Normalidad
junio 29, 2022
1 Introducción
Autor: Álvaro Alonso Fernández
Departamento de Ciencias de la Vida
Universidad de Alcalá (España)
Vamos a ver las diferentes formas que tenemos de testar si nuestros datos siguen o no una distribución normal. Hay numerosos test estadísticos que requiere normalidad de los datos y/o residuos.
2 Nuestros datos
Vamos a generra dos grupos de datos, uno de ellos serán números al azar de media y desviación estandard conocidas. El otro grupo serán datos con varios valores de cero que a priori no seguirán una distribución normal.
Veamos como son nuestros datos:
datosnormales<-rnorm(n = 1000,mean = 10, sd = 0.45)
datosnormales## [1] 10.295970 9.857355 9.305293 10.003133 9.303017 9.987121 10.321200
## [8] 9.872394 9.581399 10.034851 9.701169 9.962355 10.415475 9.797745
## [15] 10.375105 9.270550 10.409775 10.096041 9.969673 9.920380 10.419105
## [22] 10.158465 8.608593 10.431189 9.865389 10.015981 10.246173 9.837168
## [29] 10.995062 9.848294 10.176708 10.506287 10.311376 10.268122 10.133686
## [36] 9.747118 10.132191 10.153679 10.440467 11.065574 9.438836 10.165979
## [43] 9.907778 9.837899 9.644271 10.697057 10.682714 9.626395 9.950155
## [50] 10.186587 9.777230 9.750037 9.652679 9.487665 10.532319 10.210768
## [57] 10.212094 9.756629 10.062561 8.692654 9.673974 9.797770 10.109608
## [64] 10.038591 10.220059 10.507620 10.208967 9.300541 10.742831 9.381485
## [71] 10.092517 10.070510 9.122612 9.996679 9.536572 10.718169 10.092616
## [78] 9.443468 10.214696 10.533108 10.256825 9.935128 9.916472 10.426813
## [85] 9.564848 10.903202 9.913923 9.806935 10.711564 10.141823 10.107921
## [92] 9.485718 10.423500 10.323770 9.927695 9.415172 9.263486 10.101753
## [99] 10.523276 10.463575 10.300050 9.976786 10.682375 10.792971 9.895100
## [106] 10.210166 9.873166 9.620620 10.337671 10.203005 10.007168 9.891444
## [113] 10.642969 10.191641 10.356198 9.737222 10.390573 9.539454 9.769405
## [120] 10.559895 9.667627 10.059918 9.430955 9.745498 9.988461 9.942344
## [127] 10.175206 10.245493 10.933308 9.949420 9.656259 9.671953 10.072705
## [134] 10.383532 9.461047 9.668309 10.484668 9.848760 9.999440 9.517732
## [141] 10.447267 10.051700 10.003722 9.858767 10.107870 9.957337 10.109601
## [148] 9.455056 10.056186 10.318176 8.922561 11.152661 9.730648 9.325502
## [155] 9.805358 9.556084 10.304515 9.931879 9.520793 10.079929 10.825773
## [162] 9.783405 9.884875 10.122533 9.509176 10.076427 10.119417 9.776270
## [169] 9.466085 9.716567 9.397381 10.318243 9.074843 10.594745 10.444971
## [176] 9.564289 9.567693 9.775262 10.130403 10.337650 10.227448 9.655474
## [183] 10.339275 9.688805 9.885440 9.882612 9.255772 9.847249 9.988596
## [190] 10.096406 10.028312 9.212903 8.980891 10.471864 10.180920 10.364474
## [197] 10.276336 9.677986 10.025124 9.943791 10.431932 9.981811 9.236954
## [204] 10.420740 9.939102 10.399941 10.912889 9.111831 9.780320 9.985498
## [211] 10.017325 10.259489 9.955251 9.608389 10.514210 10.278564 10.404693
## [218] 9.958507 9.406287 9.935601 9.805778 10.539989 10.213461 10.548168
## [225] 10.742541 9.801501 10.248131 10.264984 9.814534 10.236626 9.735225
## [232] 8.969167 9.853249 10.128426 9.906534 9.927899 10.347502 9.768537
## [239] 9.961830 10.600423 9.938011 10.061376 10.087920 10.352575 10.461845
## [246] 9.942041 9.876297 10.003764 10.482385 9.723516 10.052272 9.338851
## [253] 10.330485 10.645809 9.363983 10.091685 9.937979 9.958564 10.055835
## [260] 9.580965 9.504669 9.951634 9.840051 9.567930 9.712488 9.654561
## [267] 10.121350 10.510418 9.999870 10.595426 10.384534 9.740472 9.715164
## [274] 9.677618 9.489291 8.577305 10.479931 9.907328 9.567009 9.912652
## [281] 9.546296 9.894230 9.989347 10.621439 10.054636 9.795087 10.467468
## [288] 10.389747 9.197952 9.663585 9.683270 9.947635 9.852849 9.564726
## [295] 8.851671 10.063275 9.946704 9.954737 9.823010 10.282010 10.020676
## [302] 9.514749 9.735082 9.990652 9.642707 10.665061 10.142606 10.123541
## [309] 9.224338 10.491445 10.379655 9.407202 9.478714 9.715384 10.037258
## [316] 8.796165 10.208959 9.620643 10.325144 10.320247 9.343811 10.302741
## [323] 9.738165 10.227669 10.840220 10.707849 10.103530 9.214197 10.173779
## [330] 9.883267 10.341659 10.518802 10.026276 9.990473 9.577667 9.836094
## [337] 10.282094 10.287319 10.186915 9.530881 10.434100 10.017583 10.573980
## [344] 10.205985 9.187321 9.388499 10.315855 9.782393 10.132577 10.640031
## [351] 10.456968 9.833849 9.725900 9.697466 9.564564 9.603984 10.599616
## [358] 10.190624 9.734998 11.094520 9.639810 10.738478 10.235786 9.253356
## [365] 9.498206 9.812388 9.366107 10.110357 9.566071 10.158435 10.240464
## [372] 9.800755 9.743201 10.027303 10.759976 9.985888 10.350340 10.291646
## [379] 9.430814 9.666790 9.456786 10.322756 9.638986 10.221919 9.715305
## [386] 11.095191 10.064295 10.115279 9.880905 9.796842 9.437747 10.865980
## [393] 9.725081 9.477940 9.469790 9.893579 9.432909 9.101299 10.302165
## [400] 9.569367 9.998979 9.487568 10.114035 9.611890 9.809704 10.579440
## [407] 9.696046 9.583319 10.387165 10.254155 10.872937 10.630639 10.007604
## [414] 10.459027 10.348528 10.166189 9.695022 9.955628 10.206817 9.551516
## [421] 10.681777 9.964483 10.120603 10.210178 10.423040 9.787264 10.747518
## [428] 10.093798 10.965782 9.350290 9.988722 9.015030 9.713009 10.027716
## [435] 10.102947 10.237092 10.244259 10.054075 9.898321 10.541675 8.937049
## [442] 9.950337 9.909238 10.368931 9.599722 9.464356 9.495818 10.043871
## [449] 10.259325 10.379811 10.817607 9.954128 9.784987 9.707893 9.201667
## [456] 9.951156 9.973741 9.927256 10.185730 9.427241 9.838032 11.332160
## [463] 10.033840 10.805221 10.287939 9.508922 10.406081 10.250663 9.998678
## [470] 9.982526 10.002256 11.061163 10.017924 10.157948 10.004731 9.165676
## [477] 10.320044 9.536735 9.511056 10.045446 9.788864 10.020231 9.443739
## [484] 9.602704 9.997087 10.060122 9.944139 9.417552 9.708753 9.960256
## [491] 10.591829 10.077249 10.156948 10.159597 10.161251 10.277795 9.624092
## [498] 10.091397 10.353151 9.667407 9.306820 10.086579 10.348480 9.840592
## [505] 9.674985 10.152364 9.652145 10.755869 9.777134 10.525716 10.236027
## [512] 9.938890 9.747996 10.090559 9.628084 9.657749 9.953885 10.409449
## [519] 9.663168 9.712370 10.041023 10.129800 9.850802 9.734019 10.584264
## [526] 8.837702 10.778926 10.063419 10.017875 10.272532 9.701961 9.829022
## [533] 9.993292 10.281820 9.584624 10.199924 9.417810 9.621313 10.225420
## [540] 10.054846 9.567971 10.048182 10.103944 9.723580 10.331991 9.642069
## [547] 9.861518 10.296201 10.025641 9.889409 9.943667 10.067682 10.559696
## [554] 10.326648 9.570321 9.211223 10.336672 10.960467 9.925100 10.180412
## [561] 10.434123 9.774761 9.227018 10.186902 10.635704 10.775762 10.138165
## [568] 9.273754 9.500666 9.782465 10.005453 9.776875 9.653372 10.695756
## [575] 10.202353 10.138337 10.175356 10.078623 10.130517 9.881256 9.987122
## [582] 9.892632 10.687759 10.530386 9.611534 9.635480 10.243151 9.828482
## [589] 10.298396 10.886623 10.466097 11.132117 10.418287 10.411582 9.422631
## [596] 10.200257 10.597220 9.514464 10.577443 9.677608 9.455992 9.870002
## [603] 9.405518 10.562593 10.416374 9.055468 10.116629 9.827472 10.289385
## [610] 9.243645 10.559902 8.967875 10.820310 10.278770 10.498279 10.030814
## [617] 10.317188 9.695227 10.394890 9.459158 10.243747 9.533522 10.118887
## [624] 10.892009 9.833027 9.040114 9.577267 9.828100 9.350231 10.192374
## [631] 9.859068 10.337901 10.494569 10.106572 9.731220 9.331566 9.510277
## [638] 10.176712 9.960764 9.122882 8.973299 9.785138 10.444078 10.147693
## [645] 9.847185 10.086111 10.588416 9.738229 10.248896 9.114623 10.448664
## [652] 10.311640 10.254722 9.343744 10.636180 10.450836 9.302734 10.003971
## [659] 9.873368 9.544969 10.050889 10.379675 9.438349 9.873107 9.411140
## [666] 10.343464 9.437073 9.966120 9.522694 10.386290 9.354050 10.534823
## [673] 9.857096 10.359522 9.243007 9.811722 9.130241 10.460111 10.276145
## [680] 10.124337 10.001964 9.380084 9.469986 9.865344 9.619132 9.422014
## [687] 10.007831 9.952706 9.731173 10.448745 10.725873 9.944416 10.647444
## [694] 10.201955 9.268187 10.171743 10.217291 10.837840 9.961439 10.236717
## [701] 10.296101 10.116118 10.329982 9.620240 10.156854 9.136740 10.054347
## [708] 10.560331 10.037259 9.630092 9.564982 10.322245 10.075911 11.111429
## [715] 10.246084 9.991461 9.997417 10.142891 9.835605 9.627983 9.374557
## [722] 9.931534 10.960582 9.614465 9.773030 9.968867 10.441718 9.902192
## [729] 9.291278 9.600500 10.034021 9.721373 10.525086 9.799498 10.069600
## [736] 10.713356 9.978633 9.659009 10.254985 9.595911 9.669755 9.766883
## [743] 9.945255 9.429791 10.384565 9.850867 10.624132 10.013841 9.366636
## [750] 9.572600 9.591709 10.059800 9.473778 10.572032 9.735210 10.024336
## [757] 9.962900 10.131422 10.626712 9.405810 10.489062 10.244921 9.391149
## [764] 9.936984 10.330909 10.689406 10.037857 10.118388 10.066776 9.859709
## [771] 9.913571 10.374148 10.083970 10.284114 9.728952 10.632234 9.846134
## [778] 9.671189 10.061165 10.785500 10.411027 9.966941 9.802870 9.902804
## [785] 9.893306 9.000205 9.050254 10.461292 9.585577 10.110582 10.926478
## [792] 10.530368 10.804091 10.458023 9.814752 9.269898 9.947572 9.964567
## [799] 11.063837 10.263230 10.167804 10.636880 10.375533 9.848782 9.474932
## [806] 9.806430 9.389937 9.681224 9.318450 9.298076 10.134966 10.124458
## [813] 10.242328 10.260726 9.549087 9.770425 9.541505 10.102897 10.018735
## [820] 9.732896 9.958749 9.342858 10.056589 9.777097 10.056545 10.133279
## [827] 10.248873 10.045156 10.362736 9.768067 10.058790 9.589299 10.252441
## [834] 10.213903 9.241155 10.733217 10.003518 9.821242 9.133716 10.060020
## [841] 10.346165 10.799885 10.358924 10.081747 9.902157 10.198043 10.213784
## [848] 9.563337 10.538672 10.113436 10.104174 9.278882 9.689426 10.335418
## [855] 10.091912 10.041327 9.894791 10.236184 9.748715 10.060174 9.772618
## [862] 9.871741 9.717028 10.024791 10.357168 10.088604 9.358618 9.562517
## [869] 9.601357 9.701349 9.932093 9.647036 10.360519 9.785206 9.821152
## [876] 10.071332 9.883470 9.746726 9.870270 9.680866 9.951949 10.256799
## [883] 10.448208 9.266913 9.965448 9.578090 9.779663 9.639246 9.715962
## [890] 9.331974 10.450582 10.520897 9.938312 9.915413 9.542871 10.263273
## [897] 9.647598 10.049000 9.636811 10.306137 9.678776 9.649678 10.064434
## [904] 10.361846 10.023157 10.021899 9.064940 9.800671 8.211926 10.024990
## [911] 9.990627 10.351999 9.822835 10.527189 9.816477 9.960885 9.703395
## [918] 10.009696 9.784520 9.805385 10.477173 9.712042 9.943959 10.700312
## [925] 10.847032 10.336346 10.084460 9.924626 10.358516 9.430717 9.656481
## [932] 10.052809 9.810107 9.218029 9.450336 10.357417 10.262779 9.697992
## [939] 9.797863 9.883414 10.314215 9.956986 10.399771 10.049326 10.352303
## [946] 10.361027 10.469547 9.233239 10.252597 10.742438 10.102765 9.586177
## [953] 9.506937 9.233165 9.795782 10.102186 10.420766 10.271001 9.060721
## [960] 10.128282 9.778179 9.914348 9.834580 8.700205 9.202625 9.838799
## [967] 9.181740 9.671105 9.611681 10.308983 9.711288 9.748158 10.396597
## [974] 10.358268 10.266324 9.527049 10.444113 10.097695 10.878456 10.146968
## [981] 10.570073 10.376159 10.700952 10.072137 10.928403 10.588540 10.951718
## [988] 9.159699 9.422561 10.009404 9.297022 9.804224 9.371126 10.124032
## [995] 9.230329 10.437440 9.309738 10.188080 9.946944 10.374032datosraros<-c(1,3,4,0,1,1,0,1,0,2,1,1,1,1,0,0,0,0,0,0,0,0,0,12,14,0,0,0,1,1,1,60,67,56,65,65,65,65)
datosraros## [1] 1 3 4 0 1 1 0 1 0 2 1 1 1 1 0 0 0 0 0 0 0 0 0 12 14
## [26] 0 0 0 1 1 1 60 67 56 65 65 65 653 Representación gráfica con histograma de frecuencias
El histograma de frecuencias nos permite ver la distribución de nuestros valores, ya las figuras nos dan mucha información sobre la distribución de nuestros datos:
par(mfrow=c(1,2))
hist(datosnormales)
hist(datosraros)
Este resultado sería una primera forma de comprobar la normalidad, uno de los histogramas muestra claramente la normalidad de los datos y el otro no.
4 Representación gráfica con qqplot
La línea central de este gráfico nos muestra la normalidad, los puntos son nuestros datos, cuanto más cerca de la línea mejor:
qqnorm(datosnormales)
qqline(datosnormales)
qqnorm(datosraros)
qqline(datosraros)
El qqplot sería otra forma de comprobar la normalidad, uno de los gráficos muestra claramente la normalidad de los datos y el otro no.
5 Test estadístico: Shapiro-Wilk
Hagamos un test estadístico para comprobar la normalidad. Aplicaremos el test de Shapiro-Wilk:
shapiro.test(datosnormales)##
## Shapiro-Wilk normality test
##
## data: datosnormales
## W = 0.99827, p-value = 0.4121shapiro.test(datosraros)##
## Shapiro-Wilk normality test
##
## data: datosraros
## W = 0.54313, p-value = 1.062e-09La p no significativa indica que nuestros datos son normales, cuando la p es significativa los datos no siguen una distribución normal. En nuestro caso se confirma lo que habíamos observado en los dos tipos de gráficos anteriores.
6 CRÉDITOS
Departamento de Ciencias de la Vida
Universidad de Alcalá (España)
