xi
yi
xi - xśr
(xi-xśr)^2
(xi-xśr)^3
(xi-xśr)^4
yi - yśr
(yi-yśr)^2
(yi-yśr)^3
(yi-yśr)^4
77
59
-46,786
2188,930
-102411,269
4791413,652
-21,038
442,597
-9311,365
195892,497
87
66
-36,786
1353,210
-49779,176
1831176,752
-14,038
197,065
-2766,405
38834,789
87
66
-36,786
1353,210
-49779,176
1831176,752
-14,038
197,065
-2766,405
38834,789
88
66
-35,786
1280,638
-45828,904
1640033,165
-14,038
197,065
-2766,405
38834,789
88
67
-35,786
1280,638
-45828,904
1640033,165
-13,038
169,989
-2216,322
28896,411
89
67
-34,786
1210,066
-42093,349
1464259,231
-13,038
169,989
-2216,322
28896,411
90
67
-33,786
1141,494
-38566,509
1303008,086
-13,038
169,989
-2216,322
28896,411
90
67
-33,786
1141,494
-38566,509
1303008,086
-13,038
169,989
-2216,322
28896,411
91
68
-32,786
1074,922
-35242,386
1155456,868
-12,038
144,913
-1744,468
20999,906
91
68
-32,786
1074,922
-35242,386
1155456,868
-12,038
144,913
-1744,468
20999,906
92
68
-31,786
1010,350
-32114,979
1020806,710
-12,038
144,913
-1744,468
20999,906
92
68
-31,786
1010,350
-32114,979
1020806,710
-12,038
144,913
-1744,468
20999,906
92
68
-31,786
1010,350
-32114,979
1020806,710
-12,038
144,913
-1744,468
20999,906
93
69
-30,786
947,778
-29178,287
898282,751
-11,038
121,837
-1344,842
14844,363
93
69
-30,786
947,778
-29178,287
898282,751
-11,038
121,837
-1344,842
14844,363
93
69
-30,786
947,778
-29178,287
898282,751
-11,038
121,837
-1344,842
14844,363
94
69
-29,786
887,206
-26426,312
787134,124
-11,038
121,837
-1344,842
14844,363
94
69
-29,786
887,206
-26426,312
787134,124
-11,038
121,837
-1344,842
14844,363
95
69
-28,786
828,634
-23853,052
686633,968
-11,038
121,837
-1344,842
14844,363
95
69
-28,786
828,634
-23853,052
686633,968
-11,038
121,837
-1344,842
14844,363
96
69
-27,786
772,062
-21452,509
596079,417
-11,038
121,837
-1344,842
14844,363
96
69
-27,786
772,062
-21452,509
596079,417
-11,038
121,837
-1344,842
14844,363
97
69
-26,786
717,490
-19218,682
514791,607
-11,038
121,837
-1344,842
14844,363
97
69
-26,786
717,490
-19218,682
514791,607
-11,038
121,837
-1344,842
14844,363
98
70
-25,786
664,918
-17145,570
442115,675
-10,038
100,761
-1011,443
10152,869
98
70
-25,786
664,918
-17145,570
442115,675
-10,038
100,761
-1011,443
10152,869
98
70
-25,786
664,918
-17145,570
442115,675
-10,038
100,761
-1011,443
10152,869
98
70
-25,786
664,918
-17145,570
442115,675
-10,038
100,761
-1011,443
10152,869
99
70
-24,786
614,346
-15227,175
377420,757
-10,038
100,761
-1011,443
10152,869
99
70
-24,786
614,346
-15227,175
377420,757
-10,038
100,761
-1011,443
10152,869
99
70
-24,786
614,346
-15227,175
377420,757
-10,038
100,761
-1011,443
10152,869
100
70
-23,786
565,774
-13457,496
320099,988
-10,038
100,761
-1011,443
10152,869
100
70
-23,786
565,774
-13457,496
320099,988
-10,038
100,761
-1011,443
10152,869
100
71
-23,786
565,774
-13457,496
320099,988
-9,038
81,685
-738,273
6672,512
100
71
-23,786
565,774
-13457,496
320099,988
-9,038
81,685
-738,273
6672,512
101
71
-22,786
519,202
-11830,532
269570,505
-9,038
81,685
-738,273
6672,512
101
71
-22,786
519,202
-11830,532
269570,505
-9,038
81,685
-738,273
6672,512
101
71
-22,786
519,202
-11830,532
269570,505
-9,038
81,685
-738,273
6672,512
101
71
-22,786
519,202
-11830,532
269570,505
-9,038
81,685
-738,273
6672,512
Miary zmienności
102
71
-21,786
474,630
-10340,285
225273,443
-9,038
81,685
-738,273
6672,512
102
71
-21,786
474,630
-10340,285
225273,443
-9,038
81,685
-738,273
6672,512
102
71
-21,786
474,630
-10340,285
225273,443
-9,038
81,685
-738,273
6672,512
102
71
-21,786
474,630
-10340,285
225273,443
-9,038
81,685
-738,273
6672,512
102
71
-21,786
474,630
-10340,285
225273,443
-9,038
81,685
-738,273
6672,512
102
71
-21,786
474,630
-10340,285
225273,443
-9,038
81,685
-738,273
6672,512
102
71
-21,786
474,630
-10340,285
225273,443
-9,038
81,685
-738,273
6672,512
102
72
-21,786
474,630
-10340,285
225273,443
-8,038
64,609
-519,331
4174,380
102
72
-21,786
474,630
-10340,285
225273,443
-8,038
64,609
-519,331
4174,380
103
72
-20,786
432,058
-8980,753
186673,939
-8,038
64,609
-519,331
4174,380
103
72
-20,786
432,058
-8980,753
186673,939
-8,038
64,609
-519,331
4174,380
103
72
-20,786
432,058
-8980,753
186673,939
-8,038
64,609
-519,331
4174,380
103
72
-20,786
432,058
-8980,753
186673,939
-8,038
64,609
-519,331
4174,380
103
72
-20,786
432,058
-8980,753
186673,939
-8,038
64,609
-519,331
4174,380
104
72
-19,786
391,486
-7745,938
153261,128
-8,038
64,609
-519,331
4174,380
104
72
-19,786
391,486
-7745,938
153261,128
-8,038
64,609
-519,331
4174,380
104
72
-19,786
391,486
-7745,938
153261,128
-8,038
64,609
-519,331
4174,380
104
72
-19,786
391,486
-7745,938
153261,128
-8,038
64,609
-519,331
4174,380
105
72
-18,786
352,914
-6629,839
124548,147
-8,038
64,609
-519,331
4174,380
105
72
-18,786
352,914
-6629,839
124548,147
-8,038
64,609
-519,331
4174,380
105
72
-18,786
352,914
-6629,839
124548,147
-8,038
64,609
-519,331
4174,380
106
72
-17,786
316,342
-5626,455
100072,132
-8,038
64,609
-519,331
4174,380
106
72
-17,786
316,342
-5626,455
100072,132
-8,038
64,609
-519,331
4174,380
106
72
-17,786
316,342
-5626,455
100072,132
-8,038
64,609
-519,331
4174,380
106
73
-17,786
316,342
-5626,455
100072,132
-7,038
49,533
-348,616
2453,562
106
73
-17,786
316,342
-5626,455
100072,132
-7,038
49,533
-348,616
2453,562
106
73
-17,786
316,342
-5626,455
100072,132
-7,038
49,533
-348,616
2453,562
106
73
-17,786
316,342
-5626,455
100072,132
-7,038
49,533
-348,616
2453,562
106
73
-17,786
316,342
-5626,455
100072,132
-7,038
49,533
-348,616
2453,562
106
73
-17,786
316,342
-5626,455
100072,132
-7,038
49,533
-348,616
2453,562
106
73
-17,786
316,342
-5626,455
100072,132
-7,038
49,533
-348,616
2453,562
107
73
-16,786
281,770
-4729,788
79394,218
-7,038
49,533
-348,616
2453,562
107
73
-16,786
281,770
-4729,788
79394,218
-7,038
49,533
-348,616
2453,562
107
73
-16,786
281,770
-4729,788
79394,218
-7,038
49,533
-348,616
2453,562
107
73
-16,786
281,770
-4729,788
79394,218
-7,038
49,533
-348,616
2453,562
107
73
-16,786
281,770
-4729,788
79394,218
-7,038
49,533
-348,616
2453,562
107
73
-16,786
281,770
-4729,788
79394,218
-7,038
49,533
-348,616
2453,562
107
73
-16,786
281,770
-4729,788
79394,218
-7,038
49,533
-348,616
2453,562
107
73
-16,786
281,770
-4729,788
79394,218
-7,038
49,533
-348,616
2453,562
107
74
-16,786
281,770
-4729,788
79394,218
-6,038
36,457
-220,130
1329,145
108
74
-15,786
249,198
-3933,836
62099,542
-6,038
36,457
-220,130
1329,145
108
74
-15,786
249,198
-3933,836
62099,542
-6,038
36,457
-220,130
1329,145
108
74
-15,786
249,198
-3933,836
62099,542
-6,038
36,457
-220,130
1329,145
108
74
-15,786
249,198
-3933,836
62099,542
-6,038
36,457
-220,130
1329,145
108
74
-15,786
249,198
-3933,836
62099,542
-6,038
36,457
-220,130
1329,145
109
74
-14,786
218,626
-3232,601
47797,239
-6,038
36,457
-220,130
1329,145
109
74
-14,786
218,626
-3232,601
47797,239
-6,038
36,457
-220,130
1329,145
109
74
-14,786
218,626
-3232,601
47797,239
-6,038
36,457
-220,130
1329,145
109
74
-14,786
218,626
-3232,601
47797,239
-6,038
36,457
-220,130
1329,145
109
74
-14,786
218,626
-3232,601
47797,239
-6,038
36,457
-220,130
1329,145
110
74
-13,786
190,054
-2620,082
36120,445
-6,038
36,457
-220,130
1329,145
110
74
-13,786
190,054
-2620,082
36120,445
-6,038
36,457
-220,130
1329,145
110
74
-13,786
190,054
-2620,082
36120,445
-6,038
36,457
-220,130
1329,145
110
74
-13,786
190,054
-2620,082
36120,445
-6,038
36,457
-220,130
1329,145
110
74
-13,786
190,054
-2620,082
36120,445
-6,038
36,457
-220,130
1329,145
110
74
-13,786
190,054
-2620,082
36120,445
-6,038
36,457
-220,130
1329,145
110
74
-13,786
190,054
-2620,082
36120,445
-6,038
36,457
-220,130
1329,145
110
74
-13,786
190,054
-2620,082
36120,445
-6,038
36,457
-220,130
1329,145
111
74
-12,786
163,482
-2090,278
26726,298
-6,038
36,457
-220,130
1329,145
111
74
-12,786
163,482
-2090,278
26726,298
-6,038
36,457
-220,130
1329,145
111
74
-12,786
163,482
-2090,278
26726,298
-6,038
36,457
-220,130
1329,145
111
75
-12,786
163,482
-2090,278
26726,298
-5,038
25,381
-127,872
644,218
111
75
-12,786
163,482
-2090,278
26726,298
-5,038
25,381
-127,872
644,218
111
75
-12,786
163,482
-2090,278
26726,298
-5,038
25,381
-127,872
644,218
111
75
-12,786
163,482
-2090,278
26726,298
-5,038
25,381
-127,872
644,218
112
75
-11,786
138,910
-1637,191
19295,931
-5,038
25,381
-127,872
644,218
112
75
-11,786
138,910
-1637,191
19295,931
-5,038
25,381
-127,872
644,218
112
75
-11,786
138,910
-1637,191
19295,931
-5,038
25,381
-127,872
644,218
112
75
-11,786
138,910
-1637,191
19295,931
-5,038
25,381
-127,872
644,218
112
75
-11,786
138,910
-1637,191
19295,931
-5,038
25,381
-127,872
644,218
112
75
-11,786
138,910
-1637,191
19295,931
-5,038
25,381
-127,872
644,218
112
75
-11,786
138,910
-1637,191
19295,931
-5,038
25,381
-127,872
644,218
112
75
-11,786
138,910
-1637,191
19295,931
-5,038
25,381
-127,872
644,218
112
75
-11,786
138,910
-1637,191
19295,931
-5,038
25,381
-127,872
644,218
112
76
-11,786
138,910
-1637,191
19295,931
-4,038
16,305
-65,841
265,868
112
76
-11,786
138,910
-1637,191
19295,931
-4,038
16,305
-65,841
265,868
112
76
-11,786
138,910
-1637,191
19295,931
-4,038
16,305
-65,841
265,868
112
76
-11,786
138,910
-1637,191
19295,931
-4,038
16,305
-65,841
265,868
112
76
-11,786
138,910
-1637,191
19295,931
-4,038
16,305
-65,841
265,868
112
76
-11,786
138,910
-1637,191
19295,931
-4,038
16,305
-65,841
265,868
113
76
-10,786
116,338
-1254,819
13534,483
-4,038
16,305
-65,841
265,868
113
76
-10,786
116,338
-1254,819
13534,483
-4,038
16,305
-65,841
265,868
113
76
-10,786
116,338
-1254,819
13534,483
-4,038
16,305
-65,841
265,868
113
76
-10,786
116,338
-1254,819
13534,483
-4,038
16,305
-65,841
265,868
113
76
-10,786
116,338
-1254,819
13534,483
-4,038
16,305
-65,841
265,868
113
76
-10,786
116,338
-1254,819
13534,483
-4,038
16,305
-65,841
265,868
113
76
-10,786
116,338
-1254,819
13534,483
-4,038
16,305
-65,841
265,868
113
76
-10,786
116,338
-1254,819
13534,483
-4,038
16,305
-65,841
265,868
114
76
-9,786
95,766
-937,164
9171,088
-4,038
16,305
-65,841
265,868
114
76
-9,786
95,766
-937,164
9171,088
-4,038
16,305
-65,841
265,868
114
76
-9,786
95,766
-937,164
9171,088
-4,038
16,305
-65,841
265,868
114
76
-9,786
95,766
-937,164
9171,088
-4,038
16,305
-65,841
265,868
114
76
-9,786
95,766
-937,164
9171,088
-4,038
16,305
-65,841
265,868
114
76
-9,786
95,766
-937,164
9171,088
-4,038
16,305
-65,841
265,868
115
76
-8,786
77,194
-678,225
5958,882
-4,038
16,305
-65,841
265,868
115
76
-8,786
77,194
-678,225
5958,882
-4,038
16,305
-65,841
265,868
115
76
-8,786
77,194
-678,225
5958,882
-4,038
16,305
-65,841
265,868
115
77
-8,786
77,194
-678,225
5958,882
-3,038
9,229
-28,039
85,183
115
77
-8,786
77,194
-678,225
5958,882
-3,038
9,229
-28,039
85,183
115
77
-8,786
77,194
-678,225
5958,882
-3,038
9,229
-28,039
85,183
115
77
-8,786
77,194
-678,225
5958,882
-3,038
9,229
-28,039
85,183
115
77
-8,786
77,194
-678,225
5958,882
-3,038
9,229
-28,039
85,183
115
77
-8,786
77,194
-678,225
5958,882
-3,038
9,229
-28,039
85,183
115
77
-8,786
77,194
-678,225
5958,882
-3,038
9,229
-28,039
85,183
115
77
-8,786
77,194
-678,225
5958,882
-3,038
9,229
-28,039
85,183
115
77
-8,786
77,194
-678,225
5958,882
-3,038
9,229
-28,039
85,183
115
77
-8,786
77,194
-678,225
5958,882
-3,038
9,229
-28,039
85,183
115
77
-8,786
77,194
-678,225
5958,882
-3,038
9,229
-28,039
85,183
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148
90
24,214
586,318
14197,099
343768,558
9,962
99,241
988,643
9848,864
149
90
25,214
635,746
16029,695
404172,717
9,962
99,241
988,643
9848,864
149
90
25,214
635,746
16029,695
404172,717
9,962
99,241
988,643
9848,864
149
90
25,214
635,746
16029,695
404172,717
9,962
99,241
988,643
9848,864
149
90
25,214
635,746
16029,695
404172,717
9,962
99,241
988,643
9848,864
150
90
26,214
687,174
18013,574
472207,826
9,962
99,241
988,643
9848,864
150
90
26,214
687,174
18013,574
472207,826
9,962
99,241
988,643
9848,864
151
90
27,214
740,602
20154,737
548491,020
9,962
99,241
988,643
9848,864
151
90
27,214
740,602
20154,737
548491,020
9,962
99,241
988,643
9848,864
151
91
27,214
740,602
20154,737
548491,020
10,962
120,165
1317,254
14439,734
152
91
28,214
796,030
22459,185
633663,436
10,962
120,165
1317,254
14439,734
152
91
28,214
796,030
22459,185
633663,436
10,962
120,165
1317,254
14439,734
155
91
31,214
974,314
30412,231
949287,373
10,962
120,165
1317,254
14439,734
155
91
31,214
974,314
30412,231
949287,373
10,962
120,165
1317,254
14439,734
156
91
32,214
1037,742
33429,814
1076908,035
10,962
120,165
1317,254
14439,734
157
92
33,214
1103,170
36640,682
1216983,599
11,962
143,089
1711,636
20474,589
157
92
33,214
1103,170
36640,682
1216983,599
11,962
143,089
1711,636
20474,589
157
92
33,214
1103,170
36640,682
1216983,599
11,962
143,089
1711,636
20474,589
158
92
34,214
1170,598
40050,833
1370299,200
11,962
143,089
1711,636
20474,589
159
92
35,214
1240,026
43666,268
1537663,975
11,962
143,089
1711,636
20474,589
160
93
36,214
1311,454
47492,988
1719911,059
12,962
168,013
2177,790
28228,517
161
94
37,214
1384,882
51536,991
1917897,589
13,962
194,937
2721,717
38000,607
162
94
38,214
1460,310
55804,279
2132504,700
13,962
194,937
2721,717
38000,607
162
94
38,214
1460,310
55804,279
2132504,700
13,962
194,937
2721,717
38000,607
165
95
41,214
1698,594
70005,845
2885220,884
14,962
223,861
3349,415
50113,946
167
96
43,214
1867,450
80699,975
3487368,741
15,962
254,785
4066,885
64915,622
171
101
47,214
2229,162
105247,645
4969162,313
20,962
439,405
9210,817
193077,144
61893
40019
0,000 122158,102
48821,402
86817816,957
0,000 18654,278 -6289,615 1596295,301
Analiza szeregu szczegółowego prostego
Kolumna xi:
Miary położenia
xśr= 123,786
średnia arytmetyczna
n= 500
liczebność
Me= 124
mediana (wartość środkowa)
dwa wzory dla parzystych lub nieparzastych
Miary zmienności
R= 94
rozrzut xmax-xmin
s= 15,631
odchylenie standardowe
sqrt(s^2)
s^2= 244,316
wariancja
Vs= 12,627
%
współczynnik zmienności
umiarkowana zmienność danych
Miary kształtu
As= 0,026
współczynnik asymetrii
niewielka asymetria prawostronna (znak +/-)
m3= 97,643
moment centralny 3-ego rzędu
K= 2,909
współczynnik koncentracji
brak koncentracji danych
m4= #########
moment centralny 4-ego rzędu
Wnioski: Średnia wartość to 123,786. Srodek naszewgo zbioru czyli mediana to 124.
Rozrzut czyli różnica pomiędzy wartością największą i najmniejszą wynosi 94.
Wartości różnią się średnio od wartości średniej o 15,631. Posiadają niewielką asymetrię prawostronną.
Wartości nie są skoncentrowane wokół średniej, czyli dość dużo się od niej różnią.
Kolumna yi
Suma yi=
40019
Miary położenia
xśr=
n=
80,038
średnia arytmetyczna
Me=
500
liczebność
Miary zmienności
0
mediana (wartość środkowa)
dwa wzory dla parzystych lub nieparzastych, chodzi o ilość elemantów
xn =
R=
s= 42
rozrzut
s^2=
6,108
odchylenie standardowe
s^2 to wariancja - różnica pomiędzy poszczególnymi daneymi srednia sredniej
Vs=
37,309
wariancja
Miary kształtu
7,631
%
współczynnik zmienności
ponizej 10 %- mala zmiennosc, 10-20%srednia zmiennosc, pow 20% duza zmiennosc
As=
m3=
-0,055
współczynnik asymetrii
niewielka asymetria lewostronna (znak +/- rodzaj asymetrii),
K=
-12,579
moment centralny 3-ego rzędu
m4=
2,294
współczynnik koncentracji
brak koncentracji danych, bo ponizej 3, wartosci odbiegaja od sredniej
3192,591
moment centralny 4-ego rzędu
Wnioski: Średnia wartość kolumny yi to 80,038. Rozrzut czyli różnica pomiędzy wartością największą i najmniejszą wynosi 42.
Wartości yi różnią się średnio od wartości średniej o 6,108. Posiadają niewielką asymetrię lewostronną
Wartości nie są skoncentrowane wokół średniej, czyli dość dużo się od niej różnią.
Zamiana szeregu szczegółowego prostego na przedziałowy.
Szereg xi
k
x0k
x1k
nk
xk
nkskum fkskum
n=
500
1
77,0
86,4
1
81,7
1
0,002
Xmin=
77
2
86,4
95,8
19
91,1
20
0,04
Xmax=
171
3
95,8
105,2
40
100,5
60
0,12
R=
94
4
105,2
114,6
73
109,9
133
0,266
k=
9,965978
5
114,6
124,0
122
119,3
255
0,51
k=
10
6
124,0
133,4
117
128,7
372
0,744
dx=
9,4
7
133,4
142,8
67
138,1
439
0,878
dx=
9,4
8
142,8
152,2
46
147,5
485
0,97
9
152,2
161,6
10
156,9
495
0,99
10
161,6
171,0
5
166,3
500
1
0
100
200
300
400
500
600
1
20
60
133
255
372
439
485
495
500
Dystrybuanta
0
5
10
15
20
25
30
140,7
150,1
159,5
168,9
178,3
187,7
197,1
206,5
Histogram
Szereg yi
k
x0k
x1k
nk
xk
nkskum fkskum
n=
500
1
59,0
63,2
1
61,1
1
0,002
Xmin=
59
2
63,2
67,4
7
65,3
8
0,016
Xmax=
101
3
67,4
71,6
38
69,5
46
0,092
R=
42
4
71,6
75,8
67
73,7
113
0,226
k=
9,965978
5
75,8
80,0
143
77,9
256
0,512
k=
10
6
80,0
84,2
123
82,1
379
0,758
dx=
4,2
7
84,2
88,4
79
86,3
458
0,916
dx=
4,2
8
88,4
92,6
35
90,5
493
0,986
9
92,6
96,8
6
94,7
499
0,998
10
96,8
101,0
1
98,9
500
1
0
20
40
60
80
100
120
140
160
61,1
65,3
69,5
73,7
77,9
82,1
86,3
90,5
94,7
98,9
Histogram
0
100
200
300
400
500
600
0,002 0,016 0,092 0,226 0,512 0,758 0,916 0,986 0,998
1
Dystrybuanta
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