Find distribution and transform to normal distribution

I have data that describes how often an event takes place during one hour (“number per hour”, nph) and how long the events last (“duration in seconds per hour”, dph).

This is the original data:

nph <- c(2.50000000003638, 3.78947368414551, 1.51456310682008, 5.84686774940732, 4.58823529414907, 5.59999999993481, 5.06666666666667, 11.6470588233699, 1.99999999998209, NA, 4.46153846149851, 18, 1.05882352939726, 9.21739130425452, 27.8399999994814, 15.3750000002237, NA, 6.00000000004109, 9.71428571436649, 12.4848484848485, 16.5034965037115, 20.6666666666667, 3.49999999997453, 4.65882352938624, 4.74999999996544, 3.99999999994522, 2.8, 14.2285714286188, 11.0000000000915, NA, 2.66666666666667, 3.76470588230138, 4.70588235287673, 13.2727272728677, 2.0000000000137, 18.4444444444444, 17.5555555555556, 14.2222222222222, 2.00000000001663, 4, 8.46153846146269, 19.2000000001788, 13.9024390245481, 13, 3.00000000004366, NA, 7.36000000006855, 1.61137440758472, 1.50000000000873, 3.36585365857481, 22.3750000003256, 10.8387096775008, 2.92307692305075, 3.48837209304214, 5.17647058827074, 37.6666666666667, 1.17647058824335, 7.45454545462435, 36.2352941171508, 6.82352941167125, 2.22222222222222, 6.13333333333333, 11.4285714286665, 42.7058823523563, 28.1052631584975, 18.3333333333333, 1.24999999999091, 5.1034482758211, 1.82857142855926, 1.30693069306629, 3.22222222222222, 17.2800000001609, 10.5714285715165, 7.81818181826456, 3.14285714288328, 4.05194805197256, 3.6, 23.0909090904203, 0.249999999998181, 10, 27.3043478258106, 2.49999999998181, 2.00000000001663, 9.14285714293317, 4.74999999996544, 29.3999999996577, 16.9999999998021, 15.7777777777778, 1.74999999998727, 3.46666666666667, 2.45161290324422, 2.05231388331614, 2.60000000001513, 15.4054054053569, 4, 12.2222222222222, 2.46153846151642, 8.15384615399219, 2.23529411761644, 15.1111111111111, 0.23529411764867, 10.5454545455661, 17.5714285715747, 2.3030303030303, 1.37931034481651, 8.32000000007749, 5.1578947368105, 24.1999999997183, 15.4782608694085, 21.8749999998408, 2.74999999997999, 9.91304347823578, 3.86206896548623, 1.16959064328441, 2.84210526319272, 12.857142856929, 4, 3.69230769227463, 2, NA, 1.88888888888889, 15.4285714283148, 0.222222222222222, 6.16666666666667, 13.1034482757569, 3.19999999996275, 4.87499999996453, 2.88000000002682, 5.12499999996271, 26.6666666666667, 9.75000000014188, 17.2048192770602, 1.99999999998545, 1.65517241377981, 3.16666666666667, 2.23529411766237, 6.82352941181143, 2.74999999991996, 2.99999999997817, 11.4929577463281, 1.59999999998137, 8.65116279074452, 5.69230769240964, 13.7777777777778, 0.222222222222222, 10.6000000002468, 13.91304347812, 2.75862068963302, NA, 4.26666666666667, 5.64705882356808, 2.74999999997999, 15.047619047619, 16.6666666666667, 1.49999999998909, 4.62499999996635, 5.71428571428571, 1.83206106868927, 2.44444444444444, 2.4, 3.9999999999709, 2.33333333333333, 3.20000000007451, 5.931034482711, 7.14285714273835, 14.7272727274286, 0.352941176465754, 8.40000000019558, 10.1250000001473, 2.66666666666667, NA, 2.66666666666667, 4.7058823529734, 4.83333333333333, 9.31034482751146, 24.5882352937809, 2.13333333333333, 10.1739130434525, 5.56521739124801, 2.12658227848728, 1.88888888888889, 5.80000000013504, 7.14285714291654, 1.71428571429997, 1.99999999994179, NA, 5.00000000007276, NA, 0.129032258062578, 8.22222222222222, 7.16666666666667, 4.13793103444954, 2.82352941178404, 3.07692307697818, 4.00000000004902, 4.74999999986176, 9.75000000014188, 20.1333333333333, 2.66666666666667, 6.78947368416893, 1.46666666666667, 1.73195876289076, 4.76923076931619, 2.88888888888889, 7.4285714286332, 5.2, 3.384615384676, 4.7727272727399, 6.59999999992317, 11.4545454546667, 1.41176470586302, 11.1999999998696, 6.08000000005662, 4, 4.71428571432492, 5.00000000004158, 6.8, 6.83870967747072, 14.2500000002074, 5.49999999983993, 2.4, 4.71910112354612, 4, 1.72185430463842, 2.44444444444444, 4.30769230776946, 6.30769230780528, 3.53846153852491, 4.35294117641097, NA, 5.99999999990022, NA, NA, 7.42857142857143, 10.1333333333333, 6.79999999992084, 5.54838709681587, 1.83333333333333, 7.06666666666667, 2.9090909091217, 10.8000000001006, NA, 2.13333333333333, NA, 5.09090909090909, 4.21052631570563, 4.00000000003326, 4.28571428571429, 4.28571428574992, 2.49999999998181, 2.76923076928037, 4.99999999985448, 3.87500000005639, NA, NA, 12.2105263159391, 5.44444444444444, 2.6249999999809, 3.74193548389907, 3.28571428574161, 4.88888888888889, 9.33333333333333, 4.21621621620295, NA, 0.8, 4.5306122448549, 4.14285714289159, 3.1137724550985, 0.266666666666667, 5.27272727261567, 1.84615384613731, 8.36363636372488, 2.42857142853104, NA, 2.42857142853104, 8.28571428578318, 1.64705882350685, 8.2, 6.88888888888889, 1.74999999998727, 7.6, 3.33333333333333, 6.24999999995453, 9.56521739120752, 4.93333333333333, 16.4, 2.53333333333333, 7.2, 1.33333333333333, 3.3962264151018, 2, 9.38461538453135, 1.57142857144164, 3.45454545458201, 5.37499999996089, 7.74193548375467, 3.38461538458508, 7, NA, 4.54545454545455, 14.5, 1.93939393939394, 4.33333333333333, 4, 6.58823529402741, 2.90909090902933, 3.32530120480995, 25.6666666666667, 2, 6.54545454545455, 4.4, 3.54378818739119, 1.62499999998818, 4.22222222222222, 2.53333333333333, 14.6666666666667, 2.96296296296296, NA, 3.00000000004366, 16.1999999998114, 1.55555555555556, 3.11111111111111, NA, 4.8, 3.99999999997339, 4, 6.37499999995362, 2.7999999999674, NA, 32.8, 2.49999999998181, 11.0561797754255, NA, 2.75229357793903, 1.7142857142572, 7.66666666666667, 7.28571428577487, 2.36363636358633, 2.14285714287496, 6.27272727274387, 3.62499999997362, 19.6666666666667, 1.71428571427431, 6.60869565210701, 5.57894736838687, 5.84615384610149, 3.03030303030303, 1.33333333333333, 4.87499999996453, 4.71428571432492, 4.74418604653732, 13.0588235292329, 3.12500000004547, NA, 3.37500000004911, 2.41525423729648, 2.37499999998272, 4.54545454550265, 6.28571428576655, 2.55555555555556, 3.17647058819179, 5.59999999993481, 5.85714285719156, 7.42857142844789, NA, 4.83333333333333, 5.33333333333333, 4.48484848484848, 2.93333333333333, 3.83333333333333, 5.52941176474375, 9.33333333333333, 5.16666666666667, 18, 2.82352941178404, 5.54838709681587, 3.55555555555556, 1.25237191650965, 2, 2.16666666666667, 7.16666666666667, 3.00000000002495, 2.83333333333333, 2.48275862068966, 4.42857142860825, 11.1428571426718, NA, 5.52380952380952, 34.3448275859312, 4.75000000006912, 3.26315789471685, 10.2857142857998, 10.5555555555556, 5.00000000004158, 19.0843373493441, 20.6153846152, 2.24999999998363, 8.59259259259259, 4.25806451616101, 2.85714285716014, 5.1578947368105, 8.66666666666667, 3.14285714280487, 6.30769230763582, 6.79999999992084, 8.07692307663376, 5.73333333333333, 8.46153846146269, 2.34482758618807, 4.31999999991953, 4.57142857135254, 2.87500000004184, 2.28571428567627, 0.857142857149985, 10.2352941175069, 3.26086956520914, NA, 13.3333333333333, 2.75000000004002, 6.45161290312889, 3.61290322575218, 1.48854961831995, 3.37499999997544, 4.0540540540413, 5.73333333333333, 3.85714285707871, 3, 6.31578947364551, 1.55555555555556, 7.84615384608358, 0.4, 7.66666666666667, NA, 7.85185185185185, 2.59090909091595, 7.28571428577487, 5.74999999995816, 3.28571428574161, 16.043478260829, 15.8000000003679, 2.50000000003638, NA, 2.06451612904776, 1.82163187855948, 0.874999999993634, 13.2000000001229, 6.92307692301493, 3.7142857143166, 3.00000000001343, 5.83333333333333, 3.86666666666667, 9.39999999989057, 2.49999999998181, 6.24000000005811, 4.58823529414907, 3.72413793109428, 3.21428571427235, 6.85714285719988, 8.42857142864151, 5.23076923086291, 10.5454545455661, 14.1428571429747, 4.00000000005821, 4.08791208795393, 8.47058823517811, 3.94422310755509, 3.62500000005275, 6.0000000001397, 1.33333333333333, 3.73333333333333, 6.31578947352942, NA, 4.53333333333333, 8.46153846169001, 0.470588235287673, 2.28571428571429, 22.7142857144746, 8.00000000012846, 2.8108108108285, 4.57142857146658, 5.87500000008549, 6.42857142862488, 19.2258064513241, 13.4666666666667, 3.46666666666667, 4.90322580648844, 3.51515151515152, 1.56862745098755, 1.53846153844776, 3.63636363636364, 4.71428571432492, 3.06666666666667, 4.61538461546728, NA, 2.83333333333333, 5.53846153841194, 1.80645161287609, 9.14285714285714, 2.42857142853104, 3.2, 5.00000000007276, 4.42857142860825, 6.12500000008913, 3.24999999990541, 4.16326530608288, 14.6666666666667, 5.37499999996089, 7.43478260867684, 9.93548387104236, 3.73205741626378, 2.24999999998363, 13.7777777777778, 4.74074074074074, 7.4285714286332, 3.61904761904762, 7.13513513511269, 5.28571428575824, 5, 2.5882352940822, 11.5000000001673, 27.1249999998026, 2.875, 2.81081081077544, 9.42857142864983, 7.05882352931509, 3.83333333333333, 16.8695652172205, 16.7692307690806, 10.1333333333333, 5.45454545455989, 7.8750000001146, 1.6883116883219, 2.66666666666667, 11.7857142856653, 3.33333333333333, 6.33333333333333, 7.39999999991385, 12.5882352942039, 4.00000000003326, 6.72727272734392, 3.03030303030303, 6, 30.6666666666667, 3.74999999997272, 3.00000000003011, 8.00000000006652, 8.00000000006009, 2.57142857144995, 10.695652173886, 14.2666666666667, 7.75000000011278, 2.51162790697674, 6.33333333333333, 3.28125000004775, 1.88888888888889, 10.4000000002421, 4.87499999996453, 13.7142857143998, 8.5, NA, 4.87499999996453, 8.181818181645, 1.24999999999091, 4.38095238095238, 27.1764705878631, 2.37499999998272, 2.94117647060838, 11.7142857143831, 5.99999999996324, 2.37499999998272, 14.7637795275455, 14.313253012008)
dph <- c(3.12500000004547, 6.69473684199041, 4.3106796117187, 11.6937354988146, 103.882352941888, 10.9999999998719, 7.33333333333333, 20.3529411761918, 5.23076923072239, NA, 4.61538461534328, 47.5555555555556, 2.94117647054795, 18.9565217389385, 44.3199999991745, 28.5000000004147, NA, 10.4705882353658, 19.000000000158, 25.8181818181818, 43.2167832173461, 51.5555555555556, 8.37499999993906, 6.91764705878563, 9.37499999993179, 5.64705882345207, 4.53333333333333, 27.4285714286627, 14.4285714286914, NA, 1.6, 5.76470588227399, 4.70588235287673, 55.2727272733122, 2.11764705883803, 30.8888888888889, 41.2222222222222, 23.4444444444444, 2.42857142859162, 6.2, 17.0769230767702, 21.2800000001982, 40.8292682931466, 14.5, 6.25000000009095, NA, 15.0400000001401, 5.68720379147547, 2.40000000001397, NA, 26.3750000003838, 18.0645161291679, 3.99999999996418, 6.13953488375417, 8.47058823535212, 128.666666666667, 2.23529411766237, 34.1818181821799, 115.999999998411, 5.99999999991782, 5.77777777777778, 10.6666666666667, 15.4285714286997, 54.8235294110138, 81.315789475428, 42.3333333333333, 1.74999999998727, 7.99999999993577, 4.34285714282825, 1.90099009900552, 5.22222222222222, 39.840000000371, 25.1428571430662, 7.81818181826456, 8.57142857149985, 15.2727272728196, 6.4, 93.0909090889387, 0.374999999997272, 23.1666666666667, 29.3913043475286, 0.874999999993634, 1.71428571429997, 13.5714285715414, 5.49999999995998, 134.799999998431, 77.7999999990943, 18, 2.24999999998363, 5.73333333333333, 3.09677419357165, 2.29376257547098, 5.70000000003318, 23.1891891891162, 14, 13.5555555555556, 1.69230769229254, 9.23076923093455, 4.35294117641097, 48.6666666666667, 0.352941176473005, 16.0000000001693, 56.7142857147573, 1.81818181818182, 1.37931034481651, 19.6800000001833, 6.63157894732779, 134.999999998428, 41.0434782604541, 26.8749999998045, 3.62499999997362, 16.5652173912624, 10.3448275861238, 1.28654970761285, 2.94736842108875, 13.4285714283481, 7.6, 3.2307692307403, 2, NA, 3.44444444444444, 93.1428571413081, 0.111111111111111, 13.6666666666667, 28.1379310342568, 2.39999999997206, 7.8749999999427, 4.00000000003725, 6.99999999994907, 60, 26.8750000003911, 30.5060240963, 3.12499999997726, 3.17241379307798, 4.83333333333333, 9.29411764712247, 12.7058823530282, 4.24999999987631, 6.99999999994907, 9.97183098578469, 2.39999999997206, 8.93023255818789, 15.3846153848909, 94, 0.111111111111111, 21.4000000004983, 29.9130434779581, 1.24137931033486, NA, 15.8666666666667, 7.17647058828444, 1.49999999998909, 37.9047619047619, 27.6666666666667, 1.74999999998727, 9.37499999993179, 17.3333333333333, 11.603053435032, 5.33333333333333, 2.8, 7.99999999994179, 3.5, 1.60000000003725, 7.31034482752751, 6.42857142846452, 56.7272727278731, 0, 21.6000000005029, 28.8750000004202, 1.6, NA, 4.5, 5.64705882356808, 7.16666666666667, 36.2068965514334, 40.235294117096, 4.8, 22.3043478260305, 8.86956521730152, 3.94936708861923, 3.33333333333333, 12.6000000002934, 20.0000000001663, 1.28571428572498, 0.749999999978172, NA, 6.25000000009095, NA, 0.258064516125156, 18.6666666666667, 17, 5.51724137926605, 2.58823529413537, 11.0769230771215, 5.26315789480134, 11.4999999996653, 34.1250000004966, 42.4, 6.53333333333333, 33.1578947366389, 4.4, 4.9484536082593, 11.2307692309704, 5.11111111111111, 23.8571428573412, 0.4, 2.30769230773364, 6.81818181819986, 8.19999999990454, 26.7272727275556, 0.352941176465754, 24.1999999997183, 7.04000000006557, 2.5, 7.14285714291654, 11.4285714286665, 12.1333333333333, 2.83870967744068, 42.7500000006221, 4.99999999985448, 3.33333333333333, 10.112359550456, 16.8, 4.23841059603303, 2.22222222222222, 14.4615384617975, 15.6923076925887, 3.23076923082709, 1.05882352939726, NA, 7.42857142844789, NA, NA, 16.952380952381, 12.4, 6.29999999992666, 85.4193548393512, 4.33333333333333, 11.8666666666667, 6.0000000000635, 19.6800000001833, NA, 3.46666666666667, NA, 13.0909090909091, 12.6315789471169, 5.14285714289991, 9.14285714285714, 12.1428571429581, 2.87499999997908, 1.692307692338, 10.2499999997017, 5.00000000007276, NA, NA, 19.578947368661, 10.4444444444444, 1.74999999998727, 4.77419354842295, 8.57142857149985, 9.66666666666667, 13.5238095238095, 7.29729729727434, NA, 1.6, 9.18367346930048, 6.85714285719988, 4.5508982036055, 0.666666666666667, 10.90909090886, 2.61538461536119, 6.1818181818836, 1.57142857140244, NA, 1.99999999996674, 24.4285714287746, 0.941176470575345, 16.6, 17.6666666666667, 0.999999999992724, 10.2666666666667, 7.5, 11.2499999999181, 11.9999999998785, 12.8, 29.7333333333333, 5.33333333333333, 13.6, 1.84615384615385, 12.7924528302168, 2.4, 23.6923076920955, 2.42857142859162, 4.90909090914286, 3.62499999997362, 11.4193548385381, 4.92307692303284, 17, NA, 16.9090909090909, 20.8333333333333, 0.96969696969697, 8, 11.8333333333333, 10.2352941175069, 5.81818181805867, 6.07228915660947, 39.3333333333333, 4.13333333333333, 9.6969696969697, 11.2, 7.94297352346302, 2.12499999998454, 4.66666666666667, 2.66666666666667, 11.3333333333333, 3.7037037037037, NA, 2.87500000004184, 24.3999999997159, 1.88888888888889, 10.4444444444444, NA, 3.73333333333333, 7.08571428566715, 15.8333333333333, 11.2499999999181, 2.59999999996973, NA, 43.6, 3.24999999997635, 22.9213483149066, NA, 5.22935779808415, 1.85714285711197, 14.3333333333333, 15.4285714286997, 4.363636363544, 1.8571428571583, 7.36363636365585, 6.37499999995362, 51.3333333333333, 3.42857142854862, 1.043478260859, 4.94736842102232, 2.76923076920597, 5.09090909090909, 2.5, 7.49999999994543, 9.71428571436649, 7.25581395352766, 29.8823529407672, 6.62500000009641, NA, 6.12500000008913, 5.59322033900236, 5.12499999996271, 5.45454545460318, 7.00000000005821, 2.44444444444444, 3.05882352936987, 16.9999999998021, 7.71428571434986, 16.8571428568625, NA, 8.83333333333333, 6.77777777777778, 2.78787878787879, 5.06666666666667, 8.83333333333333, 9.17647058829813, 14.1666666666667, 5.5, 36.6666666666667, 4.23529411767606, 7.48387096779814, 5.33333333333333, 2.73244781783923, 2.13333333333333, 2.5, 11.5, 6.42857142862488, 3, 1.79310344827586, 8.00000000006652, 24.8571428567295, NA, 6.09523809523809, 68.5517241373807, 21.2500000003092, 6.21052631575142, 19.2857142858747, 15.1111111111111, 5.5714285714749, 42.6506024095189, 42.615384615003, 4.87499999996453, 13.3333333333333, 11.8709677420246, 8.83116883122224, 6.31578947364551, 9.83333333333333, 1.99999999996674, 7.69230769223881, 4.39999999994878, 17.3076923070723, 8.13333333333333, 16.461538461391, 1.65517241377981, 7.03999999986887, 10.2857142855432, 2.12500000003092, 1.14285714283814, 1.14285714286665, 13.1764705880548, 3.7826086956426, NA, 28.1333333333333, 3.75000000005457, 8.38709677406756, 6.83870967731663, 3.20610687022758, 6.49999999995271, 6.32432432430443, 13.8666666666667, 8.42857142843125, 2.83333333333333, 13.4210526314967, 3.33333333333333, 14.1538461537194, 0.933333333333333, 15.8333333333333, NA, 8.2962962962963, 5.31818181819589, 13.5714285715414, 10.1249999999263, 6.28571428576655, 39.260869565118, 26.6000000006193, 4.00000000005821, NA, 3.74193548389907, 5.35104364326849, 0.749999999994543, 12.0000000001118, 4.30769230765373, 6.57142857148322, 6.00000000002686, 13.3333333333333, 5.33333333333333, 16.1999999998114, 1.87499999998636, 13.1200000001222, 11.0588235294875, 2.0689655172746, 5.57142857140541, 17.1428571429997, 12.8571428572498, 10.4615384617258, 27.2727272730159, 25.5714285716412, 9.25000000013461, 12.3956043957313, 20.8235294114795, 4.54183266930586, 6.25000000009095, 14.000000000326, 1.33333333333333, 8.13333333333333, 7.15789473666668, NA, 62.6666666666667, 18.0000000003224, 0.117647058821918, 6.66666666666667, 43.8571428575075, 8.55172413806835, 5.40540540543942, 7.71428571434986, 11.0000000001601, 18.2857142858663, 52.6451612895318, 26.4, 5.6, 13.1612903226795, 5.93939393939394, 2.48366013073029, 1.53846153844776, 2.36363636363636, 4.14285714289159, 1.33333333333333, 9.23076923093455, NA, 2.83333333333333, 10.9230769229791, 2.19354838706382, 18.6666666666667, 3.57142857136918, 1.6, 8.50000000012369, 9.85714285722482, 11.2500000001637, 1.74999999994907, 6.367346938715, 33, 10.8749999999209, 23.9999999999393, 23.4838709679183, 3.73205741626378, 2.74999999997999, 20.6666666666667, 4.14814814814815, 13.2857142858248, 4.57142857142857, 15.2432432431953, 5.85714285719156, 10, 2.5882352940822, 20.5000000002983, 58.3749999995753, 1.875, 5.08108108101713, 13.5714285715414, 10.8235294116165, 2.66666666666667, 27.4782608692871, 30.9230769228, 17.6, 7.77272727274784, 15.7500000002292, 2.46753246754739, 2.77777777777778, 12.6428571428046, 3.6, 11.2222222222222, 6.79999999992084, 20.705882353083, 2.85714285716662, 14.1818181819683, 3.51515151515152, 11.7777777777778, 57.8888888888889, 3.9999999999709, 5.58620689660779, 15.4285714286997, 11.3548387097627, 1.00000000000832, 23.9999999999393, 25.3333333333333, 20.1250000002929, 4.88372093023256, 13.1111111111111, 2.57812500003752, 2.66666666666667, 12.0000000002794, 7.74999999994361, 23.2857142859079, 10.3333333333333, NA, 4.74999999996544, 12.545454545189, 1.74999999998727, 8, 55.999999999233, 2.12499999998454, 5.05882352944641, 24.5714285716329, 8.21052631573917, 1.99999999998545, 29.17322834643, 30.5060240963)
par(mfrow = c(2, 2))
hist(nph)
hist(dph)
qqnorm(nph)
qqline(nph)
qqnorm(dph)
qqline(dph)

These are the distributions:

enter image description here

As the data is obviously not normally distributed, many statistical test cannot be applied to this data. But maybe I can transform the data to a normal distribution?

How can I find out which distribution this is?
And how can I transfrom the data to a normal distribution?

The goal is to do an analysis of variance (MANOVA) or some such (the data presented here are the two dependent variables).

Answer

The data looks like having an exponential distribution. For transformation, simple log seems to work fine.

hist(log(dph), freq=FALSE, ylim=c(0, .4))
lines(seq(-6, 6, by=0.01), dnorm(seq(-6, 6, by=0.01), 2, 1), col="red")
qqnorm(log(dph), ylim=c(0, 5))
qqline(log(dph), col="red")

enter image description here

Attribution
Source : Link , Question Author : Community , Answer Author : Community

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