Data-driven marketing is a strategic approach that uses data to understand consumer behavior and preferences, and to create personalized customer experiences**. It involves:

  • Gathering data

    Collecting information from customer interactions, social media, website analytics, market research, and other sources 

  • Analyzing data

    Identifying patterns, trends, and correlations to understand customers 

  • Acting on insights

    Using the insights to create personalized marketing strategies and optimize campaigns 

  • Measuring performance

    Evaluating performance in real time to refine campaigns based on changing customer needs 

Data-driven marketing can help businesses:

  • Create targeted campaigns: Use customer data to create more targeted and effective marketing efforts 

  • Improve customer experience: Enhance the customer experience to foster long-term relationships 

  • Increase ROI: Develop personalized marketing strategies to maximize return on investment 

  • Make better business decisions: Use social media intelligence to predict future audience behavior and drive revenue gains 

Some examples of data-driven marketing include:

  • Product recommendations: Ecommerce sites like Amazon use data to recommend products to customers 

  • Targeted emails: Send emails that appeal to your audience by providing content and information related to their interests 

  • Demographic data: Use demographic data to inform your next marketing campaign.

Data
data=read.csv("Dummy Data HSS.csv")
knitr::kable(data)
TV Radio Social.Media Influencer Sales
16 6.5662308 2.9079828 Mega 54.73276
13 9.2377646 2.4095672 Mega 46.67790
41 15.8864460 2.9134102 Mega 150.17783
83 30.0200283 6.9223040 Mega 298.24634
15 8.4374077 1.4059982 Micro 56.59418
29 9.6143820 1.0271635 Mega 105.88915
55 24.8938115 4.2736021 Micro 198.67982
31 17.3550418 2.2898549 Nano 108.73393
76 24.6488982 7.1301160 Macro 270.18940
13 0.4311278 2.2294230 Mega 48.28058
62 24.3451888 5.1514832 Nano 224.96102
42 15.8073374 3.1949246 Mega 145.54399
64 20.2404239 3.9211480 Micro 229.63238
NA 22.3516671 3.0318153 Mega 276.16535
34 0.2263265 2.3727056 Nano 121.33642
36 3.9001960 0.0604018 Nano 123.52611
60 22.3537208 0.9408018 Mega 212.76440
25 9.9058426 1.8532249 Micro 89.81992
100 36.1160915 3.6742960 Macro 353.80464
92 40.7369897 4.6365639 Nano 329.35054
46 12.6950923 3.8845838 Macro 162.19133
78 35.5682238 2.8324501 Macro 276.40134
37 2.8289725 4.5567012 Micro 128.48466
99 37.2638186 6.8865348 Nano 349.86158
75 20.3489880 1.2271805 Micro 272.25011
54 27.8076140 4.0607820 Nano 192.46191
NA 34.1116736 4.6241483 Nano 342.91337
27 7.2800890 2.1794780 Micro 96.74937
21 9.2826852 3.0557584 Mega 76.99293
40 13.0504108 1.9711848 Nano 139.28879
67 24.1376159 5.5730934 Mega 234.85103
90 29.0210166 6.4319953 Micro 323.03218
12 3.5676607 3.5583114 Nano 40.79220
81 27.3357751 5.2636504 Micro 290.77757
43 16.3487110 2.8140135 Mega 150.62649
47 17.3765256 1.9050403 Mega 161.78515
43 18.2301924 4.0792930 Nano 155.59920
75 23.6262384 6.1997692 Mega 263.02069
79 23.5105707 2.1889261 Macro 281.57894
65 24.5635462 3.4196681 Macro 231.67005
43 12.8421284 2.5936973 Macro 148.70697
82 27.6475854 5.6112218 Macro 294.95242
68 21.6392012 3.2236477 Micro 242.82488
98 37.5709521 8.5296044 Mega 351.77307
26 10.5995325 0.6792753 Macro 91.92230
10 6.1452483 1.5911165 Macro 32.56683
NA 34.8596371 7.7814167 Mega 318.96978
32 0.6276188 2.6760752 Micro 115.26257
29 8.8520944 0.0414943 Macro 101.87310
32 11.5241490 2.5667914 Mega 111.89971
29 14.4341063 0.4632773 Macro 104.35267
91 40.1085273 7.2361705 Mega 326.38171
83 34.2814118 8.6974874 Nano 300.73053
72 23.9226958 3.3738097 Mega 255.15517
47 24.1691319 2.0879103 Micro 161.71709
10 3.4558684 2.4001181 Micro 39.62568
49 10.0676262 1.7296674 Macro 169.00811
14 5.1978562 0.4139660 Mega 50.98848
20 10.7500506 2.4421738 Macro 70.47172
17 10.4474516 3.3561064 Nano 58.59009
68 29.8927411 6.0967437 Mega 237.77216
91 25.0455574 6.1093697 Macro 322.10050
52 17.1132398 2.6386424 Mega 181.07961
41 3.9215400 2.5278506 Nano 145.91920
65 11.6564878 0.3584715 Micro 228.56465
82 31.6076908 3.9524481 Mega 288.98224
14 18.6494418 4.5299271 Mega 49.81567
84 31.2664084 8.0807316 Nano 299.09905
99 30.1405691 6.5461580 Mega 351.35989
15 2.2342934 1.0777183 Mega 55.00971
64 24.6985219 6.7413828 Micro 223.09915
70 15.2477262 4.5877034 Macro 247.13940
77 26.8223500 5.5141872 Nano 274.70710
81 20.3919468 3.2122503 Macro 289.17576
46 17.7901345 4.1176320 Micro 168.03654
NA 6.4822934 0.8668450 Macro 91.17722
56 25.2760870 3.2392652 Mega 198.32152
98 33.0551587 6.0634577 Mega 349.74530
47 18.2339617 0.8275453 Mega 166.13722
18 8.8087028 2.4141863 Nano 59.28932
77 22.9553127 2.6425048 Mega 272.63214
22 6.0417258 2.1366324 Mega 77.41732
99 37.7178052 3.5900864 Nano 349.42397
36 9.0967799 3.8188345 Mega 127.09267
12 1.6481980 1.3057251 Mega 43.06509
94 47.1162934 9.3229578 Micro 336.59342
97 39.2323064 3.6986611 Nano 345.31316
44 16.9120343 4.0346602 Macro 154.20656
86 31.5218973 6.3982836 Mega 304.74377
93 33.0594510 5.7892984 Micro 335.20907
30 10.6451076 1.9155171 Macro 104.18097
32 0.7580667 2.3592002 Mega 112.26969
29 2.1725963 4.2083245 Mega 103.63487
72 25.4065964 3.2154554 Micro 262.40117
14 8.0012659 4.1878343 Micro 51.44298
95 38.4152250 7.6736366 Micro 336.76063
20 3.9347835 0.3284698 Nano 72.62590
49 24.3590443 3.7973673 Nano 174.19814
29 11.2136888 0.5588366 Macro 98.99964
NA 7.6358194 1.5541458 Macro 56.18673
39 13.1872564 2.7663523 Macro 140.41529
87 33.7078982 3.3272727 Nano 309.36167
38 9.4377117 4.3775878 Mega 133.56549
43 9.7267360 0.3379982 Macro 152.25467
30 8.4219947 3.2794806 Macro 112.30811
93 19.9563868 2.2614063 Mega 333.57911
86 25.2607074 6.7656639 Nano 307.61909
63 26.0359863 3.2908169 Nano 225.48623
93 28.0382088 2.9242818 Nano 333.96643
94 32.1462051 4.6109253 Macro 334.64592
63 21.8465439 2.6424102 Macro 222.46831
30 17.4690191 4.3376342 Nano 105.18531
18 7.2919555 2.3179591 Micro 62.31766
24 3.4260212 0.0031768 Micro 86.11554
56 14.8801812 2.3675568 Macro 197.22513
40 17.4288496 3.3067061 Macro 143.83591
74 29.5629510 2.0607197 Micro 263.10563
52 14.5658950 1.9982184 Mega 182.98601
74 20.2208679 4.5409045 Macro 267.73734
NA 30.4704853 6.8069190 Micro 336.81869
70 15.3541893 1.0493696 Micro 250.75139
99 33.8782435 8.0916110 Mega 356.58214
42 15.2152361 1.5014460 Nano 145.21647
85 31.7969383 2.1625024 Mega 303.39739
38 8.8722747 1.6052134 Macro 132.34437
30 15.4326725 3.9865702 Macro 104.50393
49 21.0321804 5.0870170 Micro 174.56411
27 6.4475082 1.4156382 Mega 96.43343
30 4.5613122 1.6527596 Mega 104.15036
100 36.5738082 5.2731562 Mega 354.79737
21 1.3803111 5.2866810 Mega 80.94073
38 18.9036091 4.8899934 Macro 131.10152
39 9.7227460 0.9030604 Micro 135.54432
14 4.8282798 0.2860386 Micro 49.18779
40 18.3784010 1.3200801 Mega 141.38012
33 17.4494755 0.8877616 Macro 119.33081
88 26.0096399 1.4956008 Nano 321.04669
67 28.4818210 1.5208039 Mega 240.40276
65 20.0615558 3.5292742 Nano 226.99893
95 30.7372858 8.9802344 Micro 341.42814
95 36.4223062 9.9974722 Nano 335.30143
NA 9.1644640 1.0966814 Macro 65.25919
77 31.6532001 7.7769784 Nano 273.96038
70 21.4385186 5.7081050 Mega 249.30190
34 16.1936527 0.7922030 Mega 117.05466
59 13.4521760 1.5030822 Macro 208.74951
30 12.1555835 0.9325293 Macro 106.09959
68 22.3896868 1.5061058 Micro 243.08166
43 8.5738645 3.2983622 Micro 156.95375
18 8.4771353 0.6088174 Mega 61.70849
84 33.9556193 4.4798711 Nano 299.34660
61 28.3851872 6.1577046 Micro 222.45076
24 11.8864379 1.5582091 Micro 82.54907
41 8.7152483 0.1814665 Nano 144.59209
67 18.3046411 4.8951719 Micro 234.02118
51 24.0069257 4.1520220 Nano 183.32774
100 36.7341349 7.3597143 Micro 354.10397
44 18.9479550 2.2298138 Micro 160.57964
10 0.7585694 0.5278809 Nano 35.54800
21 1.3973711 1.4445911 Nano 77.54795
32 5.4719556 0.6729645 Micro 110.37486
98 31.3639146 5.6575504 Nano 351.42667
39 4.8561230 0.8741633 Macro 131.07725
NA 38.1184237 6.6766114 Micro 328.55518
40 19.6390631 0.7957259 Micro 146.14569
44 19.0153746 6.5433070 Mega 153.58223
49 24.8710493 5.0362814 Nano 174.98616
78 30.1172653 3.3577941 Macro 278.95515
49 20.4236547 4.3467774 Mega 178.37468
55 22.3747830 5.0665779 Mega 197.01114
73 22.8673761 0.9689842 Nano 261.09088
58 17.3987751 4.0074471 Nano 204.09903
82 23.4347871 5.9148661 Micro 288.02075
27 10.3335305 1.1301225 Macro 97.61236
94 29.3446099 3.4364810 Mega 336.37262
69 23.2904511 5.6716861 Mega 245.97996
94 37.0970552 7.5962905 Nano 334.64379
70 23.8085684 3.7666030 Micro 246.49480
88 27.4614934 1.5240621 Macro 312.14296
64 23.0814880 3.0562981 Macro 219.56818
39 11.7825426 2.9567268 Micro 138.37531
99 37.7324114 7.7240527 Macro 349.04834
81 26.4254219 NA Macro 288.64944
NA 1.2870605 0.3961787 Macro 56.54529
25 0.4138494 NA Macro 92.35709
94 30.4981569 8.8233119 Macro 331.48409
73 25.3402089 NA Mega 258.35897
36 8.8401125 0.9629417 Nano 125.38706
78 26.4067649 2.4751016 Macro 271.02644
89 29.6823838 NA Macro 320.26440
44 6.3131734 1.3100279 Macro 152.83125
79 27.8832065 7.4679417 Nano 282.74810
41 13.1426574 NA Nano 142.62686
26 15.2667331 2.2058799 Macro 88.96783
17 8.1979325 2.1198723 Mega 63.25378
34 12.6603982 NA Mega 117.11414
13 8.0775016 0.0132301 Nano 48.30375
98 NA 6.3997296 Nano 345.95284
12 3.5565766 0.3600149 Micro 37.24327
11 3.7852547 1.5696446 Micro 45.55251
89 NA 4.5354580 Micro 316.72562
39 13.2212374 3.6605661 Micro 135.77315
86 27.1481728 4.4093994 Nano 308.39305
22 NA 4.1325256 Macro 78.03150
76 26.1926733 4.1276009 Macro 269.60301
33 13.3658674 0.0121384 Macro 116.23750
12 NA 1.2300262 Micro 50.00921
83 28.4091534 4.0852161 Micro 292.85717
77 25.5983792 5.5147866 Nano NA
37 10.2529162 0.8653287 Nano 135.53008
NA 17.0070746 5.1996348 Macro 208.81638
92 16.7037689 1.8884033 Macro 325.51371
100 32.9201940 9.9042861 Nano 357.12457
94 30.3727060 5.9461129 Mega 334.08924
100 36.4667534 5.6359925 Mega NA
27 3.4388245 0.7495188 Mega 97.50796
53 19.0445033 2.7150957 Macro 191.61804
73 28.4645976 5.5456844 Mega 256.42523
78 26.3940701 3.8373699 Mega 279.59064
59 25.6872865 6.9031092 Macro 209.89565
44 15.7263398 5.4703166 Mega 159.63094
74 24.2206336 0.2858976 Micro NA
82 33.6746953 8.2430451 Nano 290.01985
47 14.1623244 4.3881145 Nano 170.12619
22 3.5984743 4.3256205 Micro 78.81257
78 24.8902829 3.5310933 Macro 277.09407
19 8.7267827 0.4978385 Mega NA
15 9.8096473 2.1750992 Macro 51.15174
78 36.1347793 6.8125289 Micro 275.81541
63 19.1020433 4.7730497 Micro 225.79172
45 12.2273996 1.3060538 Nano 162.30391
22 6.8097517 0.0114505 Macro NA
31 8.8439246 0.1048109 Micro 111.12775
23 2.2017524 0.3765479 Nano 80.45444
19 4.8944107 0.1471960 Macro 69.99071
40 11.9078925 1.9580362 Nano 138.67700
27 1.3844155 2.3981292 Nano NA
35 5.3926243 0.0206400 Macro 118.50116
63 20.7746817 6.2591533 Micro 229.09032
78 23.8095914 1.9150471 Nano 276.91906
91 22.2767137 3.2900798 Mega 322.66777
44 16.6093140 5.1590666 Mega 157.49007
16 12.2513576 1.4631182 Mega 56.81534
40 12.7084063 5.8382519 Nano 139.79239
60 21.4388896 3.1477514 Nano 212.16912
49 16.3190367 0.4633115 Micro 177.50255
71 26.0239494 3.1164285 Mega 253.98611
59 21.2729121 3.3851215 Nano 215.90548
92 33.2958596 1.2294542 Macro 327.65364
60 19.0155872 5.8951653 Mega 214.45761
28 10.9320086 1.8676461 Nano 97.20055
29 9.1970323 3.2466677 Micro 99.10934
97 31.7070101 4.3651667 Nano 347.15476
11 9.3988392 3.2065085 Mega 41.35123
49 13.7619832 2.5026530 Nano 173.15509
38 4.3388430 0.0280732 Mega 139.83436
18 12.3844781 1.6371575 Nano 65.12530
42 10.0190347 0.1910937 Nano 152.94960
76 32.3192637 6.4945179 Mega 270.63785
86 29.9196270 6.3129030 Mega 303.60965
70 30.4738874 5.7043604 Nano 248.05148
69 24.3624606 5.4631760 Macro 239.73289
47 14.5951699 0.5732358 Nano 162.14795
46 17.8748317 3.9189419 Micro 167.53323
98 33.5556744 5.4017153 Micro 343.70471
63 18.3172130 5.1064101 Mega 222.45381
66 15.8100767 0.4951697 Mega 229.55825
76 21.4302502 3.6083514 Nano 269.03733
12 12.8482798 2.8103753 Nano 42.65152
75 38.2714320 8.5864502 Micro 263.52381
85 25.2418171 5.9695380 Nano 296.24860
12 3.1886683 0.2761359 Nano 46.84860
90 33.5117751 4.6635268 Macro 322.46976
88 35.4551909 4.0913966 Macro 312.31077
35 18.8147265 3.0414687 Mega 123.95214
89 26.1180611 4.0200191 Nano 320.58830
54 22.4500150 3.3894996 Nano 188.82864
16 1.0950547 1.7847117 Mega 55.33417
98 32.0790944 4.1583388 Mega 342.65223
23 7.9897874 0.5089551 Mega 84.70734
93 33.7474266 2.7980532 Mega 333.07737
52 17.4528804 3.9098536 Nano 188.11638
27 8.3013931 0.2977357 Mega 98.16192
13 5.3588217 3.1035571 Nano 47.50814
83 21.9360125 4.4362596 Micro 294.49334
66 21.5782335 8.8742376 Macro 231.99429
19 7.7895072 4.0678220 Macro 62.07952
85 23.7734612 4.5426610 Mega 304.90422
63 19.7190662 5.4819152 Nano 220.70613
62 25.0790811 5.0661012 Nano 225.46953
45 15.5373290 0.1258132 Mega 157.68730
23 6.6216353 0.1136312 Mega 79.93999
50 11.3639870 2.3212959 Mega 185.31029
60 19.4999565 4.9123362 Macro 216.26816
25 13.0045515 5.2771244 Micro 91.95955
31 12.2259659 0.3972073 Macro 111.14311
77 21.0319654 1.8672881 Micro 275.38867
27 9.2358457 2.9632095 Nano 93.35176
77 25.5834396 3.8709919 Macro 276.08390
95 25.5463677 1.9634901 Micro 333.36825
48 25.3772310 3.9970279 Micro 172.88546
14 4.6431757 0.6373758 Micro 53.97461
24 4.0143131 1.2589688 Macro 91.49657
65 23.2903100 7.3898277 Mega 229.19697
35 9.5314522 0.1035036 Nano 125.33934
86 24.2932031 8.0412123 Micro 306.60659
48 14.4187533 3.1174135 Nano 170.28332
59 19.3408643 1.6023405 Micro 210.57177
88 26.3006566 4.2040399 Nano 314.18055
78 23.3224512 3.8667395 Mega 280.83162
79 31.0574086 3.8069528 Micro 286.67751
22 1.0606444 1.0938823 Macro 73.09319
46 17.6133302 3.4804297 Macro 160.61509
83 23.7789827 2.9306812 Micro 295.49153
26 12.1053762 4.0074347 Macro 94.00246
34 2.0899999 0.2177521 Nano 123.12193
50 3.4972003 3.3370714 Nano 174.74503
79 19.1558441 0.0147119 Macro 281.59289
75 25.2719440 7.7171053 Nano 267.80361
32 13.4574603 0.9176561 Mega 116.78384
29 9.4879936 0.2504328 Micro 105.77671
96 36.7230774 3.9503926 Nano 343.08775
74 27.9632954 2.8375850 Nano 258.37543
49 20.2658572 3.4461326 Micro 171.85667
34 16.8567829 3.8860053 Mega 118.91638
37 12.2287160 1.7492901 Mega 129.01484
37 8.8235251 3.4058395 Mega 132.54961
59 15.2920501 3.3404362 Macro 208.27445
48 14.1156171 3.3283784 Mega 175.54616
80 24.5546348 4.8323258 Micro 285.59623
28 5.3137065 2.4735988 Nano 91.49753
64 25.5282825 3.1408700 Nano 226.27696
50 21.1748042 4.4864045 Macro 180.31954
46 18.5484213 2.4555829 Nano 164.75620
86 24.7862440 6.3499623 Mega 309.20433
69 29.5944021 8.2437150 Nano 246.09152
85 29.5010222 2.5182221 Mega 300.55845
52 9.8375925 2.0801822 Macro 183.71187
54 17.4430453 5.5492195 Nano 191.00347
17 2.2492038 0.3951503 Nano 61.86531
93 26.5028165 3.5376186 Mega 331.64786
85 24.4839792 5.9189856 Micro 297.05649
21 1.5059433 2.4389042 Mega 73.87342
88 37.9692970 4.7396822 Macro 314.31137
75 29.4842347 7.2377840 Nano 268.79978
38 3.0463088 1.7436294 Macro 132.32604
95 36.3657020 1.9446649 Macro 339.11528
46 25.2819994 2.7987575 Mega 163.63962
75 26.0754986 5.1437064 Mega 269.44499
64 10.9308993 3.0853751 Mega 225.06405
35 8.9784759 0.0536270 Nano 125.32892
30 6.1223179 2.6068065 Mega 104.52468
86 27.7380058 5.2935404 Macro 307.89333
79 23.1834530 0.7621600 Nano 282.71085
16 16.9241711 6.4474763 Nano 54.65467
63 22.8160751 4.7170421 Nano 228.42413
78 22.7681093 2.9641017 Micro 277.39933
24 12.7944845 3.9731630 Nano 86.22926
34 14.4453545 1.2702840 Nano 124.59571
36 14.7586798 0.8074131 Micro 125.07369
64 17.6171414 5.1166769 Micro 228.54247
79 37.3790367 4.7675272 Macro 281.11479
11 7.2283544 5.2501819 Micro 46.52970
22 5.1208168 2.2052997 Micro 74.74160
46 3.1598299 4.2618097 Nano 161.18295
49 15.4455226 0.0624101 Mega 174.55676
39 8.4379645 2.9263669 Mega 141.59009
38 0.8572806 0.2187394 Nano 137.69737
23 4.0379142 2.8765055 Micro 78.65146
17 8.4826979 0.9461485 Nano 58.92532
57 17.5093738 3.9129105 Macro 204.45338
24 8.8226046 0.3883146 Mega 86.78603
38 13.9738116 2.9372555 Nano 133.38763
55 13.2367514 0.6039183 Mega 199.13170
84 27.9335234 3.9785259 Macro 300.56783
23 3.4244220 1.2548873 Nano 82.71103
25 21.5716003 0.1825941 Micro 89.09325
61 24.0267015 4.4685400 Macro 214.50674
92 36.2813941 7.6930239 Mega 334.25482
52 14.7187578 5.0679181 Micro 184.28327
42 25.0434430 7.9541628 Macro 151.13023
87 26.2634461 4.6662544 Mega 314.08678
23 3.5110585 1.6686675 Nano 82.12966
16 1.1142156 0.1243055 Micro 55.74976
97 31.5975858 6.0961670 Macro 345.60024
83 22.1231839 4.5484734 Mega 301.02134
32 7.2351697 4.1659782 Micro 114.22255
81 30.5167855 4.2537659 Mega 286.33617
81 35.7440086 8.4855793 Mega 290.19918
32 9.4450321 0.9047104 Nano 115.35924
24 11.0410077 0.5818032 Nano 82.90997
93 28.2746577 2.4041387 Mega 328.32173
52 20.5974419 5.6875173 Mega 183.61200
21 2.0277184 0.5427714 Nano 70.90877
15 7.8882108 3.9983621 Macro 53.54401
62 19.9397494 1.5816684 Mega 222.46925
45 14.7651467 4.9737305 Mega 159.81160
84 29.2810358 4.1823465 Micro 300.75393
12 8.4667090 4.0765836 Mega 41.86992
66 23.1625976 2.0513335 Mega 238.37144
71 21.6847852 4.3628890 Mega 253.46133
30 2.7631619 1.4912920 Nano 108.70799
96 36.1367947 8.7243524 Nano 340.79798
81 36.0106228 9.3079830 Micro 291.87025
61 13.6238392 1.0816300 Mega 218.79674
88 31.2965668 6.2685637 Nano 312.23363
95 25.7217025 3.6539188 Micro 338.72657
32 17.7787456 6.0107966 Macro 108.83527
23 3.1591491 1.8380384 Mega 84.46858
19 11.8430838 1.4117445 Macro 63.78917
99 35.3081482 8.2336288 Macro 347.88214
21 17.7284981 1.3166703 Micro 72.79319
83 22.6476699 1.8153031 Macro 294.37293
34 9.5290389 0.0463245 Macro 122.39803
65 21.8286820 3.8523649 Macro 226.06810
42 11.6668523 3.1241142 Nano 147.67032
28 11.7428859 1.5457905 Nano 108.65424
84 23.1588378 6.4933950 Macro 301.97969
74 31.2006318 5.9561476 Macro 262.00001
58 31.4993678 5.3744368 Micro 203.05789
63 21.7928057 0.3935890 Nano 223.84189
46 21.9598093 0.8805932 Micro 164.79101
51 24.9164862 4.6282894 Mega 180.37461
42 9.3529637 0.6061266 Mega 146.35779
98 31.2897755 4.6233061 Nano 343.94009
18 10.1378144 3.1819717 Macro 66.54840
72 19.4643893 1.7675311 Nano 258.01784
90 29.9419018 8.3489981 Mega 320.71534
47 13.8547061 3.7236195 Macro 167.08740
11 15.2596558 1.9380912 Micro 42.71996
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25 9.7880569 0.5036825 Macro 92.84837
43 12.5124372 2.4963334 Mega 149.23243
55 12.2069595 5.3851056 Micro 196.45693
97 38.1569212 8.1046440 Micro 338.26148
32 3.5237504 0.0284310 Macro 115.12200
71 17.5505090 4.0414514 Macro 252.37321
66 24.5200668 7.7894710 Macro 235.62250
58 17.1100988 1.2342604 Nano 211.99273
64 20.3620385 1.5276418 Macro 228.78367
73 20.3651267 4.0103555 Nano 259.57726
45 10.5228612 1.8555802 Micro 162.75409
78 20.7240078 5.5034998 Macro 274.79331
54 23.6576358 3.8469972 Mega 186.28532
15 5.9930828 0.0529006 Macro 54.19133
81 34.2589880 2.9334852 Mega 286.82380
16 5.3819210 1.3040314 Nano 54.26762
11 0.4016489 1.1607680 Nano 35.10151
50 22.7281766 3.7941203 Mega 184.04021
41 12.2617186 1.6229137 Mega 146.65952
41 11.5335511 0.5067401 Macro 148.13004
54 22.2813146 1.4938313 Nano 190.97754
28 6.0086560 2.7894988 Micro 100.80504
23 7.6965239 0.0889229 Macro 84.19134
13 12.3724134 6.5158271 Micro 46.12760
26 0.2056395 3.9803178 Mega 90.03771
98 32.9706347 7.8909097 Micro 349.06383
64 22.6932307 0.3496204 Micro 226.01791
24 8.5267784 1.8727407 Nano 83.14711
38 10.8333665 0.8307509 Nano 137.66056
41 17.5067939 1.2585082 Micro 148.06008
32 16.2904663 5.2128561 Mega 112.85829
75 21.5369552 1.2324236 Micro 264.99899
82 28.9204304 3.2130668 Micro 285.39214
12 2.5953162 0.3341691 Mega 44.02357
91 30.7294123 8.1039854 Nano 327.00559
30 6.7672661 2.4553748 Nano 110.21890
36 15.3930559 3.6011620 Nano 119.22071
72 24.3254729 3.6713703 Nano 259.99271
90 34.6183590 4.3617351 Nano 321.95734
65 8.8802655 1.0101249 Macro 231.37021
91 27.7078343 3.1691812 Mega 326.51199
95 27.5364328 8.9813259 Macro 336.43185
13 4.0746717 2.9698461 Nano 48.53912
77 25.1213177 5.5181153 Nano 272.53678
100 26.4520153 3.4385020 Nano 353.74701
59 15.3520850 0.2014421 Mega 210.34935
53 23.7893674 2.3634669 Micro 187.81291
14 0.9363491 0.3893045 Mega 47.04218
71 26.9205593 5.6250464 Micro 248.16053
18 4.8555495 2.0780035 Mega 64.19338
21 17.9170008 4.6557226 Micro 77.38880
87 35.1061072 7.5941338 Mega 312.35126
65 23.2050652 2.0704536 Nano 229.99870
67 18.2241086 3.2498784 Nano 233.50762
72 23.9201259 4.5955661 Micro 260.14327
49 13.3862938 0.8121753 Mega 171.44530
67 20.0109313 6.3733115 Micro 239.52581
44 11.6834720 0.3913814 Nano 155.33766
98 39.5276035 3.7786635 Mega 347.98786
46 8.3371106 0.7994691 Mega 162.37147
60 20.7891157 7.6011707 Mega 211.21081
61 20.7895442 2.4739382 Macro 225.82215
23 9.6446103 2.9801511 Micro 80.44199
12 3.8125784 0.3463778 Nano 41.90671
28 14.9275791 4.4414203 Mega 98.41240
26 15.3934094 2.0250594 Nano 91.25693
59 19.6343774 3.4606425 Mega 214.53822
57 21.0660924 1.3677077 Mega 199.47833
77 24.5090091 5.5155984 Mega 272.54901
32 16.9784932 3.0377091 Mega 119.22348
89 26.2371939 2.7296470 Macro 316.05898
67 30.7136045 3.6445509 Micro 242.65199
23 7.1525988 2.8537515 Mega 84.82326
81 29.2563413 6.1742865 Mega 290.43890
28 7.5151690 1.3845536 Mega 101.45291
34 9.1030245 2.0794360 Macro 124.76251
94 33.4164161 4.9950884 Mega 333.05504
30 15.0037342 1.2584130 Macro 104.95492
81 24.0051528 5.3282891 Micro 288.92350
66 14.5609133 6.5555281 Nano 234.68489
60 23.5291290 2.9157143 Nano 214.56067
43 13.1122320 1.4823395 Mega 151.69210
71 25.1967662 8.0509980 Mega 249.57775
42 14.7528840 5.1931925 Mega 153.40227
36 20.6338737 3.4427149 Micro 129.52272
90 25.3298962 4.3015066 Nano 318.95923
37 11.4274043 0.0145941 Macro 130.26341
11 2.5656489 2.0352968 Mega 37.59293
10 3.4936455 0.4553956 Nano 32.82551
14 4.2483464 2.6791459 Micro 50.93200
33 1.0292642 1.5126970 Macro 112.31541
31 6.3048453 0.8022797 Micro 110.43937
77 17.3313826 0.4144728 Micro 272.78122
60 16.6884177 2.9464579 Macro 211.47915
54 21.0192920 4.4587442 Macro 194.38727
41 10.0061917 2.9103359 Mega 146.98446
82 28.9836200 5.7782609 Mega 292.95789
40 4.5889349 2.2987536 Mega 141.71064
65 18.7429314 7.6437301 Mega 233.29271
26 9.1750472 0.9067207 Micro 92.18424
88 31.6481837 2.0219446 Nano 311.91197
85 32.7858424 3.2532164 Macro 303.06067
77 23.0779622 2.2156389 Macro 273.26031
45 12.9876769 2.9773297 Mega 158.53877
16 5.3319428 0.3263016 Macro 59.78506
35 13.4620270 1.4288429 Nano 119.12176
27 15.9640731 2.8073014 Macro 98.35031
36 10.8604315 0.8987131 Micro 130.90075
32 10.3989144 1.1748287 Mega 115.49000
64 23.6214026 3.3706046 Nano 226.50812
63 19.2960734 2.4826204 Nano 226.56921
23 10.6907255 3.6357555 Nano 84.80777
31 17.0743379 4.8192763 Mega 109.59430
40 12.1535313 4.0605355 Micro 144.98588
72 17.9103839 1.8641675 Micro 255.41990
43 10.5012716 0.5459578 Micro 153.88754
34 20.2724833 4.3408422 Mega 122.29171
11 7.3985274 1.3131584 Micro 37.99292
44 22.6888972 1.2090242 Mega 161.10697
51 14.5983348 1.9868408 Micro 178.52264
16 8.8163207 1.7827896 Nano 53.21425
14 17.1393303 1.2339746 Mega 46.91758
77 27.9060964 4.1449172 Macro 277.73341
44 19.0028474 4.3620363 Mega 154.23303
65 28.9453671 3.9959540 Mega 235.60333
60 12.8405937 2.4829083 Nano 214.78664
37 10.5269893 0.8303112 Macro 134.40925
79 24.0631302 2.1732796 Nano 284.70874
11 3.3357334 1.3147621 Mega 39.78812
11 2.6111005 0.0101887 Micro 37.72268
67 23.6283028 4.8053572 Mega 235.92043
53 24.7022264 0.1694673 Mega 192.12491
43 14.2341046 7.5569285 Nano 154.99818
42 13.6820266 1.1535916 Macro 151.73200
61 18.0879436 3.1242336 Micro 217.39245
60 13.9930063 2.4102184 Macro 210.95922
60 15.5698130 0.5399631 Macro 219.14852
18 9.8454508 3.2526284 Mega 65.27424
66 16.7867864 3.7874612 Micro 238.71154
88 35.5445611 7.5225552 Macro 309.90390
85 24.1811044 2.5425312 Mega 301.99412
55 23.7184621 4.2228978 Mega 194.20773
12 6.2409496 2.1948137 Mega 41.71037
41 6.7239931 2.7627726 Nano 149.34562
14 0.0218826 0.3011501 Micro 53.70202
60 21.8418643 5.0925283 Macro 210.68002
54 26.0775975 3.8081787 Macro 188.72320
93 25.2851489 2.8058396 Macro 327.46629
99 36.0241739 4.2887550 Macro 355.80712
19 1.4901923 1.4856841 Micro 66.39380
16 2.3392222 0.5631206 Mega 52.42380
26 4.4723600 0.7170903 Micro 94.68587
71 20.6106854 6.5455731 Nano 249.10192
44 19.8000724 5.0961919 Micro 163.63146
71 17.5346403 1.9408732 Macro 253.61041
42 15.9666875 5.0465476 Micro 148.20241

The data contains 4572 observations/rows and 5 columns/variables. Following are the columns (variables):

  • TV promotion budget (in million)

  • Social Media promotion budget (in million)

  • Radio promotion budget (in million)

  • Influencer: Whether the promotion collaborate with Mega, Macro, Nano, Micro influencer

  • Sales (in million)

Variable Sales is a decision variable (Y) and other variables are predictor variables (Xs). In this case study we will predict Sales based on given values of predictors from a machine learning model.

library(skimr)
skim(data)
Data summary
Name data
Number of rows 4572
Number of columns 5
_______________________
Column type frequency:
character 1
numeric 4
________________________
Group variables None
Variable type: character
skim_variable n_missing complete_rate min max empty n_unique whitespace
Influencer 0 1 4 5 0 4 0
Variable type: numeric
skim_variable n_missing complete_rate mean sd p0 p25 p50 p75 p100 hist
TV 10 1 54.07 26.13 10.0 32.00 53.00 77.00 100.00 ▇▇▇▇▇
Radio 4 1 18.16 9.68 0.0 10.53 17.86 25.65 48.87 ▆▇▇▃▁
Social.Media 6 1 3.32 2.21 0.0 1.53 3.06 4.81 13.98 ▇▆▂▁▁
Sales 6 1 192.47 93.13 31.2 112.32 189.23 272.51 364.08 ▇▇▇▇▆ 
library(ggplot2)
library(GGally)

ggpairs(data[,-4])

There are only 26 missing values which is 0.57 % of total observations. We can remove the missing values.

data=na.omit(data)
sum(is.na(data))
[1] 0

TO build a machine learning model, we need to meet certain assumptions. One of the assumptions is that data should be normally distributed. We can check the normality of dat from density plots and statistics like skewness and kurtosis.

Let’s check the distribution of each variable:

Budget: TV
Budget: Radio
Budget: Social Media
Sales

We can observe that the skewness of each variable is between +1 and -1 hence we will not transform these variables.

We can see that the Influencer variable is a character variable but it carries the 4 levels repeating. We need to transform this variable into a factor variable.

data$Influencer=as.factor(data$Influencer)

table(data$Influencer)

Macro  Mega Micro  Nano 
 1112  1152  1148  1134

Machine learning model building starts with splitting data into train and validation sets. We divide the data into 80:20 split. The 80% data will be used to train the machine learning model and 20% data will be used to evaluate the machine learning model.

set.seed(1234)

s=sample(1:nrow(data), replace = FALSE, 0.8*nrow(data))

train=data[s,]

val=data[-s,]

paste("No. of observations in train set: ", nrow(train))
[1] "No. of observations in train set:  3636"
paste("No. of observations in val set: ", nrow(val))
[1] "No. of observations in val set:  910"
paste("No. of observations in original data: ", nrow(data))
[1] "No. of observations in original data:  4546"

To train the machine learning model, we need to choose machine learning algorithm. The selection of machine learning algorithm depends on nature of predictors (Xs) and decision variable (Y). In this case, we have four predictors - TV, Radio, Social.Media and Influencer. The decision variable (Y) is Sales.

The predictors- TV, Radio, Social.Media and Sales are numeric variables measured in USD millions. The variable - Influencer is a factor variable.

We can select Multiple Linear Regression for building machine learning model to estimate a numeric variable. The decision variable - Sales is a numeric variable and we will predict Sales through our model.

\[ y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \cdots + \beta_kx_k + \epsilon \] In the equation, \(y\) is decision variable, \(\beta_0\) is an Intercept, \(\beta_0,\beta_1,...\) etc. are regression coefficients for corresponding predictor (\(X_1,X_2,...\)) respectively and \(\epsilon\) is an error term.

The \(\beta\) coefficients well be estimated through linear multiple regression algorithm. The \(\epsilon\) is calculated by finding the difference between actual \(y\) and predicted \(\hat{y}\). Formula for error term is as follows:

\[\epsilon_i = y_i - \hat{y}_i = y_i - \left(\beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \dots + \beta_p x_{ip}\right)\]

We will build machine learning model with the train set. After building base model, we will test the model on val set. Follow the command chunk:

model=lm(Sales ~ ., data= train)
summary(model)

Call:
lm(formula = Sales ~ ., data = train)

Residuals:
     Min       1Q   Median       3Q      Max 
-11.5341  -2.0552  -0.0018   2.0543  11.2942 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)     -0.077105   0.145907  -0.528    0.597    
TV               3.563706   0.003827 931.252   <2e-16 ***
Radio           -0.011349   0.011008  -1.031    0.303    
Social.Media     0.025227   0.028093   0.898    0.369    
InfluencerMega  -0.023436   0.140649  -0.167    0.868    
InfluencerMicro -0.130672   0.140244  -0.932    0.352    
InfluencerNano  -0.152569   0.139723  -1.092    0.275    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.975 on 3629 degrees of freedom
Multiple R-squared:  0.999, Adjusted R-squared:  0.999 
F-statistic: 5.916e+05 on 6 and 3629 DF,  p-value: < 2.2e-16
Interpretation

Write interpretation on your own.

  • Significant Variables

  • Insignificant Variables

  • Model Parameters (Adjusted R-squared, F- Statistic, Degree of Freedom, p-value and standard error.)

We can evaluate machine learning model in two ways:

    1. comparing the predicted \(\hat{y}\) with actual \(y\) values of the val set. For this, we will use Root Mean Squared Error (RMSE). We did not use the val set while training machine learning model but we will use it for evaluating the model. Formula for RMSE is:

\[\text{RMSE} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2}\] Here,

\(y_i\) : The actual value for the \(i\)-th data point. \(\hat{y}\) : The predicted value for the \(i\)-th data point. \(n\) : The total number of observations. \((y_i - \hat{y}_i)^2\) : The square of difference between the actual and predicted values.

The RMSE should be as minimum as possible for a good machine learning model.
    1. using model diagnostic plots. We will be using four different types of plots to evaluate model.
Predicting with Model
preds= predict(model, newdata = val)
library(Metrics)
rmse(val$Sales, preds)
[1] 2.855351

The RMSE value is 2.86 which is quite low. This indicates that model is a good fit. We can compare the actual Sales and Predicted Sales by creating following table:

pred_data=data.frame(Sales=round(val$Sales, digits = 2), 
                     Predicted_Sales=round(preds, digits = 2), 
                     Diff=round(preds-val$Sales, digits = 2))
Actual Sales vs Predicted Sales
knitr::kable(pred_data)
Sales Predicted_Sales Diff
12 145.54 149.48 3.93
19 353.80 355.98 2.17
21 162.19 163.81 1.62
31 234.85 238.53 3.68
46 32.57 35.53 2.96
55 161.72 167.06 5.35
63 181.08 185.08 4.00
73 274.71 274.01 -0.70
77 198.32 199.26 0.94
78 349.75 348.92 -0.82
80 59.29 63.88 4.59
81 272.63 274.11 1.48
85 43.07 42.68 -0.39
86 336.59 334.48 -2.11
87 345.31 345.10 -0.22
89 304.74 306.18 1.44
92 112.27 113.99 1.72
93 103.63 103.33 -0.31
94 262.40 256.17 -6.23
104 152.25 153.06 0.81
106 333.58 331.15 -2.42
109 333.97 330.95 -3.02
113 62.32 63.91 1.60
128 96.43 96.08 -0.35
136 119.33 117.35 -1.98
140 341.43 338.22 -3.21
149 156.95 153.02 -3.94
150 61.71 63.97 2.26
153 82.55 85.23 2.68
158 160.58 156.44 -4.14
170 197.01 195.78 -1.23
177 334.64 334.53 -0.11
178 246.49 249.08 2.58
200 45.55 38.99 -6.56
202 135.77 138.72 2.95
205 269.60 270.57 0.97
219 279.59 277.67 -1.92
226 277.09 277.70 0.60
236 138.68 142.23 3.56
243 56.82 56.82 0.00
258 152.95 149.34 -3.61
266 222.45 224.33 1.88
269 42.65 42.46 -0.19
273 322.47 320.39 -2.08
283 98.16 96.03 -2.13
285 294.49 295.44 0.95
286 231.99 235.11 3.11
295 91.96 88.87 -3.09
302 53.97 49.65 -4.33
311 286.68 281.07 -5.61
313 160.62 163.74 3.13
321 105.78 103.04 -2.74
325 118.92 120.97 2.06
327 132.55 131.74 -0.81
329 175.55 170.88 -4.67
334 164.76 163.55 -1.20
336 246.09 245.54 -0.55
339 191.00 192.15 1.15
343 73.87 74.78 0.91
345 268.80 266.90 -1.90
348 163.64 163.61 -0.03
360 125.07 127.94 2.86
361 228.54 227.80 -0.74
362 281.11 281.15 0.04
371 204.45 202.95 -1.50
384 55.75 56.80 1.05
390 115.36 113.72 -1.63
401 253.46 252.79 -0.67
405 218.80 217.16 -1.64
416 147.67 149.39 1.72
420 203.06 206.27 3.21
421 223.84 224.05 0.20
428 320.72 320.50 -0.21
429 167.09 167.35 0.27
430 42.72 38.87 -3.85
433 178.30 181.55 3.25
434 173.73 174.43 0.70
446 210.40 210.02 -0.38
447 311.61 313.27 1.66
449 201.16 195.70 -5.45
450 80.79 81.63 0.84
455 71.40 74.69 3.29
456 52.04 53.40 1.36
458 153.66 153.05 -0.61
461 355.05 355.87 0.82
462 220.61 220.77 0.16
469 242.59 238.64 -3.94
472 291.14 288.29 -2.86
477 319.44 320.48 1.05
488 154.23 152.89 -1.33
489 240.23 242.08 1.85
492 210.91 213.62 2.72
509 355.52 356.10 0.58
524 264.54 263.48 -1.07
526 313.26 313.35 0.09
530 179.73 177.89 -1.84
532 33.51 42.71 9.20
534 212.60 213.44 0.83
550 319.45 316.85 -2.60
553 349.63 345.30 -4.33
562 351.81 352.48 0.67
563 227.32 227.84 0.53
564 253.94 249.21 -4.73
574 61.87 60.31 -1.56
581 292.09 295.52 3.43
583 241.24 241.98 0.74
590 340.84 345.28 4.43
597 180.76 181.47 0.72
601 297.07 298.89 1.82
607 347.68 345.24 -2.44
618 267.74 266.82 -0.93
631 217.44 217.09 -0.35
642 207.14 206.42 -0.71
645 117.45 117.39 -0.06
652 151.07 149.44 -1.63
663 166.62 160.09 -6.53
667 61.85 63.87 2.02
669 147.86 149.57 1.71
672 97.14 99.72 2.59
673 107.21 110.35 3.14
676 150.53 149.49 -1.04
681 287.90 284.66 -3.24
689 129.64 131.75 2.12
692 280.50 277.68 -2.82
701 166.75 167.35 0.60
715 259.25 259.84 0.58
718 145.41 145.87 0.46
724 176.95 174.25 -2.70
727 297.36 295.49 -1.87
737 192.52 192.21 -0.31
744 158.86 160.15 1.28
747 124.08 124.62 0.54
748 298.90 298.95 0.05
750 159.73 160.06 0.33
760 155.37 156.65 1.28
764 219.55 220.75 1.21
769 176.59 178.08 1.49
773 217.23 213.34 -3.89
788 236.76 235.08 -1.67
792 344.93 341.66 -3.27
804 169.40 163.85 -5.55
808 264.24 263.44 -0.80
822 240.27 231.29 -8.98
825 228.82 231.37 2.55
826 39.16 35.58 -3.58
830 147.92 149.55 1.63
835 127.06 124.52 -2.54
838 217.37 217.10 -0.27
840 271.23 263.28 -7.95
846 128.75 128.02 -0.73
848 283.08 284.61 1.52
852 295.52 291.91 -3.61
855 190.46 188.68 -1.78
858 74.81 78.24 3.43
863 62.50 60.50 -2.00
870 266.51 266.99 0.48
893 342.58 348.87 6.30
894 237.99 238.42 0.43
904 349.21 348.72 -0.49
906 156.98 153.11 -3.87
913 283.98 284.82 0.84
914 69.01 74.65 5.64
916 192.11 188.48 -3.63
918 41.02 39.12 -1.90
921 257.39 256.34 -1.05
923 207.72 209.88 2.16
927 193.56 195.74 2.18
931 73.43 74.74 1.31
932 286.10 295.40 9.31
940 189.61 188.65 -0.96
942 128.07 124.66 -3.41
949 307.34 309.62 2.28
955 203.41 206.30 2.89
958 238.60 242.13 3.53
959 140.53 142.22 1.69
961 269.62 274.03 4.41
964 199.56 199.19 -0.36
970 75.27 78.22 2.96
976 237.65 241.97 4.32
981 84.80 85.31 0.51
991 350.57 348.87 -1.70
1004 250.36 252.79 2.43
1011 216.63 213.54 -3.09
1013 312.56 313.33 0.77
1014 313.00 313.31 0.31
1018 240.19 241.87 1.68
1020 352.69 352.34 -0.35
1025 38.77 35.47 -3.30
1031 229.30 227.95 -1.35
1062 104.41 106.77 2.36
1069 115.61 113.86 -1.75
1074 325.26 327.38 2.12
1075 44.47 46.23 1.75
1076 300.41 299.17 -1.24
1078 268.49 266.95 -1.54
1080 256.22 252.77 -3.45
1083 121.03 121.02 -0.01
1087 122.69 120.93 -1.76
1089 306.44 306.26 -0.18
1090 134.79 135.13 0.34
1094 223.68 217.10 -6.58
1096 273.04 270.53 -2.52
1097 318.05 316.70 -1.34
1108 247.78 252.73 4.95
1111 38.05 35.47 -2.58
1115 293.26 295.59 2.32
1119 158.76 160.09 1.33
1135 101.31 99.55 -1.76
1136 207.31 206.40 -0.91
1140 57.90 60.56 2.66
1141 326.68 327.54 0.86
1142 159.17 160.15 0.98
1166 200.35 199.40 -0.94
1173 210.88 206.29 -4.59
1177 164.21 167.41 3.19
1182 312.43 306.24 -6.19
1187 84.12 81.76 -2.36
1188 152.60 152.99 0.39
1189 252.85 252.59 -0.26
1190 62.50 63.95 1.45
1194 99.48 96.11 -3.37
1196 220.30 227.73 7.42
1199 131.00 128.13 -2.86
1201 281.54 284.76 3.22
1210 98.65 95.86 -2.80
1211 266.37 266.89 0.52
1215 239.17 238.51 -0.66
1218 235.94 238.61 2.67
1219 72.92 74.68 1.77
1220 170.45 170.91 0.46
1228 284.79 288.38 3.59
1239 87.87 85.18 -2.69
1240 134.59 135.17 0.58
1243 208.18 202.84 -5.34
1248 345.05 341.76 -3.29
1250 205.65 202.92 -2.74
1252 150.88 149.54 -1.34
1255 179.78 181.46 1.68
1257 152.22 156.66 4.44
1259 133.30 135.23 1.93
1262 260.98 263.47 2.49
1269 329.03 327.36 -1.67
1275 50.74 53.34 2.61
1276 170.05 174.46 4.42
1277 232.62 234.94 2.32
1278 111.83 110.19 -1.64
1279 202.99 199.36 -3.62
1282 122.82 124.59 1.77
1285 136.80 138.69 1.89
1290 285.48 281.33 -4.15
1295 295.50 298.94 3.44
1297 57.41 56.73 -0.68
1298 253.46 252.75 -0.71
1307 100.28 96.10 -4.18
1309 279.69 281.14 1.45
1312 72.31 71.12 -1.19
1321 34.05 35.39 1.34
1323 279.41 281.34 1.93
1342 353.45 352.47 -0.97
1343 257.18 256.26 -0.92
1346 222.60 217.06 -5.54
1353 209.19 209.88 0.69
1357 94.29 88.83 -5.46
1364 317.58 320.47 2.89
1373 197.67 195.80 -1.87
1380 243.50 249.24 5.74
1381 326.23 327.40 1.18
1382 255.97 252.84 -3.13
1383 60.38 56.84 -3.55
1386 178.59 181.39 2.80
1387 209.03 202.76 -6.27
1391 152.42 152.87 0.45
1400 95.28 92.38 -2.90
1402 306.42 302.49 -3.93
1411 57.41 53.18 -4.23
1413 301.49 299.16 -2.33
1419 175.90 174.39 -1.51
1421 266.41 270.44 4.02
1428 331.01 331.19 0.18
1430 236.18 238.49 2.31
1436 242.99 242.10 -0.89
1438 249.05 249.19 0.14
1450 213.98 209.95 -4.03
1459 311.11 313.00 1.89
1461 125.61 128.06 2.45
1469 296.81 295.57 -1.24
1473 267.74 270.48 2.74
1486 58.53 49.57 -8.96
1489 62.19 67.48 5.29
1490 170.81 177.97 7.16
1498 101.44 99.58 -1.86
1500 273.61 277.50 3.89
1504 164.34 160.16 -4.17
1507 154.40 152.95 -1.45
1513 284.43 288.29 3.86
1521 237.76 238.41 0.65
1532 40.14 39.25 -0.89
1537 308.06 313.35 5.29
1542 152.05 153.13 1.08
1544 175.99 181.38 5.39
1554 282.62 284.66 2.05
1555 93.02 92.60 -0.42
1560 273.47 274.03 0.56
1571 73.83 74.65 0.82
1573 321.75 320.32 -1.42
1574 54.81 53.24 -1.57
1587 148.92 149.38 0.45
1590 113.40 113.76 0.37
1592 80.51 78.18 -2.33
1598 147.56 149.46 1.90
1599 119.02 117.25 -1.77
1603 63.36 60.26 -3.09
1604 138.75 138.68 -0.07
1607 243.57 241.97 -1.60
1609 171.19 174.41 3.22
1619 163.35 167.29 3.93
1620 353.13 356.04 2.91
1632 119.97 120.88 0.91
1633 90.93 88.99 -1.94
1635 202.36 202.79 0.42
1636 349.06 352.43 3.37
1639 56.11 56.71 0.60
1642 95.39 99.72 4.33
1645 303.30 299.06 -4.24
1647 255.85 252.82 -3.02
1649 91.22 92.34 1.12
1662 330.77 331.14 0.37
1664 343.01 341.64 -1.37
1665 262.28 259.92 -2.36
1668 307.39 309.56 2.17
1673 217.34 217.08 -0.26
1675 336.90 334.54 -2.36
1681 186.61 188.53 1.91
1684 355.41 356.04 0.63
1696 213.55 209.81 -3.73
1699 330.15 331.15 1.00
1701 78.67 85.29 6.62
1707 181.37 181.58 0.22
1708 136.57 131.49 -5.08
1714 90.85 92.44 1.59
1716 213.84 213.60 -0.24
1718 324.20 323.92 -0.29
1720 293.58 291.87 -1.71
1734 42.19 42.64 0.45
1735 205.85 206.58 0.73
1740 162.92 163.72 0.80
1743 292.80 291.95 -0.85
1747 136.26 138.77 2.51
1759 272.14 274.09 1.95
1762 193.84 195.83 1.99
1763 146.06 142.39 -3.67
1774 341.10 334.66 -6.44
1780 318.42 313.37 -5.05
1786 277.71 277.61 -0.09
1787 221.81 224.09 2.28
1809 36.31 39.08 2.77
1810 122.13 124.47 2.34
1811 189.98 185.14 -4.84
1814 38.22 38.99 0.78
1821 85.89 81.90 -3.99
1826 56.32 56.97 0.65
1832 214.24 209.76 -4.48
1842 111.98 113.75 1.77
1843 275.53 274.16 -1.37
1846 289.04 288.37 -0.66
1851 142.49 142.26 -0.23
1862 116.33 120.94 4.62
1868 111.32 110.32 -1.00
1873 304.43 306.28 1.85
1876 146.56 146.12 -0.44
1881 273.67 274.15 0.47
1890 296.71 295.34 -1.37
1894 66.22 67.54 1.32
1895 314.89 316.72 1.84
1897 131.02 131.57 0.55
1902 266.94 270.58 3.63
1904 316.57 316.80 0.23
1907 85.90 89.07 3.16
1908 322.37 324.05 1.68
1912 227.60 224.16 -3.44
1917 306.81 306.12 -0.69
1920 101.29 103.20 1.91
1935 293.15 291.77 -1.38
1936 216.90 224.27 7.37
1939 324.05 324.04 -0.01
1940 293.30 288.40 -4.90
1943 151.66 153.03 1.37
1944 277.62 274.03 -3.60
1952 249.20 249.14 -0.05
1954 325.06 327.33 2.28
1961 272.62 277.63 5.01
1962 59.22 56.94 -2.28
1965 170.99 167.25 -3.75
1971 159.24 156.46 -2.78
1991 38.78 39.00 0.21
1995 183.07 185.08 2.01
2001 261.62 256.31 -5.31
2010 246.90 245.53 -1.37
2012 93.29 99.58 6.29
2023 219.88 216.98 -2.89
2044 154.53 153.10 -1.43
2052 120.07 121.02 0.95
2057 120.63 117.17 -3.47
2058 77.69 74.58 -3.11
2061 196.72 195.85 -0.87
2067 148.22 152.93 4.72
2069 207.62 213.46 5.83
2071 178.47 181.48 3.01
2072 311.74 309.59 -2.15
2073 177.93 181.67 3.73
2074 210.47 213.56 3.09
2085 145.30 142.35 -2.95
2087 316.27 316.88 0.61
2092 277.34 277.64 0.30
2097 200.81 203.08 2.27
2102 291.60 291.95 0.36
2115 288.66 291.94 3.27
2116 181.05 177.95 -3.11
2120 328.24 327.53 -0.71
2130 195.42 195.61 0.19
2133 150.19 156.66 6.47
2135 252.71 256.43 3.73
2137 223.44 217.08 -6.36
2142 37.75 39.01 1.27
2150 174.88 170.86 -4.02
2155 310.22 309.67 -0.55
2158 182.89 181.51 -1.38
2163 210.73 209.82 -0.92
2164 254.96 252.59 -2.37
2166 169.23 167.06 -2.17
2167 167.41 163.70 -3.71
2177 229.29 234.98 5.69
2178 71.40 67.46 -3.94
2181 303.04 309.55 6.50
2182 72.12 74.75 2.63
2185 46.84 46.19 -0.65
2192 106.81 103.14 -3.67
2196 147.03 142.25 -4.78
2207 299.42 302.42 3.01
2215 193.06 192.02 -1.04
2217 72.22 74.62 2.40
2218 111.20 113.67 2.47
2221 102.24 106.70 4.45
2227 94.68 92.31 -2.37
2232 54.33 53.37 -0.96
2235 179.19 177.99 -1.20
2237 153.87 156.59 2.72
2246 229.72 231.35 1.63
2247 123.91 124.59 0.68
2253 43.30 49.84 6.55
2255 312.23 313.21 0.98
2272 187.45 188.62 1.17
2273 121.79 124.57 2.78
2274 243.29 241.86 -1.43
2275 256.77 256.27 -0.50
2281 330.05 327.35 -2.71
2284 117.17 117.52 0.35
2288 31.20 35.38 4.18
2292 269.53 266.82 -2.71
2299 259.70 256.27 -3.43
2305 138.00 135.04 -2.96
2306 219.93 213.46 -6.47
2309 328.33 327.38 -0.94
2311 215.12 220.79 5.67
2318 96.05 92.44 -3.61
2326 154.49 152.86 -1.63
2329 144.30 142.25 -2.04
2335 113.45 110.34 -3.11
2338 311.67 306.04 -5.63
2339 112.42 110.11 -2.30
2341 138.14 138.59 0.45
2342 95.10 96.08 0.98
2347 306.81 306.32 -0.49
2348 334.56 338.09 3.53
2364 96.10 96.00 -0.10
2368 307.63 306.22 -1.41
2372 56.04 53.31 -2.74
2375 320.88 320.32 -0.56
2376 113.42 113.81 0.39
2378 357.11 352.48 -4.63
2382 125.54 128.05 2.51
2393 295.46 295.33 -0.13
2396 103.92 103.24 -0.68
2404 353.00 352.51 -0.49
2407 238.78 234.93 -3.85
2420 75.10 74.75 -0.35
2427 129.02 128.05 -0.96
2428 152.60 149.50 -3.10
2435 193.26 192.02 -1.24
2436 252.05 252.68 0.63
2439 325.14 327.61 2.47
2443 357.59 355.88 -1.71
2466 75.67 74.49 -1.18
2468 335.74 338.23 2.49
2480 304.78 302.52 -2.26
2481 129.70 131.44 1.74
2499 351.79 345.15 -6.64
2503 125.06 124.57 -0.49
2506 298.12 299.05 0.93
2507 125.82 128.12 2.31
2511 258.45 259.70 1.26
2512 355.11 355.88 0.77
2513 39.78 39.17 -0.61
2514 184.10 188.67 4.57
2532 274.12 274.00 -0.12
2535 88.89 88.89 0.00
2537 303.72 306.21 2.50
2539 279.98 281.14 1.16
2540 116.14 113.72 -2.43
2542 350.40 348.76 -1.64
2572 66.50 74.68 8.18
2581 37.11 39.13 2.02
2586 123.13 124.44 1.31
2592 158.89 160.04 1.14
2601 133.39 138.66 5.27
2604 347.62 341.74 -5.88
2606 82.49 81.87 -0.61
2617 310.82 309.50 -1.32
2625 344.85 338.11 -6.74
2634 44.26 42.64 -1.63
2641 112.99 113.68 0.69
2644 119.00 120.85 1.85
2651 317.39 320.30 2.91
2659 146.91 149.42 2.51
2661 349.54 348.76 -0.78
2666 53.90 53.34 -0.56
2675 139.20 138.77 -0.43
2679 294.20 299.04 4.84
2694 223.53 227.79 4.25
2702 219.39 217.03 -2.36
2711 84.36 85.17 0.81
2714 331.87 323.84 -8.02
2715 234.95 231.29 -3.67
2722 298.97 298.95 -0.02
2727 222.50 224.10 1.60
2745 91.77 92.56 0.79
2749 188.46 191.95 3.49
2750 64.50 64.05 -0.45
2754 281.53 281.16 -0.37
2761 291.32 291.84 0.52
2775 36.88 35.62 -1.26
2777 136.56 131.65 -4.90
2780 225.16 224.37 -0.79
2781 86.45 85.37 -1.08
2788 322.00 320.41 -1.59
2802 37.48 39.08 1.60
2806 238.07 234.94 -3.13
2815 186.94 188.57 1.63
2819 85.56 92.43 6.87
2820 172.80 174.27 1.48
2821 118.23 117.32 -0.91
2824 362.04 355.99 -6.05
2829 223.25 216.91 -6.33
2841 290.62 291.81 1.19
2843 80.42 78.13 -2.29
2852 355.90 355.90 0.00
2861 39.06 38.95 -0.11
2867 317.71 316.76 -0.96
2872 166.65 163.69 -2.97
2875 109.15 113.74 4.59
2880 348.55 345.44 -3.11
2881 126.77 124.51 -2.27
2892 244.18 245.51 1.33
2894 70.21 71.01 0.80
2895 140.37 142.38 2.00
2897 109.08 106.81 -2.28
2902 263.38 266.90 3.52
2920 356.75 355.93 -0.81
2921 224.77 220.66 -4.11
2924 115.35 113.71 -1.65
2929 54.52 49.65 -4.87
2930 140.77 135.03 -5.74
2933 266.79 266.99 0.20
2934 70.97 67.60 -3.38
2940 330.72 334.52 3.80
2941 67.58 63.95 -3.64
2942 178.15 174.52 -3.63
2944 184.10 185.03 0.94
2945 94.43 92.54 -1.89
2953 345.22 348.87 3.65
2955 299.93 295.58 -4.35
2962 178.23 181.58 3.35
2967 47.58 49.66 2.08
2972 200.38 199.23 -1.15
2982 119.54 117.41 -2.13
2986 351.11 348.85 -2.26
2991 301.74 302.47 0.73
3006 313.13 313.28 0.15
3007 301.65 302.69 1.04
3009 56.89 53.28 -3.61
3011 335.51 330.92 -4.58
3012 59.80 60.34 0.54
3017 182.91 188.69 5.78
3019 141.01 138.88 -2.13
3022 128.13 128.07 -0.07
3025 129.95 127.98 -1.96
3029 298.37 298.87 0.50
3038 169.25 167.19 -2.06
3040 194.23 199.35 5.11
3049 117.05 117.51 0.47
3054 208.88 206.26 -2.62
3057 159.06 160.21 1.15
3058 305.80 306.00 0.20
3064 205.54 206.64 1.10
3069 166.07 163.66 -2.40
3077 136.95 135.17 -1.78
3078 338.80 341.78 2.99
3082 211.04 209.99 -1.04
3103 244.93 242.08 -2.85
3112 284.23 284.71 0.48
3116 108.49 106.78 -1.71
3121 52.68 53.20 0.51
3125 162.43 163.62 1.19
3126 317.49 316.90 -0.59
3127 267.29 263.45 -3.84
3131 132.16 135.20 3.04
3141 170.62 170.68 0.07
3144 305.71 309.58 3.87
3145 326.43 330.91 4.47
3155 178.56 181.42 2.86
3158 99.36 96.14 -3.22
3160 351.86 348.88 -2.99
3176 268.89 270.45 1.56
3177 106.65 106.78 0.13
3192 191.96 192.34 0.38
3197 165.23 163.56 -1.67
3200 322.39 320.27 -2.12
3205 95.04 92.54 -2.50
3207 164.60 163.61 -0.99
3208 97.74 92.38 -5.36
3211 94.64 96.12 1.48
3216 85.94 81.84 -4.09
3217 340.62 338.27 -2.35
3218 207.57 206.55 -1.02
3220 109.98 106.77 -3.21
3229 295.71 298.95 3.23
3231 302.83 302.57 -0.26
3232 230.12 224.32 -5.81
3236 157.66 160.00 2.35
3242 248.31 249.19 0.88
3244 156.46 160.18 3.71
3252 268.81 266.87 -1.95
3253 188.21 188.71 0.50
3259 81.24 78.31 -2.92
3265 77.29 74.56 -2.73
3270 331.75 327.53 -4.22
3271 299.26 299.08 -0.18
3279 331.54 331.20 -0.34
3282 81.29 85.42 4.13
3283 322.47 320.48 -1.99
3285 114.67 113.68 -0.99
3287 139.99 142.14 2.15
3289 164.75 167.34 2.60
3295 351.42 348.97 -2.45
3298 238.72 238.54 -0.18
3310 287.31 284.68 -2.64
3311 157.89 156.56 -1.32
3313 333.42 334.45 1.03
3319 54.87 56.73 1.86
3323 188.82 188.53 -0.29
3324 52.32 49.69 -2.63
3331 133.54 135.18 1.64
3339 360.11 356.03 -4.08
3354 115.34 113.81 -1.53
3361 152.46 156.45 3.99
3363 208.86 210.18 1.32
3366 72.62 71.05 -1.57
3367 272.55 273.90 1.35
3376 132.17 128.08 -4.09
3413 314.65 316.86 2.21
3419 306.25 309.82 3.57
3425 92.67 92.37 -0.30
3428 90.53 88.75 -1.78
3434 128.72 131.71 2.99
3442 82.73 88.93 6.20
3448 151.22 149.33 -1.90
3450 69.43 70.98 1.55
3463 265.94 263.52 -2.42
3469 52.14 53.28 1.14
3476 347.32 345.38 -1.94
3481 256.95 259.73 2.78
3486 139.78 138.71 -1.07
3502 327.30 324.10 -3.20
3503 267.46 263.34 -4.11
3509 180.27 181.57 1.30
3511 321.56 316.83 -4.72
3525 303.30 302.65 -0.65
3527 184.97 185.11 0.14
3528 114.70 120.91 6.21
3535 117.55 117.46 -0.10
3540 56.97 60.30 3.33
3547 236.89 235.02 -1.87
3557 131.67 131.47 -0.19
3566 138.43 138.90 0.47
3575 129.89 131.61 1.73
3583 267.85 266.99 -0.86
3596 51.34 46.17 -5.17
3618 314.25 316.98 2.73
3619 225.78 227.77 1.99
3623 77.79 81.72 3.93
3626 118.50 117.38 -1.12
3627 345.95 345.36 -0.59
3630 164.26 163.70 -0.55
3637 166.45 167.34 0.89
3643 346.95 348.86 1.91
3657 298.88 299.07 0.19
3665 49.50 46.09 -3.40
3668 265.22 267.05 1.84
3670 37.07 35.50 -1.57
3672 97.93 96.01 -1.92
3674 210.07 213.57 3.50
3678 124.18 124.42 0.24
3679 177.31 174.24 -3.07
3686 273.80 274.06 0.26
3688 317.16 313.20 -3.96
3690 33.72 35.45 1.73
3691 298.31 302.49 4.18
3694 90.63 88.96 -1.66
3696 274.91 274.09 -0.81
3704 123.57 124.50 0.94
3709 185.05 181.41 -3.64
3712 313.12 313.19 0.06
3721 132.88 131.56 -1.31
3723 342.96 341.71 -1.25
3724 204.11 199.29 -4.83
3726 120.11 120.83 0.72
3732 185.03 188.46 3.43
3733 308.22 309.70 1.48
3746 220.84 217.16 -3.68
3747 212.41 209.93 -2.48
3765 248.03 249.22 1.18
3768 256.37 256.17 -0.21
3777 160.50 167.36 6.86
3780 330.31 334.64 4.34
3794 158.31 160.22 1.91
3796 213.00 213.54 0.54
3801 67.57 71.15 3.58
3809 232.11 231.23 -0.88
3811 210.49 210.01 -0.48
3820 250.15 252.58 2.43
3822 257.44 256.32 -1.12
3823 352.82 352.37 -0.45
3830 165.58 167.27 1.69
3840 61.40 60.43 -0.97
3843 310.05 309.64 -0.40
3845 89.76 88.84 -0.92
3847 56.35 53.25 -3.10
3851 122.07 124.38 2.31
3852 207.28 206.42 -0.86
3855 223.86 227.76 3.90
3857 134.78 135.40 0.62
3860 315.09 313.36 -1.73
3861 60.91 64.03 3.12
3878 68.21 64.02 -4.19
3885 114.68 117.49 2.82
3893 209.29 206.36 -2.94
3900 333.05 334.67 1.62
3902 98.93 95.86 -3.07
3903 201.08 202.86 1.78
3917 348.27 352.53 4.26
3918 166.35 167.24 0.89
3921 190.68 188.70 -1.98
3925 80.76 81.55 0.79
3931 48.20 53.28 5.08
3937 342.37 338.18 -4.19
3959 49.77 49.66 -0.11
3975 310.48 313.13 2.65
3979 137.57 135.11 -2.46
3980 177.50 181.39 3.89
3983 134.19 135.08 0.89
3989 151.64 153.01 1.37
4000 188.36 192.09 3.73
4006 63.09 63.85 0.76
4007 343.77 345.37 1.60
4014 231.30 231.37 0.07
4020 117.10 113.79 -3.31
4022 82.40 85.31 2.91
4026 163.34 167.26 3.92
4027 272.41 274.09 1.69
4028 100.28 103.25 2.98
4033 307.30 306.15 -1.15
4036 75.89 74.65 -1.25
4052 247.15 249.06 1.91
4053 226.99 227.79 0.80
4056 290.46 291.84 1.38
4058 97.32 99.54 2.22
4060 103.31 103.28 -0.04
4066 261.36 259.66 -1.70
4067 266.66 263.30 -3.35
4069 231.14 231.18 0.03
4070 275.49 277.59 2.10
4071 296.33 295.56 -0.77
4075 261.95 259.68 -2.28
4076 226.44 227.86 1.42
4083 268.92 270.49 1.57
4086 341.25 338.19 -3.06
4091 340.20 341.62 1.42
4097 263.45 267.04 3.59
4098 271.54 273.93 2.39
4099 318.30 316.83 -1.47
4101 301.37 302.66 1.29
4103 63.20 60.46 -2.74
4104 212.27 210.07 -2.20
4106 258.38 256.25 -2.13
4113 286.07 291.91 5.84
4114 314.60 313.28 -1.33
4116 236.98 238.60 1.62
4120 117.25 120.96 3.71
4130 314.82 316.91 2.09
4131 71.64 71.17 -0.47
4132 146.05 145.74 -0.31
4134 84.68 81.61 -3.07
4139 220.04 217.05 -2.99
4144 209.19 206.34 -2.85
4146 123.87 124.46 0.59
4148 194.53 199.37 4.85
4155 201.58 199.24 -2.34
4159 173.49 170.69 -2.80
4186 106.45 113.94 7.49
4190 274.04 270.57 -3.47
4191 152.44 153.03 0.59
4192 62.97 63.99 1.02
4199 64.98 63.84 -1.14
4202 88.89 85.42 -3.47
4207 94.79 88.97 -5.81
4208 182.02 184.85 2.83
4209 336.33 331.06 -5.27
4213 66.90 63.89 -3.01
4216 129.33 128.14 -1.19
4218 354.76 355.89 1.13
4221 268.54 270.49 1.95
4222 100.87 99.71 -1.16
4226 59.34 56.93 -2.41
4232 232.15 234.93 2.78
4233 50.60 46.22 -4.38
4237 256.14 256.34 0.20
4241 351.38 348.79 -2.59
4242 236.10 234.77 -1.32
4244 132.83 131.58 -1.25
4245 287.15 288.48 1.33
4251 170.67 167.20 -3.47
4256 270.67 270.57 -0.10
4271 147.57 142.21 -5.36
4275 265.97 266.90 0.93
4283 101.04 102.98 1.94
4284 193.65 195.67 2.02
4297 84.99 85.21 0.22
4306 103.77 99.67 -4.10
4308 310.90 309.63 -1.26
4320 92.56 89.00 -3.56
4325 305.18 302.63 -2.55
4328 332.59 334.62 2.03
4331 255.86 263.30 7.44
4335 202.89 206.36 3.47
4339 85.69 81.75 -3.94
4340 222.91 224.24 1.33
4341 272.94 270.42 -2.52
4342 242.80 242.01 -0.79
4343 337.82 338.24 0.41
4345 300.52 295.50 -5.02
4347 157.73 156.44 -1.29
4358 350.26 348.83 -1.43
4362 109.82 110.33 0.50
4366 54.32 53.33 -1.00
4370 168.50 167.12 -1.38
4374 179.24 181.49 2.25
4378 350.17 348.90 -1.26
4380 178.64 177.93 -0.71
4381 283.20 284.81 1.61
4382 309.74 306.17 -3.57
4393 279.71 281.11 1.40
4394 67.66 67.60 -0.06
4398 110.02 106.64 -3.37
4406 71.99 78.09 6.10
4407 99.43 99.51 0.07
4408 89.41 85.41 -3.99
4413 146.06 138.73 -7.33
4418 149.23 153.06 3.83
4424 211.99 206.30 -5.69
4428 274.79 277.80 3.00
4430 54.19 53.31 -0.88
4433 35.10 39.00 3.89
4437 190.98 192.00 1.02
4439 84.19 81.80 -2.39
4440 46.13 46.14 0.02
4442 349.06 348.86 -0.20
4455 321.96 320.22 -1.74
4467 77.39 74.54 -2.84
4474 155.34 156.45 1.11
4479 80.44 81.72 1.28
4486 119.22 113.82 -5.40
4487 316.06 316.86 0.80
4488 242.65 238.30 -4.35
4490 290.44 288.38 -2.06
4495 288.92 288.31 -0.61
4496 234.68 234.98 0.29
4501 129.52 127.94 -1.58
4507 112.32 117.55 5.24
4526 226.51 227.66 1.16
4527 226.57 224.13 -2.44
4531 255.42 256.22 0.80
4535 161.11 156.48 -4.63
4537 53.21 56.73 3.52
4539 277.73 274.12 -3.62
4542 214.79 213.51 -1.28
4549 155.00 153.04 -1.96
4552 210.96 213.65 2.69
4555 238.71 234.90 -3.81
4564 327.47 331.13 3.67
4571 253.61 252.80 -0.81
Residual vs Fitted Plot

We can see that the Residuals (Y Axis variable) are randomly and uniformly spread across the fitted value. This shows that the model is a good fit.

QQ Residuals Plot

We can see that the residual points are majorly coninciding on QQ line except for a few outliers. Overall, it indicates that model is quite stable.

Scale-Location Plot

We can see that the standardized residuals are randomly and uniformly spread across the fitted values. It indicates that the model is robust.

Cook’s Distance Plot

This plot shows how each observation is measured on Cook’s distance. If a point is farther from the X axis, the more it should be considered as outlier. Except for few points, major part of the data is in the acceptable Cook’s distance range.

Dr Amita Sharma
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