imprimir y llevar al parcial, si tu cátedra lo permite.

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Estadística · Hoja de fórmulas y tablas

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Estadística descriptiva

Media
xˉ=∑xin\displaystyle \bar x = \frac{\sum x_i}{n}
Media, datos agrupados
xˉ=∑fi min\displaystyle \bar x = \frac{\sum f_i\, m_i}{n}
mᵢ: marca de clase
Mediana, datos agrupados
Me=Li+n/2−Fi−1fi a\displaystyle Me = L_i + \frac{n/2 - F_{i-1}}{f_i}\, a
Lᵢ: límite inferior de la clase de la mediana; a: ancho
Varianza muestral
s2=∑(xi−xˉ)2n−1\displaystyle s^2 = \frac{\sum (x_i - \bar x)^2}{n - 1}
Desvío y coeficiente de variación
s=s2CV=sxˉ⋅100\displaystyle s = \sqrt{s^2} \qquad CV = \frac{s}{\bar x} \cdot 100
Puntaje z
z=x−xˉs\displaystyle z = \frac{x - \bar x}{s}
Rango intercuartílico
RIC=Q3−Q1\displaystyle RIC = Q_3 - Q_1
atípico: fuera de Q₁ − 1,5·RIC y Q₃ + 1,5·RIC

Dos variables

Covarianza
sxy=∑(xi−xˉ)(yi−yˉ)n−1\displaystyle s_{xy} = \frac{\sum (x_i - \bar x)(y_i - \bar y)}{n - 1}
Correlación de Pearson
r=sxysx sy\displaystyle r = \frac{s_{xy}}{s_x\, s_y}
Recta de mínimos cuadrados
b1=sxysx2=r sysxb0=yˉ−b1xˉ\displaystyle b_1 = \frac{s_{xy}}{s_x^2} = r\,\frac{s_y}{s_x} \qquad b_0 = \bar y - b_1 \bar x
Coeficiente de determinación
R2=r2\displaystyle R^2 = r^2
en la regresión simple

Probabilidad

Regla de la suma
P(A∪B)=P(A)+P(B)−P(A∩B)\displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B)
Probabilidad condicional
P(A∣B)=P(A∩B)P(B)\displaystyle P(A \mid B) = \frac{P(A \cap B)}{P(B)}
Regla del producto
P(A∩B)=P(A) P(B∣A)\displaystyle P(A \cap B) = P(A)\, P(B \mid A)
independientes: P(A ∩ B) = P(A)·P(B)
Probabilidad total
P(B)=∑iP(B∣Ai) P(Ai)\displaystyle P(B) = \sum_i P(B \mid A_i)\, P(A_i)
Teorema de Bayes
P(Aj∣B)=P(B∣Aj) P(Aj)∑iP(B∣Ai) P(Ai)\displaystyle P(A_j \mid B) = \frac{P(B \mid A_j)\, P(A_j)}{\sum_i P(B \mid A_i)\, P(A_i)}
Conteo
Pn=n!Vn,k=n!(n−k)!Cn,k=n!k! (n−k)!\displaystyle P_n = n! \qquad V_{n,k} = \frac{n!}{(n-k)!} \qquad C_{n,k} = \frac{n!}{k!\,(n-k)!}

Variables aleatorias

Esperanza y varianza
E(X)=∑x p(x)V(X)=E(X2)−E(X)2\displaystyle E(X) = \sum x\, p(x) \qquad V(X) = E(X^2) - E(X)^2
Transformación lineal
E(aX+b)=a E(X)+bV(aX+b)=a2 V(X)\displaystyle E(aX + b) = a\,E(X) + b \qquad V(aX + b) = a^2\, V(X)
Binomial
P(X=k)=(nk)pk(1−p)n−kμ=npσ2=np(1−p)\displaystyle P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \qquad \mu = np \quad \sigma^2 = np(1-p)
Poisson
P(X=k)=e−λλkk!μ=σ2=λ\displaystyle P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \qquad \mu = \sigma^2 = \lambda
λ se reescala al intervalo pedido
Normal: tipificación
z=x−μσx=μ+z σ\displaystyle z = \frac{x - \mu}{\sigma} \qquad x = \mu + z\,\sigma
Aproximación normal a la binomial
μ=npσ=np(1−p)\displaystyle \mu = np \qquad \sigma = \sqrt{np(1-p)}
con corrección por continuidad de ±0,5

Distribuciones muestrales

Error estándar de la media
σxˉ=σnz=xˉ−μσ/n\displaystyle \sigma_{\bar x} = \frac{\sigma}{\sqrt n} \qquad z = \frac{\bar x - \mu}{\sigma / \sqrt n}
Error estándar de la proporción
σp^=p (1−p)n\displaystyle \sigma_{\hat p} = \sqrt{\frac{p\,(1-p)}{n}}

Intervalos de confianza y tamaño de muestra

Media, σ conocido
xˉ±zα/2 σn\displaystyle \bar x \pm z_{\alpha/2}\, \frac{\sigma}{\sqrt n}
Media, σ desconocido
xˉ±tα/2; n−1 sn\displaystyle \bar x \pm t_{\alpha/2;\, n-1}\, \frac{s}{\sqrt n}
Proporción
p^±zα/2p^ (1−p^)n\displaystyle \hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p\,(1-\hat p)}{n}}
Diferencia de medias
(xˉ1−xˉ2)±tα/2s12n1+s22n2\displaystyle (\bar x_1 - \bar x_2) \pm t_{\alpha/2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}
Tamaño para una media
n=(zα/2 σE)2\displaystyle n = \left( \frac{z_{\alpha/2}\, \sigma}{E} \right)^2
siempre se redondea hacia arriba
Tamaño para una proporción
n=zα/22 p (1−p)E2\displaystyle n = \frac{z_{\alpha/2}^2\, p\,(1-p)}{E^2}
sin dato previo, p = 0,5

Pruebas de hipótesis

Una media
t=xˉ−μ0s/n\displaystyle t = \frac{\bar x - \mu_0}{s / \sqrt n}
gl = n − 1; con σ conocido, z
Una proporción
z=p^−p0p0(1−p0)/n\displaystyle z = \frac{\hat p - p_0}{\sqrt{p_0 (1 - p_0) / n}}
Dos medias, varianzas iguales
t=xˉ1−xˉ2sp1n1+1n2sp2=(n1−1)s12+(n2−1)s22n1+n2−2\displaystyle t = \frac{\bar x_1 - \bar x_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \qquad s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}
gl = n₁ + n₂ − 2
Dos medias, Welch
t=xˉ1−xˉ2s12n1+s22n2\displaystyle t = \frac{\bar x_1 - \bar x_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
no supone varianzas iguales
Chi-cuadrado
Eij=fi cjnχ2=∑(O−E)2E\displaystyle E_{ij} = \frac{f_i\, c_j}{n} \qquad \chi^2 = \sum \frac{(O - E)^2}{E}
independencia: gl = (r − 1)(c − 1); bondad de ajuste: gl = k − 1
Decisión
p-valor<α  ⇒  se rechaza H0\displaystyle \text{p-valor} < \alpha \;\Rightarrow\; \text{se rechaza } H_0
bilateral: el p-valor suma las dos colas

ANOVA y regresión

Sumas de cuadrados
SCE=∑nj(xˉj−xˉ)2SCD=∑(nj−1) sj2\displaystyle SC_E = \sum n_j (\bar x_j - \bar x)^2 \qquad SC_D = \sum (n_j - 1)\, s_j^2
Estadístico F
F=SCE/(k−1)SCD/(N−k)\displaystyle F = \frac{SC_E / (k - 1)}{SC_D / (N - k)}
k grupos, N casos en total
Pendiente: prueba e intervalo
t=b1EE(b1)b1±tα/2; n−2  EE(b1)\displaystyle t = \frac{b_1}{EE(b_1)} \qquad b_1 \pm t_{\alpha/2;\, n-2}\; EE(b_1)
gl = n − 2
Error estándar de la pendiente
EE(b1)=se∑(xi−xˉ)2se=∑(yi−y^i)2n−2\displaystyle EE(b_1) = \frac{s_e}{\sqrt{\sum (x_i - \bar x)^2}} \qquad s_e = \sqrt{\frac{\sum (y_i - \hat y_i)^2}{n - 2}}

Normal estándar: Φ(z) = P(Z ≤ z)

Fila: entero y décima de z; columna: centésima. Para z negativo, Φ(−z) = 1 − Φ(z).

z0,000,010,020,030,040,050,060,070,080,09
0,00,50000,50400,50800,51200,51600,51990,52390,52790,53190,5359
0,10,53980,54380,54780,55170,55570,55960,56360,56750,57140,5753
0,20,57930,58320,58710,59100,59480,59870,60260,60640,61030,6141
0,30,61790,62170,62550,62930,63310,63680,64060,64430,64800,6517
0,40,65540,65910,66280,66640,67000,67360,67720,68080,68440,6879
0,50,69150,69500,69850,70190,70540,70880,71230,71570,71900,7224
0,60,72570,72910,73240,73570,73890,74220,74540,74860,75170,7549
0,70,75800,76110,76420,76730,77040,77340,77640,77940,78230,7852
0,80,78810,79100,79390,79670,79950,80230,80510,80780,81060,8133
0,90,81590,81860,82120,82380,82640,82890,83150,83400,83650,8389
1,00,84130,84380,84610,84850,85080,85310,85540,85770,85990,8621
1,10,86430,86650,86860,87080,87290,87490,87700,87900,88100,8830
1,20,88490,88690,88880,89070,89250,89440,89620,89800,89970,9015
1,30,90320,90490,90660,90820,90990,91150,91310,91470,91620,9177
1,40,91920,92070,92220,92360,92510,92650,92790,92920,93060,9319
1,50,93320,93450,93570,93700,93820,93940,94060,94180,94290,9441
1,60,94520,94630,94740,94840,94950,95050,95150,95250,95350,9545
1,70,95540,95640,95730,95820,95910,95990,96080,96160,96250,9633
1,80,96410,96490,96560,96640,96710,96780,96860,96930,96990,9706
1,90,97130,97190,97260,97320,97380,97440,97500,97560,97610,9767
2,00,97720,97780,97830,97880,97930,97980,98030,98080,98120,9817
2,10,98210,98260,98300,98340,98380,98420,98460,98500,98540,9857
2,20,98610,98640,98680,98710,98750,98780,98810,98840,98870,9890
2,30,98930,98960,98980,99010,99040,99060,99090,99110,99130,9916
2,40,99180,99200,99220,99250,99270,99290,99310,99320,99340,9936
2,50,99380,99400,99410,99430,99450,99460,99480,99490,99510,9952
2,60,99530,99550,99560,99570,99590,99600,99610,99620,99630,9964
2,70,99650,99660,99670,99680,99690,99700,99710,99720,99730,9974
2,80,99740,99750,99760,99770,99770,99780,99790,99790,99800,9981
2,90,99810,99820,99820,99830,99840,99840,99850,99850,99860,9986
3,00,99870,99870,99870,99880,99880,99890,99890,99890,99900,9990
3,10,99900,99910,99910,99910,99920,99920,99920,99920,99930,9993
3,20,99930,99930,99940,99940,99940,99940,99940,99950,99950,9995
3,30,99950,99950,99950,99960,99960,99960,99960,99960,99960,9997
3,40,99970,99970,99970,99970,99970,99970,99970,99970,99970,9998

t de Student: valor que deja el área α a la derecha

En un intervalo o una prueba bilateral se busca α/2: para el 95 %, la columna 0,025.

gl0,10,050,0250,010,0050,001
13,0786,31412,70631,82163,657318,309
21,8862,9204,3036,9659,92522,327
31,6382,3533,1824,5415,84110,215
41,5332,1322,7763,7474,6047,173
51,4762,0152,5713,3654,0325,893
61,4401,9432,4473,1433,7075,208
71,4151,8952,3652,9983,4994,785
81,3971,8602,3062,8963,3554,501
91,3831,8332,2622,8213,2504,297
101,3721,8122,2282,7643,1694,144
111,3631,7962,2012,7183,1064,025
121,3561,7822,1792,6813,0553,930
131,3501,7712,1602,6503,0123,852
141,3451,7612,1452,6242,9773,787
151,3411,7532,1312,6022,9473,733
161,3371,7462,1202,5832,9213,686
171,3331,7402,1102,5672,8983,646
181,3301,7342,1012,5522,8783,610
191,3281,7292,0932,5392,8613,579
201,3251,7252,0862,5282,8453,552
211,3231,7212,0802,5182,8313,527
221,3211,7172,0742,5082,8193,505
231,3191,7142,0692,5002,8073,485
241,3181,7112,0642,4922,7973,467
251,3161,7082,0602,4852,7873,450
261,3151,7062,0562,4792,7793,435
271,3141,7032,0522,4732,7713,421
281,3131,7012,0482,4672,7633,408
291,3111,6992,0452,4622,7563,396
301,3101,6972,0422,4572,7503,385
401,3031,6842,0212,4232,7043,307
501,2991,6762,0092,4032,6783,261
601,2961,6712,0002,3902,6603,232
801,2921,6641,9902,3742,6393,195
1001,2901,6601,9842,3642,6263,174
1201,2891,6581,9802,3582,6173,160
∞1,2821,6451,9602,3262,5763,090

Chi-cuadrado: valor que deja el área α a la derecha

Para la prueba de independencia, gl = (filas − 1)(columnas − 1) y α = 0,05 suele ser la columna que se usa.

gl0,9950,990,9750,950,90,10,050,0250,010,005
10,0000,0000,0010,0040,0162,7063,8415,0246,6357,879
20,0100,0200,0510,1030,2114,6055,9917,3789,21010,597
30,0720,1150,2160,3520,5846,2517,8159,34811,34512,838
40,2070,2970,4840,7111,0647,7799,48811,14313,27714,860
50,4120,5540,8311,1451,6109,23611,07012,83315,08616,750
60,6760,8721,2371,6352,20410,64512,59214,44916,81218,548
70,9891,2391,6902,1672,83312,01714,06716,01318,47520,278
81,3441,6462,1802,7333,49013,36215,50717,53520,09021,955
91,7352,0882,7003,3254,16814,68416,91919,02321,66623,589
102,1562,5583,2473,9404,86515,98718,30720,48323,20925,188
112,6033,0533,8164,5755,57817,27519,67521,92024,72526,757
123,0743,5714,4045,2266,30418,54921,02623,33726,21728,300
133,5654,1075,0095,8927,04219,81222,36224,73627,68829,819
144,0754,6605,6296,5717,79021,06423,68526,11929,14131,319
154,6015,2296,2627,2618,54722,30724,99627,48830,57832,801
165,1425,8126,9087,9629,31223,54226,29628,84532,00034,267
175,6976,4087,5648,67210,08524,76927,58730,19133,40935,718
186,2657,0158,2319,39010,86525,98928,86931,52634,80537,156
196,8447,6338,90710,11711,65127,20430,14432,85236,19138,582
207,4348,2609,59110,85112,44328,41231,41034,17037,56639,997
218,0348,89710,28311,59113,24029,61532,67135,47938,93241,401
228,6439,54210,98212,33814,04130,81333,92436,78140,28942,796
239,26010,19611,68913,09114,84832,00735,17238,07641,63844,181
249,88610,85612,40113,84815,65933,19636,41539,36442,98045,559
2510,52011,52413,12014,61116,47334,38237,65240,64644,31446,928
2611,16012,19813,84415,37917,29235,56338,88541,92345,64248,290
2711,80812,87914,57316,15118,11436,74140,11343,19546,96349,645
2812,46113,56515,30816,92818,93937,91641,33744,46148,27850,993
2913,12114,25616,04717,70819,76839,08742,55745,72249,58852,336
3013,78714,95316,79118,49320,59940,25643,77346,97950,89253,672
4020,70722,16424,43326,50929,05151,80555,75859,34263,69166,766
5027,99129,70732,35734,76437,68963,16767,50571,42076,15479,490
6035,53437,48540,48243,18846,45974,39779,08283,29888,37991,952
8051,17253,54057,15360,39164,27896,578101,879106,629112,329116,321
10067,32870,06574,22277,92982,358118,498124,342129,561135,807140,169

F de Fisher-Snedecor: valor que deja 0,05 a la derecha

Columna: gl del numerador (k − 1 en el ANOVA); fila: gl del denominador (N − k).

gl₂ \ gl₁123456789101215203060
1161,45199,50215,71224,58230,16233,99236,77238,88240,54241,88243,91245,95248,01250,10252,20
218,5119,0019,1619,2519,3019,3319,3519,3719,3819,4019,4119,4319,4519,4619,48
310,139,559,289,129,018,948,898,858,818,798,748,708,668,628,57
47,716,946,596,396,266,166,096,046,005,965,915,865,805,755,69
56,615,795,415,195,054,954,884,824,774,744,684,624,564,504,43
65,995,144,764,534,394,284,214,154,104,064,003,943,873,813,74
75,594,744,354,123,973,873,793,733,683,643,573,513,443,383,30
85,324,464,073,843,693,583,503,443,393,353,283,223,153,083,01
95,124,263,863,633,483,373,293,233,183,143,073,012,942,862,79
104,964,103,713,483,333,223,143,073,022,982,912,852,772,702,62
114,843,983,593,363,203,093,012,952,902,852,792,722,652,572,49
124,753,893,493,263,113,002,912,852,802,752,692,622,542,472,38
134,673,813,413,183,032,922,832,772,712,672,602,532,462,382,30
144,603,743,343,112,962,852,762,702,652,602,532,462,392,312,22
154,543,683,293,062,902,792,712,642,592,542,482,402,332,252,16
164,493,633,243,012,852,742,662,592,542,492,422,352,282,192,11
174,453,593,202,962,812,702,612,552,492,452,382,312,232,152,06
184,413,553,162,932,772,662,582,512,462,412,342,272,192,112,02
194,383,523,132,902,742,632,542,482,422,382,312,232,162,071,98
204,353,493,102,872,712,602,512,452,392,352,282,202,122,041,95
214,323,473,072,842,682,572,492,422,372,322,252,182,102,011,92
224,303,443,052,822,662,552,462,402,342,302,232,152,071,981,89
234,283,423,032,802,642,532,442,372,322,272,202,132,051,961,86
244,263,403,012,782,622,512,422,362,302,252,182,112,031,941,84
254,243,392,992,762,602,492,402,342,282,242,162,092,011,921,82
264,233,372,982,742,592,472,392,322,272,222,152,071,991,901,80
274,213,352,962,732,572,462,372,312,252,202,132,061,971,881,79
284,203,342,952,712,562,452,362,292,242,192,122,041,961,871,77
294,183,332,932,702,552,432,352,282,222,182,102,031,941,851,75
304,173,322,922,692,532,422,332,272,212,162,092,011,931,841,74
404,083,232,842,612,452,342,252,182,122,082,001,921,841,741,64
604,003,152,762,532,372,252,172,102,041,991,921,841,751,651,53
1203,923,072,682,452,292,182,092,021,961,911,831,751,661,551,43
∞3,843,002,602,372,212,102,011,941,881,831,751,671,571,461,32

Qué tiene y qué no

Las fórmulas siguen el orden del programa de 14 semanas: descriptiva, probabilidad, variables aleatorias, muestreo, estimación, pruebas, ANOVA y regresión. Es lo que piden los simulacros de parcial y el recuperatorio integrador, y nada más: una hoja de fórmulas que lo trae todo es difícil de usar con el reloj corriendo.

La F está al 5 %, que es el nivel que se usa en casi todos los ejercicios. Para otros niveles, y para las tablas binomial y de Poisson, están las tablas interactivas: F, binomial y Poisson.

De dónde salen los números

Las tablas se calculan con el mismo motor que las calculadoras del sitio, y antes de publicar se comparan valor por valor contra SciPy con el redondeo impreso. El mismo control revisa que el PDF traiga las tablas de hoy: si una cambia y el PDF no se regenera, la verificación falla.