Estdesc(2)

www.esmaschido.com Estadística sergiodls@yahoo.com

A
X F Fa FX X2 FX2 rango modas 1
36 6 6 216 1296 7776 1 --- 6
52 4 10 208 2704 10816 7 --- 10
68 6 16 408 4624 27744 11 --- 16
84 5 21 420 7056 35280 17 --- 21
100 6 27 600 10000 60000 22 --- 27
116 3 30 348 13456 40368 28 --- 30
132 5 35 660 17424 87120 31 --- 35
148 5 40 740 21904 109520 36 --- 40
164 2 42 328 26896 53792 41 --- 42
180 7 49 1260 32400 226800 43 --- 49 Moda = 180
n = 49 ΣX = 5188 ΣX 2 = 659216 36 0
52 0

X0 = 36 n = 49 68 0

DX= 16 MEDIA= 105.88 84 0
MODA= 180 100 0
mediana MEDIANA= 100 116 0
Dato a buscar s = 47.85 132 0
25 σ = 47.36 148 0

180 F. Max s = 2290.07
2
164 0

7 σ2 = 2243.33 180 0
x 7

HISTOGRAMA DE FRECUENCIA
8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
36 52 68 84 100 116 132 148 164 180
DATOS

ING. Sergio De Los Santos. 1 México D.F.


A
X F Fa FX X2 FX2 rango modas
36 6 ---
52 4 ---
68 6 ---
84 5 ---
100 6 ---
116 3 ---
132 5 ---
148 5 ---
164 2 ---
180 7 ---
n= ΣX = ΣX 2 =
2
n n= n 
∑xi 
∑xi MEDIA= n  
x= n i =1
MODA=
∑xi 2 − i =1n
i =1
s=
mediana MEDIANA= n −1
Dato a buscar F. Max s =
2
F. Max σ = n 
∑ xi 
n i =1 
s2 =
∑ xi 2 − N
σ2 = i =1
σ=
N
8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
36 52 68 84 100 116 132 148 164 180 x
DATOS



B
35 5 5 175 1225 6125 1 --- 5
47 2 7 94 2209 4418 6 --- 7
59 7 14 413 3481 24367 8 --- 14 Moda = 59
71 1 15 71 5041 5041 15 --- 15
83 7 22 581 6889 48223 16 --- 22 Moda = 83
95 3 25 285 9025 27075 23 --- 25
107 2 27 214 11449 22898 26 --- 27
119 7 34 833 14161 99127 28 --- 34 Moda = 119
131 3 37 393 17161 51483 35 --- 37
143 1 38 143 20449 20449 38 --- 38
n = 38 ΣX = 3202 ΣX 2 = 309206 35 0
47 0

X0 = 35 n = 38 59 0

DX= 12 MEDIA= 84.26 71 0
MODA= multimodal 83 0
Promedio de datos s = 32.63 107 0
19 20 σ = 32.2 119 0

261 F. Max s = 1064.74
2
131 0

7 σ2 = 1036.72 24 143 0
x 7

8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
35 47 59 71 83 95 107 119 131 143
DATOS



B
35 5 ---
47 2 ---
59 7 ---
71 1 ---
83 7 ---
95 3 ---
107 2 ---
119 7 ---
131 3 ---
143 1 ---
n= ΣX = ΣX 2 =
2
n n= n 
∑xi 
x= n i =1
MODA=
∑xi 2 − i =1n
i =1
s=
Promedio de datos F. Max s =
2 2
∑ xi 
n i =1 
s2 =
∑ xi 2 − N
i =1
σ2 =
σ=
N
8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
35 47 59 71 83 95 107 119 131 143 x
DATOS



C
22 2 2 44 484 968 1 --- 2
39 5 7 195 1521 7605 3 --- 7
56 2 9 112 3136 6272 8 --- 9
73 4 13 292 5329 21316 10 --- 13
90 1 14 90 8100 8100 14 --- 14
107 2 16 214 11449 22898 15 --- 16
124 5 21 620 15376 76880 17 --- 21
141 7 28 987 19881 139167 22 --- 28 Moda = 141
158 1 29 158 24964 24964 29 --- 29
175 5 34 875 30625 153125 30 --- 34
n = 34 ΣX = 3587 ΣX 2 = 461295 22 0
39 0

X0 = 22 n = 34 56 0

DX= 17 MEDIA= 105.5 73 0
MODA= 141 90 0
17 18 σ = 49.37 141 0

141 F. Max s = 2511.11
2
158 0

7 σ2 = 2437.25 18 175 0
x 7

8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
22 39 56 73 90 107 124 141 158 175
DATOS



C
22 2 ---
39 5 ---
56 2 ---
73 4 ---
90 1 ---
107 2 ---
124 5 ---
141 7 ---
158 1 ---
175 5 ---
n= ΣX = ΣX 2 =
2
n n= n 
∑xi 
x= n i =1
MODA=
∑xi 2 − i =1n
i =1
s=
2
∑ xi 
n i =1 
s2 =
∑ xi 2 − N
σ2 = i =1
σ=
N
8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
22 39 56 73 90 107 124 141 158 175 x
DATOS



D
34 3 3 102 1156 3468 1 --- 3
50 4 7 200 2500 10000 4 --- 7
66 1 8 66 4356 4356 8 --- 8
82 7 15 574 6724 47068 9 --- 15 Moda = 82
98 3 18 294 9604 28812 16 --- 18
114 3 21 342 12996 38988 19 --- 21
130 4 25 520 16900 67600 22 --- 25
146 5 30 730 21316 106580 26 --- 30
162 6 36 972 26244 157464 31 --- 36
178 6 42 1068 31684 190104 37 --- 42
n = 42 ΣX = 4868 ΣX 2 = 654440 34 0
50 0

X0 = 34 n = 42 66 0

DX= 16 MEDIA= 115.9 82 0
MODA= 82 98 0
21 22 σ = 46.35 146 0

82 F. Max s = 2200.38
2
162 0

7 σ2 = 2147.99 41 178 0
x 7

8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
34 50 66 82 98 114 130 146 162 178
DATOS



D
34 3 ---
50 4 ---
66 1 ---
82 7 ---
98 3 ---
114 3 ---
130 4 ---
146 5 ---
162 6 ---
178 6 ---
n= ΣX = ΣX 2 =
2
n n= n 
∑xi 
x= n i =1
MODA=
∑xi 2 − i =1n
i =1
s=
2
∑ xi 
n i =1 
s2 =
∑ xi 2 − N
σ2 = i =1
σ=
N
8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
34 50 66 82 98 114 130 146 162 178 x
DATOS



E
23 5 5 115 529 2645 1 --- 5
43 6 11 258 1849 11094 6 --- 11 Moda = 43
63 6 17 378 3969 23814 12 --- 17 Moda = 63
83 5 22 415 6889 34445 18 --- 22
103 4 26 412 10609 42436 23 --- 26
123 3 29 369 15129 45387 27 --- 29
143 5 34 715 20449 102245 30 --- 34
163 1 35 163 26569 26569 35 --- 35
183 4 39 732 33489 133956 36 --- 39
203 4 43 812 41209 164836 40 --- 43
n = 43 ΣX = 4369 ΣX 2 = 587427 23 0
43 0

X0 = 23 n = 43 63 0

DX= 20 MEDIA= 101.6 83 0
22 σ = 57.77 163 0

106 F. Max s = 3417.05
2
183 0

6 σ2 = 3337.59 38 203 0
x 6

7

6

F 5
R
E 4
C
U
E 3
N
C
2
I
A
1

0
23 43 63 83 103 123 143 163 183 203
DATOS



E
23 5 ---
43 6 ---
63 6 ---
83 5 ---
103 4 ---
123 3 ---
143 5 ---
163 1 ---
183 4 ---
203 4 ---
n= ΣX = ΣX 2 =
2
n n= n 
∑xi 
x= n i =1
MODA=
∑xi 2 − i =1n
i =1
s=
2
∑ xi 
n i =1 
s2 =
∑ xi 2 − N
σ2 = i =1
σ=
N
7

6

F 5
R
E 4
C
U
E 3
N
C
2
I
A
1

0
23 43 63 83 103 123 143 163 183 203 x
DATOS



F
2 7 7 14 4 28 1 --- 7 Moda = 2
15 6 13 90 225 1350 8 --- 13
28 2 15 56 784 1568 14 --- 15
41 5 20 205 1681 8405 16 --- 20
54 2 22 108 2916 5832 21 --- 22
67 6 28 402 4489 26934 23 --- 28
80 1 29 80 6400 6400 29 --- 29
93 6 35 558 8649 51894 30 --- 35
106 2 37 212 11236 22472 36 --- 37
119 4 41 476 14161 56644 38 --- 41
n = 41 ΣX = 2201 ΣX 2 = 181527 2 0
15 0

X0 = 2 n = 41 28 0

DX= 13 MEDIA= 53.68 41 0
MODA= 2 54 0
21 σ = 39.31 93 0

2 F. Max s = 1584.27
2
106 0

7 σ2 = 1545.63 18 119 0
x 7

8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
2 15 28 41 54 67 80 93 106 119
DATOS



F
2 7 ---
15 6 ---
28 2 ---
41 5 ---
54 2 ---
67 6 ---
80 1 ---
93 6 ---
106 2 ---
119 4 ---
n= ΣX = ΣX 2 =
2
n n= n 
∑xi 
x= n i =1
MODA=
∑xi 2 − i =1n
i =1
s=
2
∑ xi 
n i =1 
s2 =
∑ xi 2 − N
σ2 = i =1
σ=
N
8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
2 15 28 41 54 67 80 93 106 119 x
DATOS



G
20 7 7 140 400 2800 1 --- 7 Moda = 20
31 6 13 186 961 5766 8 --- 13
42 1 14 42 1764 1764 14 --- 14
53 1 15 53 2809 2809 15 --- 15
64 5 20 320 4096 20480 16 --- 20
75 2 22 150 5625 11250 21 --- 22
86 4 26 344 7396 29584 23 --- 26
97 6 32 582 9409 56454 27 --- 32
108 2 34 216 11664 23328 33 --- 34
119 6 40 714 14161 84966 35 --- 40
n = 40 ΣX = 2747 ΣX 2 = 239201 20 0
31 0

X0 = 20 n = 40 42 0

DX= 11 MEDIA= 68.68 53 0
MODA= 20 64 0
mediana MEDIANA= 69.5 75 0
Promedio de datos s = 36 86 0
20 21 σ = 35.55 97 0

20 F. Max s = 1296.17
2
108 0

7 σ2 = 1263.77 19 119 0
x 7

8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
20 31 42 53 64 75 86 97 108 119
DATOS



G
20 7 ---
31 6 ---
42 1 ---
53 1 ---
64 5 ---
75 2 ---
86 4 ---
97 6 ---
108 2 ---
119 6 ---
n= ΣX = ΣX 2 =
2
n n= n 
∑xi 
x= n i =1
MODA=
∑xi 2 − i =1n
i =1
s=
2
∑ xi 
n i =1 
s2 =
∑ xi 2 − N
σ2 = i =1
σ=
N
8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
20 31 42 53 64 75 86 97 108 119 x
DATOS



H
9 3 3 27 81 243 1 --- 3
22 1 4 22 484 484 4 --- 4
35 3 7 105 1225 3675 5 --- 7
48 3 10 144 2304 6912 8 --- 10
61 6 16 366 3721 22326 11 --- 16
74 6 22 444 5476 32856 17 --- 22
87 7 29 609 7569 52983 23 --- 29 Moda = 87
100 6 35 600 10000 60000 30 --- 35
113 1 36 113 12769 12769 36 --- 36
126 4 40 504 15876 63504 37 --- 40
n = 40 ΣX = 2934 ΣX 2 = 255752 9 0
22 0

X0 = 9 n = 40 35 0

DX= 13 MEDIA= 73.35 48 0
MODA= 87 61 0
20 21 σ = 31.84 100 0

87 F. Max s = 1039.57
2
113 0

7 σ2 = 1013.58 6 126 0
x 7

8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
9 22 35 48 61 74 87 100 113 126
DATOS



H
9 3 ---
22 1 ---
35 3 ---
48 3 ---
61 6 ---
74 6 ---
87 7 ---
100 6 ---
113 1 ---
126 4 ---
n= ΣX = ΣX 2 =
2
n n= n 
∑xi 
x= n i =1
MODA=
∑xi 2 − i =1n
i =1
s=
2
∑ xi 
n i =1 
s2 =
∑ xi 2 − N
σ2 = i =1
σ=
N
8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
9 22 35 48 61 74 87 100 113 126 x
DATOS



I
8 4 4 32 64 256 1 --- 4
28 5 9 140 784 3920 5 --- 9
48 2 11 96 2304 4608 10 --- 11
68 4 15 272 4624 18496 12 --- 15
88 5 20 440 7744 38720 16 --- 20
108 4 24 432 11664 46656 21 --- 24
128 1 25 128 16384 16384 25 --- 25
148 6 31 888 21904 131424 26 --- 31 Moda = 148
168 1 32 168 28224 28224 32 --- 32
188 6 38 1128 35344 212064 33 --- 38 Moda = 188
n = 38 ΣX = 3724 ΣX 2 = 500752 8 0
28 0

X0 = 8 n = 38 48 0

DX= 20 MEDIA= 98 68 0
19 20 σ = 59.78 148 0

336 F. Max s = 3670.27
2
168 0

6 σ2 = 3573.68 41 188 0
x 6

7

6

F 5
R
E 4
C
U
E 3
N
C
2
I
A
1

0
8 28 48 68 88 108 128 148 168 188
DATOS



I
8 4 ---
28 5 ---
48 2 ---
68 4 ---
88 5 ---
108 4 ---
128 1 ---
148 6 ---
168 1 ---
188 6 ---
n= ΣX = ΣX 2 =
2
n n= n 
∑xi 
x= n i =1
MODA=
∑xi 2 − i =1n
i =1
s=
2
∑ xi 
n i =1 
s2 =
∑ xi 2 − N
σ2 = i =1
σ=
N
7

6

F 5
R
E 4
C
U
E 3
N
C
2
I
A
1

0
8 28 48 68 88 108 128 148 168 188 x
DATOS



J
1 5 5 5 1 5 1 --- 5
8 5 10 40 64 320 6 --- 10
15 5 15 75 225 1125 11 --- 15
22 5 20 110 484 2420 16 --- 20
29 5 25 145 841 4205 21 --- 25
36 5 30 180 1296 6480 26 --- 30
43 1 31 43 1849 1849 31 --- 31
50 7 38 350 2500 17500 32 --- 38 Moda = 50
57 5 43 285 3249 16245 39 --- 43
64 4 47 256 4096 16384 44 --- 47
n = 47 ΣX = 1489 ΣX 2 = 66533 1 0
8 0

X0 = 1 n = 47 15 0

DX= 7 MEDIA= 31.68 22 0
MODA= 50 29 0
24 σ = 20.3 50 0

50 F. Max s = 420.87
2
57 0

7 σ2 = 411.92 11 64 0
x 7

8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
1 8 15 22 29 36 43 50 57 64
DATOS



J
1 5 ---
8 5 ---
15 5 ---
22 5 ---
29 5 ---
36 5 ---
43 1 ---
50 7 ---
57 5 ---
64 4 ---
n= ΣX = ΣX 2 =
2
n n= n 
∑xi 
x= n i =1
MODA=
∑xi 2 − i =1n
i =1
s=
2
∑ xi 
n i =1 
s2 =
∑ xi 2 − N
σ2 = i =1
σ=
N
8

7

6
F
R 5
E
C
U 4
E
N 3
C
I 2
A
1

0
1 8 15 22 29 36 43 50 57 64 x
DATOS



K
28 6 6 168 784 4704 1 --- 6 Moda = 28
37 4 10 148 1369 5476 7 --- 10
46 6 16 276 2116 12696 11 --- 16 Moda = 46
55 3 19 165 3025 9075 17 --- 19
64 5 24 320 4096 20480 20 --- 24
73 6 30 438 5329 31974 25 --- 30 Moda = 73
82 3 33 246 6724 20172 31 --- 33
91 1 34 91 8281 8281 34 --- 34
100 5 39 500 10000 50000 35 --- 39
109 1 40 109 11881 11881 40 --- 40
n = 40 ΣX = 2461 ΣX 2 = 174739 28 0
37 0

X0 = 28 n = 40 46 0

DX= 9 MEDIA= 61.53 55 0
20 21 σ = 24.15 91 0

147 F. Max s = 598.1
2
100 0

6 σ2 = 583.15 44 109 0
x 6

7

6

F 5
R
E 4
C
U
E 3
N
C
2
I
A
1

0
28 37 46 55 64 73 82 91 100 109
DATOS



K
28 6 ---
37 4 ---
46 6 ---
55 3 ---
64 5 ---
73 6 ---
82 3 ---
91 1 ---
100 5 ---
109 1 ---
n= ΣX = ΣX 2 =
2
n n= n 
∑xi 
x= n i =1
MODA=
∑xi 2 − i =1n
i =1
s=
2
∑ xi 
n i =1 
s2 =
∑ xi 2 − N
σ2 = i =1
σ=
N
7

6

F 5
R
E 4
C
U
E 3
N
C
2
I
A
1

0
28 37 46 55 64 73 82 91 100 109 x
DATOS


Estdesc(2)

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