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BASIC
STATISTICS
We have gather the following real data of different ages from
our families
8 12 4 26 10 16 15 14 29 32
40 19 20 28 41 48 16 5 13 22
3 89 75 63 82 71 54 35 69 86
94 70 56 18 44 23 78 99 84 27
72 65 59 47 58 93 66 53 90 88
Solution:
𝑥 𝑚𝑎𝑥 = 99
𝑥 𝑚𝑖𝑛 = 3
𝑹𝒂𝒏𝒈𝒆 = 𝑹 = 𝒙 𝒎𝒂𝒙– 𝒙 𝒎𝒊𝒏
= 99 – 3
= 96
𝑁𝑜. 𝑜𝑓 𝑐𝑙𝑎𝑠𝑠𝑒𝑠 = 𝑚 = 9
𝑾𝒊𝒅𝒕𝒉 = 𝒉 = 𝑹/𝒎
= 96/9
= 10.66 ≅ 11
𝑪. 𝑰 𝑻𝒂𝒍𝒍𝒚 𝑴𝒂𝒓𝒌𝒔 𝒇
𝒙
𝑴𝒊𝒅 𝑷𝒐𝒊𝒏𝒕
𝑪𝒍𝒂𝒔𝒔 𝑩𝒐𝒖𝒏𝒅𝒂𝒓𝒊𝒆𝒔
3 – 13 IIII II 7 8 2.5 – 13.5
14 – 24 IIII IIII 9 19 13.5 – 24.5
25 – 35 IIII I 6 30 24.5 – 35.5
36 – 46 III 3 41 35.5 – 46.5
47 – 57 IIII 5 52 46.5 – 57.5
58 – 68 IIII 5 63 57.5 – 68.5
69 – 79 IIII I 6 74 68.5 – 79.5
80 – 90 IIII I 6 85 79.5 – 90.5
91 – 101 III 3 96 90.5 – 101
50
Relative
Frequency
Cumulative
Frequency
Cumulative Frequency
Distribution
Less than More than
0.14 7 2.5- 0 2.5- 50
0.18 16 13.5- 7 13.5- 43
0.12 22 24.5- 16 24.5- 34
0.06 25 35.5- 22 35.5- 28
0.1 30 46.5- 25 46.5- 25
0.1 35 57.5- 30 57.5- 20
0.12 41 68.5- 35 68.5- 15
0.12 47 79.5- 41 79.5- 9
0.06 50 90.5- 47 90.5- 3
1 101.5- 50 101.5- 0
HISTOGRAM & FREQUENCY CURVE
0
1
2
3
4
5
6
7
8
9
10
CLASS BOUNDRIES
FREQUENCY
HISTOGRAM & FREQUENCY
POLYGON
0
1
2
3
4
5
6
7
8
9
10
CLASS BOUNDRIES
FREQUENCY
0
10
20
30
40
50
60
2.5 13.5 24.5 35.5 46.5 57.5 68.5 79.5 90.5 101.5
LESS THAN
CLASS
BOUNDRIES
FREQUENCY
MORE THAN
0
10
20
30
40
50
60
2.5 13.5 24.5 35.5 46.5 57.5 68.5 79.5 90.5 101.5
CLASS
BOUNDRIES
FREQUENCY
𝒇𝒙 log 𝒙 𝒇log 𝒙 𝟏
𝒙 𝒇 𝟏
𝒙
56 0.903 6.321 0.125 0.875
171 1.2787 11.5083 0.0526 0.4734
180 1.4771 8.8626 0.0333 0.1998
123 1.6127 4.8381 0.0243 0.0729
260 1.7160 8.58 0.0192 0.096
315 1.7993 8.9965 0.0158 0.079
444 1.8692 11.2152 0.0135 0.081
510 1.9294 11.5764 0.0117 0.0702
288 1.9822 5.9468 0.0104 0.0312
2347 77.8447 1.9785
ARITHMETIC MEAN
𝒙 = 𝝁 =
Ʃ𝒇𝒙
Ʃ𝒇
=
2347
50
A.M = 46.94
GEOMETRIC MEAN
𝑮. 𝑴 = 𝑨𝒏𝒕𝒊𝒍𝒐𝒈
𝒇𝒍𝒐𝒈𝒙
𝒇
= 𝐴𝑛𝑡𝑖𝑙𝑜𝑔
77.8447
50
= 36.049
HARMONIC MEAN
𝑯. 𝑴 =
𝒇
𝒇 𝟏
𝒙
=
50
1.9785
= 25.2716
MEDIAN
𝒙 = ℓ +
𝒉
𝒇
𝑵
𝟐
− 𝑪
𝑁
2
=
50
2
= 25
= 35.5 +
11
3
25 − 22
= 35.5 + 11
𝑥 = 46.5
MODE
𝒙 = ℓ + 𝒉
𝒇 𝒎 − 𝒇 𝟏
𝟐𝒇 𝒎 − 𝒇 𝟐 − 𝒇 𝟏
= 13.5 + 11
9−7
2 9 −6−7
= 13.5 + 11 0.4
= 13.5 + 4.4
𝑥 = 17.9
QAURTILE
𝑸𝒊 = ℓ +
𝒉
𝒇
𝑵
𝟒
− 𝑪
=
𝑁
4
=
50
4
= 12.5
= 13.5 +
11
9
12.5 − 7
= 13.5 + 1.22 5.5
= 13.5 + 6.722
𝑄𝑖 = 20.22
𝑸 𝟑 = ℓ +
𝒉
𝒇
𝟑𝑵
𝟒
− 𝑪
3𝑁
4
=
3 × 50
4
= 37.5
𝑄3 = 68.5 +
11
6
37.5 − 35
= 68.5 + 1.833 2.5
= 68.5 + 4.5833
= 73
DECILE
𝑫 𝟏𝟎 = ℓ +
𝒉
𝒇
𝟑𝟎𝑵
𝟏𝟎
− 𝑪
=
30𝑁
10
=
30×10
10
= 30
= 46.5 +
11
5
30 − 25
= 46.5 + 2.2 5
= 46.5 + 4.84
= 51.34
PERCENTILE
𝑷 𝟒𝟓 = ℓ +
𝒉
𝒇
𝟒𝟓𝑵
𝟏𝟎𝟎
− 𝑪
45𝑁
100
=
45 × 50
100
= 22.5
= 35.5 +
11
3
22.5 − 22
= 35.5 + 3.666 0.5
= 35.5 + 1.8333
𝑃45 = 37.333
OGIVE : LESS THAN
0
10
20
30
40
50
60
2.5 13.5 24.5 35.5 46.5 57.5 68.5 79.5 90.5 101.5
𝐷_10
𝑃_45
𝑄_1
𝑄3
CLASS
FREQUENC
Y
OGIVE: MORE THAN
0
10
20
30
40
50
60
2.5 13.5 24.5 35.5 46.5 57.5 68.5 79.5 90.5 101.5
𝑃
𝐷
𝑄_3
𝑄1
FREQUENC
Y
CLASS BOUNDRIES
QUARTILE DEVIATION
𝑸𝑫 =
𝑸 𝟑 − 𝑸 𝟏
𝟐
=
73 −20.22
2
= 26.39
𝒙𝒊 − 𝒙 𝒙𝒊 − 𝒙 𝒇 𝒙𝒊 − 𝒙 𝒙𝒊 − 𝒙 𝒙𝒊 − 𝒙 𝒇 𝒙𝒊 − 𝒙
-38.94 38.94 272.5 -38.5 38.5 269.5
-27.94 27.94 251.46 -27.5 27.5 247.5
-16.94 16.94 101.64 -16.6 16.6 99
-5.94 5.94 17.82 -5.5 5.5 16.5
5.06 5.06 25.3 5.5 5.5 27.5
16.06 16.06 80.3 16.5 16.5 82.5
27.06 27.06 162.36 27.5 27.5 165
38.06 38.06 228.36 38.5 38.5 231
49.06 49.06 147.18 49.5 49.5 148.5
1286.92 1287
MEAN DEVIATION
𝑴. 𝑫 𝒙 =
𝒇 𝒙𝒊 − 𝒙
𝒇
𝑀. 𝐷 𝑥 =
1286.92
50
= 25.7384
MEDIAN DEVIATION
𝑴. 𝑫 𝒙 =
𝒇 𝒙𝒊 − 𝒙
𝒇
𝑀. 𝐷 𝑥 =
1287
50
= 25.74
𝒇 𝒙 𝒙𝒊 − 𝒙 𝒙𝒊 − 𝒙 𝟐
𝒇 𝒙𝒊 − 𝒙 𝟐
𝒙 𝟐
𝒇𝒙 𝟐
7 8 -38.94 1516.3 10614.1 64 448
9 19 -27.94 780.6 7025.4 361 3249
6 30 -16.94 286.96 1721.76 900 5400
3 41 -5.94 35.28 105.84 1681 5043
5 52 5.06 25.6 128 2704 13520
5 63 16.06 257.9 1289.5 3969 19845
6 74 27.06 732.24 4393.44 5476 32856
6 85 38.06 1448.56 8691.36 7225 43350
3 96 49.06 2406.88 7220.65 9216 27648
50 41190.05 151359
STANDARD DEVIATION
𝝈 =
𝒇 𝒙𝒊 − 𝒙 𝟐
𝒇
=
4119.05
50
= 823.801
= 28.7
STANDARD DEVIATION FROM
COMPUTATIONAL
𝝈 =
𝒇𝒙 𝟐
𝒇
−
𝒇𝒙
𝒇
𝟐
=
151359
50
−
2347
50
2
= 3027.18 − 46.94 2
= 3027.18 − 2203.36
= 823.869
= 28.7
VARIANCE
𝝈 𝟐 =
𝒇 𝒙𝒊 − 𝒙 𝟐
𝒇
=
41190.05
50
= 823.8
VARIANCE FROM
COMPUTATIONAL
𝝈 𝟐
=
𝒇 𝒙 𝟐
𝒇
−
𝒇𝒙
𝒇
𝟐
=
151359
50
−
2347
50
2
= 3027.18 − 46.94 2
= 3027.18 − 2203.36
= 823.8
𝒇𝒙 𝒙 𝟐 𝒇𝒙 𝟐 𝒙 𝟑 𝒇𝒙 𝟑
𝒙 𝟒 𝒇𝒙 𝟒
56 64 448 512 3684 4096 28672
171 361 3249 6859 61731 130321 1172889
180 900 5400 27000 162000 810000 4860000
123 1681 5043 68921 206763 2825761 8477283
260 2704 13520 140608 703040 7311616 36558080
315 3969 19845 250047 1250235 15752961 78764805
444 5476 32856 405224 2431344 29986576 179919456
510 7225 43350 614125 3634750 52200625 313203750
288 9216 27648 884736 2654208 84934656 254803968
2347 151359 11157655 87778903
MOMENT ABOUT ORIGIN
𝝁 𝟏 =
𝒇𝒙
𝒇
=
2347
50
= 46.94
𝝁 𝟐 =
𝒇𝒙 𝟐
𝒇
=
151359
50
= 3027.18
𝝁 𝟑 =
𝒇𝒙 𝟑
𝒇
=
11157655
50
= 223153.1
𝝁 𝟒 =
𝒇𝒙 𝟒
𝒇
=
877788903
50
= 17555778.06
𝒙𝒊 − 𝒙 𝒇 𝒙𝒊 − 𝒙 𝒙𝒊 − 𝒙 𝟐
𝒇 𝒙𝒊 − 𝒙 𝟐
-38.94 -272.58 1516.3 10614.1
-27.94 -251.46 780.6 7025.4
-16.94 -101.64 286.96 1721.76
-5.94 -17.82 35.28 105.84
5.06 25.3 25.6 128
16.06 80.3 257.9 1289.5
27.06 162.36 732.24 4393.44
38.06 228.36 1448.56 8691.36
49.06 147.18 2406.88 7220.65
0 41190.05
𝒙𝒊 − 𝒙 𝟑
𝒇 𝒙𝒊 − 𝒙 𝟑
𝒙𝒊 − 𝒙 𝟒
𝒇 𝒙𝒊 − 𝒙 𝟒
-59045.64098 -413319.4869 2299237.28 16094680.82
-21811.18218 -196300.6897 609404.4301 548439.871
-4861.163384 -29166.9798 82348.1063 494088.6378
-209.584584 -628.753752 1244.93193 3734.795
129.5542 647.77108 655.544 3277.72126
4142.2530 20711.26508 66524.583 332622.9159
19814.51182 118887.0709 536180.6897 3217084.139
55132.33062 330793.9837 2098336.500 12590019.02
118081.7094 354245.1282 5793088.664 17379265.99
185869.359 55599393.91
MOMENTS ABOUT IT MEAN
𝝁 𝟏 =
𝒇 𝒙 − 𝒙
𝒇
= 𝟎
=
0
50
= 0
𝝁 𝟐 =
𝒇 𝒙 − 𝒙 𝟐
𝒇
= 𝝈 𝟐
=
41190.79
50
= 823.8
𝝁 𝟑 =
𝒇 𝒙 − 𝒙 𝟑
𝒇
= 𝝈 𝟑
=
185869.359
50
= 3717.3871
𝝁 𝟒 =
𝒇 𝒙 − 𝒙 𝟒
𝒇
= 𝝈 𝟒
=
55599393.91
50
= 1111987.878
COFFICIENT OF MD 𝒙
𝑪𝒐𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝑴𝑫( 𝒙) =
𝑴. 𝑫 𝒙
𝒙
× 𝟏𝟎𝟎
=
25.7384
46.94
= 0.5483
COFFICIENT OF MD 𝒙
𝑪𝒐𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝑴𝑫 𝒙 =
𝑴. 𝑫 𝒙
𝒙
× 𝟏𝟎𝟎
=
25.74
46.5
= 0.5535
𝜷 𝟏 =
𝝁 𝟑
𝟐
𝝁 𝟐
𝟑
=
3717.3871 2
823.8 3
=
13818966.85
559068937.3
= 0.0247
COFFICIENT OF SKEWNES
𝜷 𝟏 =
𝝁 𝟒
𝝁 𝟐
𝟐
=
1111987.878
823.8 2
=
1111987.878
678646.44
= 1.638537849
COFICIENT OF KURTOSIS

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BASIC STATISTIC

  • 2. We have gather the following real data of different ages from our families 8 12 4 26 10 16 15 14 29 32 40 19 20 28 41 48 16 5 13 22 3 89 75 63 82 71 54 35 69 86 94 70 56 18 44 23 78 99 84 27 72 65 59 47 58 93 66 53 90 88
  • 3. Solution: 𝑥 𝑚𝑎𝑥 = 99 𝑥 𝑚𝑖𝑛 = 3 𝑹𝒂𝒏𝒈𝒆 = 𝑹 = 𝒙 𝒎𝒂𝒙– 𝒙 𝒎𝒊𝒏 = 99 – 3 = 96 𝑁𝑜. 𝑜𝑓 𝑐𝑙𝑎𝑠𝑠𝑒𝑠 = 𝑚 = 9 𝑾𝒊𝒅𝒕𝒉 = 𝒉 = 𝑹/𝒎 = 96/9 = 10.66 ≅ 11
  • 4. 𝑪. 𝑰 𝑻𝒂𝒍𝒍𝒚 𝑴𝒂𝒓𝒌𝒔 𝒇 𝒙 𝑴𝒊𝒅 𝑷𝒐𝒊𝒏𝒕 𝑪𝒍𝒂𝒔𝒔 𝑩𝒐𝒖𝒏𝒅𝒂𝒓𝒊𝒆𝒔 3 – 13 IIII II 7 8 2.5 – 13.5 14 – 24 IIII IIII 9 19 13.5 – 24.5 25 – 35 IIII I 6 30 24.5 – 35.5 36 – 46 III 3 41 35.5 – 46.5 47 – 57 IIII 5 52 46.5 – 57.5 58 – 68 IIII 5 63 57.5 – 68.5 69 – 79 IIII I 6 74 68.5 – 79.5 80 – 90 IIII I 6 85 79.5 – 90.5 91 – 101 III 3 96 90.5 – 101 50
  • 5. Relative Frequency Cumulative Frequency Cumulative Frequency Distribution Less than More than 0.14 7 2.5- 0 2.5- 50 0.18 16 13.5- 7 13.5- 43 0.12 22 24.5- 16 24.5- 34 0.06 25 35.5- 22 35.5- 28 0.1 30 46.5- 25 46.5- 25 0.1 35 57.5- 30 57.5- 20 0.12 41 68.5- 35 68.5- 15 0.12 47 79.5- 41 79.5- 9 0.06 50 90.5- 47 90.5- 3 1 101.5- 50 101.5- 0
  • 6. HISTOGRAM & FREQUENCY CURVE 0 1 2 3 4 5 6 7 8 9 10 CLASS BOUNDRIES FREQUENCY
  • 8. 0 10 20 30 40 50 60 2.5 13.5 24.5 35.5 46.5 57.5 68.5 79.5 90.5 101.5 LESS THAN CLASS BOUNDRIES FREQUENCY
  • 9. MORE THAN 0 10 20 30 40 50 60 2.5 13.5 24.5 35.5 46.5 57.5 68.5 79.5 90.5 101.5 CLASS BOUNDRIES FREQUENCY
  • 10. 𝒇𝒙 log 𝒙 𝒇log 𝒙 𝟏 𝒙 𝒇 𝟏 𝒙 56 0.903 6.321 0.125 0.875 171 1.2787 11.5083 0.0526 0.4734 180 1.4771 8.8626 0.0333 0.1998 123 1.6127 4.8381 0.0243 0.0729 260 1.7160 8.58 0.0192 0.096 315 1.7993 8.9965 0.0158 0.079 444 1.8692 11.2152 0.0135 0.081 510 1.9294 11.5764 0.0117 0.0702 288 1.9822 5.9468 0.0104 0.0312 2347 77.8447 1.9785
  • 11. ARITHMETIC MEAN 𝒙 = 𝝁 = Ʃ𝒇𝒙 Ʃ𝒇 = 2347 50 A.M = 46.94
  • 12. GEOMETRIC MEAN 𝑮. 𝑴 = 𝑨𝒏𝒕𝒊𝒍𝒐𝒈 𝒇𝒍𝒐𝒈𝒙 𝒇 = 𝐴𝑛𝑡𝑖𝑙𝑜𝑔 77.8447 50 = 36.049
  • 13. HARMONIC MEAN 𝑯. 𝑴 = 𝒇 𝒇 𝟏 𝒙 = 50 1.9785 = 25.2716
  • 14. MEDIAN 𝒙 = ℓ + 𝒉 𝒇 𝑵 𝟐 − 𝑪 𝑁 2 = 50 2 = 25 = 35.5 + 11 3 25 − 22 = 35.5 + 11 𝑥 = 46.5
  • 15. MODE 𝒙 = ℓ + 𝒉 𝒇 𝒎 − 𝒇 𝟏 𝟐𝒇 𝒎 − 𝒇 𝟐 − 𝒇 𝟏 = 13.5 + 11 9−7 2 9 −6−7 = 13.5 + 11 0.4 = 13.5 + 4.4 𝑥 = 17.9
  • 16. QAURTILE 𝑸𝒊 = ℓ + 𝒉 𝒇 𝑵 𝟒 − 𝑪 = 𝑁 4 = 50 4 = 12.5 = 13.5 + 11 9 12.5 − 7 = 13.5 + 1.22 5.5 = 13.5 + 6.722 𝑄𝑖 = 20.22
  • 17. 𝑸 𝟑 = ℓ + 𝒉 𝒇 𝟑𝑵 𝟒 − 𝑪 3𝑁 4 = 3 × 50 4 = 37.5 𝑄3 = 68.5 + 11 6 37.5 − 35 = 68.5 + 1.833 2.5 = 68.5 + 4.5833 = 73
  • 18. DECILE 𝑫 𝟏𝟎 = ℓ + 𝒉 𝒇 𝟑𝟎𝑵 𝟏𝟎 − 𝑪 = 30𝑁 10 = 30×10 10 = 30 = 46.5 + 11 5 30 − 25 = 46.5 + 2.2 5 = 46.5 + 4.84 = 51.34
  • 19. PERCENTILE 𝑷 𝟒𝟓 = ℓ + 𝒉 𝒇 𝟒𝟓𝑵 𝟏𝟎𝟎 − 𝑪 45𝑁 100 = 45 × 50 100 = 22.5 = 35.5 + 11 3 22.5 − 22 = 35.5 + 3.666 0.5 = 35.5 + 1.8333 𝑃45 = 37.333
  • 20. OGIVE : LESS THAN 0 10 20 30 40 50 60 2.5 13.5 24.5 35.5 46.5 57.5 68.5 79.5 90.5 101.5 𝐷_10 𝑃_45 𝑄_1 𝑄3 CLASS FREQUENC Y
  • 21. OGIVE: MORE THAN 0 10 20 30 40 50 60 2.5 13.5 24.5 35.5 46.5 57.5 68.5 79.5 90.5 101.5 𝑃 𝐷 𝑄_3 𝑄1 FREQUENC Y CLASS BOUNDRIES
  • 22. QUARTILE DEVIATION 𝑸𝑫 = 𝑸 𝟑 − 𝑸 𝟏 𝟐 = 73 −20.22 2 = 26.39
  • 23. 𝒙𝒊 − 𝒙 𝒙𝒊 − 𝒙 𝒇 𝒙𝒊 − 𝒙 𝒙𝒊 − 𝒙 𝒙𝒊 − 𝒙 𝒇 𝒙𝒊 − 𝒙 -38.94 38.94 272.5 -38.5 38.5 269.5 -27.94 27.94 251.46 -27.5 27.5 247.5 -16.94 16.94 101.64 -16.6 16.6 99 -5.94 5.94 17.82 -5.5 5.5 16.5 5.06 5.06 25.3 5.5 5.5 27.5 16.06 16.06 80.3 16.5 16.5 82.5 27.06 27.06 162.36 27.5 27.5 165 38.06 38.06 228.36 38.5 38.5 231 49.06 49.06 147.18 49.5 49.5 148.5 1286.92 1287
  • 24. MEAN DEVIATION 𝑴. 𝑫 𝒙 = 𝒇 𝒙𝒊 − 𝒙 𝒇 𝑀. 𝐷 𝑥 = 1286.92 50 = 25.7384
  • 25. MEDIAN DEVIATION 𝑴. 𝑫 𝒙 = 𝒇 𝒙𝒊 − 𝒙 𝒇 𝑀. 𝐷 𝑥 = 1287 50 = 25.74
  • 26. 𝒇 𝒙 𝒙𝒊 − 𝒙 𝒙𝒊 − 𝒙 𝟐 𝒇 𝒙𝒊 − 𝒙 𝟐 𝒙 𝟐 𝒇𝒙 𝟐 7 8 -38.94 1516.3 10614.1 64 448 9 19 -27.94 780.6 7025.4 361 3249 6 30 -16.94 286.96 1721.76 900 5400 3 41 -5.94 35.28 105.84 1681 5043 5 52 5.06 25.6 128 2704 13520 5 63 16.06 257.9 1289.5 3969 19845 6 74 27.06 732.24 4393.44 5476 32856 6 85 38.06 1448.56 8691.36 7225 43350 3 96 49.06 2406.88 7220.65 9216 27648 50 41190.05 151359
  • 27. STANDARD DEVIATION 𝝈 = 𝒇 𝒙𝒊 − 𝒙 𝟐 𝒇 = 4119.05 50 = 823.801 = 28.7
  • 28. STANDARD DEVIATION FROM COMPUTATIONAL 𝝈 = 𝒇𝒙 𝟐 𝒇 − 𝒇𝒙 𝒇 𝟐 = 151359 50 − 2347 50 2 = 3027.18 − 46.94 2 = 3027.18 − 2203.36 = 823.869 = 28.7
  • 29. VARIANCE 𝝈 𝟐 = 𝒇 𝒙𝒊 − 𝒙 𝟐 𝒇 = 41190.05 50 = 823.8
  • 30. VARIANCE FROM COMPUTATIONAL 𝝈 𝟐 = 𝒇 𝒙 𝟐 𝒇 − 𝒇𝒙 𝒇 𝟐 = 151359 50 − 2347 50 2 = 3027.18 − 46.94 2 = 3027.18 − 2203.36 = 823.8
  • 31. 𝒇𝒙 𝒙 𝟐 𝒇𝒙 𝟐 𝒙 𝟑 𝒇𝒙 𝟑 𝒙 𝟒 𝒇𝒙 𝟒 56 64 448 512 3684 4096 28672 171 361 3249 6859 61731 130321 1172889 180 900 5400 27000 162000 810000 4860000 123 1681 5043 68921 206763 2825761 8477283 260 2704 13520 140608 703040 7311616 36558080 315 3969 19845 250047 1250235 15752961 78764805 444 5476 32856 405224 2431344 29986576 179919456 510 7225 43350 614125 3634750 52200625 313203750 288 9216 27648 884736 2654208 84934656 254803968 2347 151359 11157655 87778903
  • 32. MOMENT ABOUT ORIGIN 𝝁 𝟏 = 𝒇𝒙 𝒇 = 2347 50 = 46.94 𝝁 𝟐 = 𝒇𝒙 𝟐 𝒇 = 151359 50 = 3027.18 𝝁 𝟑 = 𝒇𝒙 𝟑 𝒇 = 11157655 50 = 223153.1 𝝁 𝟒 = 𝒇𝒙 𝟒 𝒇 = 877788903 50 = 17555778.06
  • 33. 𝒙𝒊 − 𝒙 𝒇 𝒙𝒊 − 𝒙 𝒙𝒊 − 𝒙 𝟐 𝒇 𝒙𝒊 − 𝒙 𝟐 -38.94 -272.58 1516.3 10614.1 -27.94 -251.46 780.6 7025.4 -16.94 -101.64 286.96 1721.76 -5.94 -17.82 35.28 105.84 5.06 25.3 25.6 128 16.06 80.3 257.9 1289.5 27.06 162.36 732.24 4393.44 38.06 228.36 1448.56 8691.36 49.06 147.18 2406.88 7220.65 0 41190.05
  • 34. 𝒙𝒊 − 𝒙 𝟑 𝒇 𝒙𝒊 − 𝒙 𝟑 𝒙𝒊 − 𝒙 𝟒 𝒇 𝒙𝒊 − 𝒙 𝟒 -59045.64098 -413319.4869 2299237.28 16094680.82 -21811.18218 -196300.6897 609404.4301 548439.871 -4861.163384 -29166.9798 82348.1063 494088.6378 -209.584584 -628.753752 1244.93193 3734.795 129.5542 647.77108 655.544 3277.72126 4142.2530 20711.26508 66524.583 332622.9159 19814.51182 118887.0709 536180.6897 3217084.139 55132.33062 330793.9837 2098336.500 12590019.02 118081.7094 354245.1282 5793088.664 17379265.99 185869.359 55599393.91
  • 35. MOMENTS ABOUT IT MEAN 𝝁 𝟏 = 𝒇 𝒙 − 𝒙 𝒇 = 𝟎 = 0 50 = 0 𝝁 𝟐 = 𝒇 𝒙 − 𝒙 𝟐 𝒇 = 𝝈 𝟐 = 41190.79 50 = 823.8 𝝁 𝟑 = 𝒇 𝒙 − 𝒙 𝟑 𝒇 = 𝝈 𝟑 = 185869.359 50 = 3717.3871 𝝁 𝟒 = 𝒇 𝒙 − 𝒙 𝟒 𝒇 = 𝝈 𝟒 = 55599393.91 50 = 1111987.878
  • 36. COFFICIENT OF MD 𝒙 𝑪𝒐𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝑴𝑫( 𝒙) = 𝑴. 𝑫 𝒙 𝒙 × 𝟏𝟎𝟎 = 25.7384 46.94 = 0.5483
  • 37. COFFICIENT OF MD 𝒙 𝑪𝒐𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝑴𝑫 𝒙 = 𝑴. 𝑫 𝒙 𝒙 × 𝟏𝟎𝟎 = 25.74 46.5 = 0.5535
  • 38. 𝜷 𝟏 = 𝝁 𝟑 𝟐 𝝁 𝟐 𝟑 = 3717.3871 2 823.8 3 = 13818966.85 559068937.3 = 0.0247 COFFICIENT OF SKEWNES
  • 39. 𝜷 𝟏 = 𝝁 𝟒 𝝁 𝟐 𝟐 = 1111987.878 823.8 2 = 1111987.878 678646.44 = 1.638537849 COFICIENT OF KURTOSIS
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