I generated 5 quite well-separated random spherical normal clusters:
cluster_ v1 v2 v3 v4 v5 v6 v7 v8 v9 v10
1 -.35889 -.80415 -.95355 .09492 .09878 -.57061 .04876 .32444 .43613 -.75848
1 -.15922 -.33836 -.85863 .20295 -.02157 -.05508 .22093 -.20209 .12078 -.78535
1 -.04343 -.57511 -.48917 -.11231 -.72780 -.26187 .28009 .12036 .74661 -1.28238
1 -.63787 -.37796 -.85504 .19031 -.42068 .10327 .10764 .42621 .37841 -.31551
1 -.26072 -.27328 -.23779 -.72944 -.00055 .02082 -.56056 .44621 .18630 -.44929
1 -.69277 -.73115 -.98069 -.52607 -.19524 .93742 .22414 .02783 -.04566 .04284
1 -.86744 -.24549 -.94066 .43189 -.31535 -.16167 -.23409 .37400 .29167 -.27242
1 -.61638 -.77716 -1.01088 .41879 .25247 .06539 .93231 .17475 .06597 -.83623
1 -.39131 -.71233 -.66181 -.11544 -.60630 -.24238 .74176 -.22127 .29012 -.48975
1 -.70326 -.37406 -1.01975 .08320 .25287 -.54553 .37006 .56295 -.08655 -.42384
1 -.58532 -.78090 -.52363 .34909 -.13363 -.03828 -.54699 .58049 .40722 -1.02372
1 -.37289 -.30283 -.96715 -.25034 -.44174 -.14580 .11320 .40625 1.14946 -.71669
1 -.07412 -1.06823 -.50983 .02912 .29590 -.57374 .21784 .96741 .39346 -.55247
1 -.64557 -.28752 -1.06065 .91234 .33351 .11040 -.18679 .41887 1.31325 -.60888
1 -.57587 -.18204 -.87064 -1.02938 -.32472 -.21606 .38594 .24866 .78516 -.47058
1 -.18757 -.45480 -.85457 .28213 .09075 .01695 .09179 -.15911 .05814 -.64389
1 -.48608 -.46857 -.94948 -.18636 -.45284 -.13831 -.04654 .20109 .23887 -.53225
1 -.74734 -.08176 -.40716 -.08287 .12918 .15853 .74587 .27689 .54414 -.13492
1 -.49310 -.66883 -.96552 .12861 .17556 .30066 .44258 -.05136 .20040 -.94728
1 -.29509 -.04972 -.96719 .15146 -.15333 -.14490 -.22875 .21457 .49441 -.50094
1 -.44203 -.65198 -1.31387 .14698 -.50402 .14900 -.22044 .63406 .66311 -.18015
1 .36509 -.33656 -.44303 .29778 .19249 -.17983 -.20325 -.11391 .61558 -.71136
1 -.58545 -.71832 -.58915 -.01282 .19542 .44213 .80000 .25524 .50858 -.15348
1 -.61502 -.23686 -1.40380 .32189 -.28770 .16869 .06525 -.10767 .65105 .10628
1 -.85519 -.56026 -.87019 -.15699 -.11682 -.26702 .42833 .61952 .54453 -.30528
1 -.27144 -.47066 -.50293 -.04654 .20285 -.78603 -.05650 .04141 .43587 -.38720
1 -.66833 -.85999 -1.39358 -.17907 -.35803 .26559 -.15205 -.06139 .07303 -.21759
1 -.92891 -.83885 -.87182 -.57503 .34250 -.09976 -.49820 .15678 .56266 -.55564
1 -.24621 -.57793 -.36508 .10734 .02193 -.03344 .42350 .44285 .15249 -.28804
1 -.72920 -.46724 -1.04991 -.09380 -.59355 -.16053 -.35367 -.12637 .88971 -.36361
1 -.95095 -.59221 -.75699 -.35541 -.45564 -.67606 .27492 .01158 .60059 -.36376
1 -.42749 -.19381 -.11913 .66490 .53474 -.20897 .08913 .73044 .57282 -.71333
1 -.35771 -.06046 -.61185 -.12414 -.33521 -.18176 .27938 .51311 .09970 -.78810
2 .19264 -.39227 -.45144 -.34946 -.50718 .37317 -.68262 -1.21150 -.27218 .26079
2 -.26762 -1.14943 -.24108 -.61609 -.53647 .54013 .37995 -.14258 -.92335 .20464
2 .71585 -.57530 -.42672 -.33682 -.17193 .02049 -.11441 -1.01213 -.07658 -.01067
2 -.18252 -.40495 -.72124 -.10062 -.86607 .93464 .37502 -.98797 -.28578 .25459
2 -.27979 -.19103 -.36350 -.06228 .13197 .60294 -.38538 -.61523 -.70839 .49169
2 .28744 -.05596 -.28775 .08955 -.44324 .26238 .68400 -.76522 -1.35614 .10266
2 -.29865 -.34482 .19768 .05101 -.71033 .63751 .60821 -.87805 -.09203 -.32935
2 .33705 -.29259 -.09520 -.20307 -.22723 .46284 -.15910 -.35052 -.79023 .15426
2 -.17243 -.03626 -.08962 .04984 -.28323 .36873 .15831 -1.02778 -.43664 .04473
2 .52032 .16039 .05790 .05482 -.58955 .57874 .15957 -.65148 -.60509 -.04410
2 -.49298 -.14708 -.02757 -.29101 -.13378 .16864 .17596 -.90204 -.62725 .00444
2 .34508 -.52929 .16544 -.23151 -.35914 -.39765 .06071 -.72253 -.26575 -.01780
2 .18039 -.68884 -.17237 -.11827 -.76440 .88507 .31510 -.85731 -.55658 -.08526
2 -.06160 .14602 -.10263 -1.22261 -.35885 -.08512 -.70837 -.49577 -.41909 -.21043
2 .61226 -.48731 .17086 .01094 -.01556 .89763 -.09912 -.95948 -1.24026 .18679
2 .27781 -.11220 .19625 -.80179 -1.05123 .34191 -.14444 -.92017 -.72394 -.38116
2 -.13057 -.25444 -.29116 -.46638 -.45466 -.23273 -.35136 -.58928 -1.19391 -.21628
2 .53544 -.07734 .18806 -.29863 -.35207 .16474 -.39829 -1.04920 -.04360 .27012
2 .09011 .14510 -.13585 -.29632 -1.40790 -.54687 -.35850 -1.09756 .01308 .27056
2 -.27727 -.35648 .18564 -.20406 -.62323 .42556 .04233 -.89799 -.95699 .07104
2 .02676 -.00119 -.31353 -.47823 -.74560 .37957 -.37345 -.55019 -.72160 .78464
2 .42959 -.37294 -.61238 .27562 -.57481 .36761 -.07886 -.19893 -.51787 .08761
2 .84452 -1.02129 -.20080 -.14087 -.45586 -.14549 -.51673 -.95325 -.33927 .32113
2 .45584 -.97896 -.29851 .26187 -.00994 .23504 .39993 -.47371 -1.43147 .22279
2 .45992 .25763 -.50511 -.24222 -.47895 -.56162 -.09741 -.39730 -.77720 .34671
2 .67029 -.80599 -.21568 -.20057 -.57457 .46333 .42951 -1.46737 -.87203 .14155
2 .28548 .14793 .20718 -.62599 -.81032 .50961 .14414 -.87512 -1.33294 -.29246
2 .02684 -.74806 .03674 .35342 -.02703 .47246 .00974 -.96049 -.61683 -.15511
2 .06397 -.67032 .20367 -.71854 -1.07909 .13888 -.15452 -.93031 -.84782 .05661
2 .08814 -.41217 -.16592 -.78108 -.53232 .38670 -.08234 -1.20331 -.70393 .09167
2 .47916 .57869 .04185 .05530 -.11371 .06468 -.10352 -.58962 -.47385 -.23243
2 .30212 -.30889 .02052 -.00621 .06990 .56321 -.41854 -.63496 -.72044 .01441
2 .28807 .14702 -.11912 -.71629 -.37887 .05496 .08769 -.93546 -.65293 -.30587
2 -.00337 -.23993 .15920 -.39460 -.17969 .39506 -.26512 -.58096 -.53665 -.08089
2 -.24791 .35754 -.10902 .14933 -.70239 .45544 -.23488 -.36102 -.53181 .06614
2 -.06024 -.18187 .14727 -.37599 .15671 .49404 -.09129 -.44988 -.08485 .31208
2 .20165 -.20496 -.16680 -.27965 -.49118 .50500 -.13241 -.55039 .02795 -.52744
2 .65964 -.36595 -.71629 -.52986 -.38274 .19508 -.00661 -.17767 -.16829 .44395
2 .39324 -.06167 -.15007 .26749 -.12583 .21957 .24441 -1.12913 -.34984 .12255
2 .34302 -.74048 -.23877 -.40451 -.28469 .59656 -.09161 -.98368 -.26619 .48246
2 -.36048 -.66483 -.06230 -.61534 -.03873 .24258 .20101 -1.02728 -.51428 .92295
2 -.38574 -.25897 -.06990 -.02799 -.08942 .08306 .33846 -.44243 -.67854 -.29403
2 -.36852 -.44039 -.27738 -.42919 -.13750 .31308 .14585 -.56002 -.40077 .15878
2 .21363 -.09843 .35279 -.28396 -.39179 -.38450 -.19170 -1.10894 -.79801 .09346
2 .26392 .02051 -.03783 -.37110 -.57680 .39785 -.58526 -1.15637 -.86041 .48684
3 -.24395 .16123 -.52569 -.54756 -.46173 .78313 -.32469 .86162 .45817 .18950
3 -.45149 .27338 .33748 -.23151 -.40367 -.01772 -.28303 .32204 -.46198 .21000
3 .09345 .71764 .17051 -.00560 -.90011 .16755 -.24581 .74744 -.31104 -.19755
3 .00171 .78304 -.53418 .24089 -.04693 .76982 -.09617 .95082 -.72464 .00413
3 .14384 .67336 .00242 -.01307 .17794 .38635 -1.37637 1.09934 .43910 -.59356
3 .45663 .46030 -.71005 -.42481 .27294 .25727 .38373 1.27638 .03208 -.38087
3 -.35099 .57253 .37563 -.46467 -.15921 .08879 -1.06144 .69125 -.17132 -.42077
3 -.50472 .69089 -.06146 -.20484 -.67984 .49757 -.56882 .80419 -.75014 -.31072
3 -.17699 1.50970 .12051 -.59455 -.00184 .50617 .26104 .63712 -.63130 -.31656
3 .19155 .54132 .12707 -.42273 .08332 .14384 -.59585 .29190 -.47406 -.10795
3 .59290 .17106 .08535 -.01423 -.54075 .94399 -.60894 .43943 -.38056 .14853
3 -.17509 .79728 -.06453 -.95079 -.23326 .51459 -.47951 .63232 -.59717 -.09292
3 -.02898 .42072 -.35931 -.68248 -.30044 .33594 -.67137 .19582 -.33122 .05719
3 .06747 .89734 .80501 .10828 -.34636 .63065 -.28847 .71106 -.17164 -.47439
3 .22291 .30021 .42112 .05373 -.46780 .39958 -.14513 .19257 -.60642 -.51323
3 -.67178 .12969 -.35625 -.06941 -.90086 -.09086 -.70433 .08350 -.21895 -.03402
3 .06226 1.30616 .06839 -.02109 .25937 .37931 -.01058 .64054 -.04348 -.43756
3 .34702 1.32103 -.13883 -.55438 -.12072 .11828 -.70851 .50123 .00989 -.39402
3 -.17608 .38942 .09153 .23233 -.30488 .69906 -.07760 1.15005 -.60196 -.72459
3 -.03706 .20870 -.20816 .14613 .02804 .00570 .48920 .47652 -.44141 -.56459
3 .11554 .72674 -.28102 .61072 -.54430 .74248 -.65831 1.08916 -.52031 -.46012
3 -.08616 .76170 .60656 -.20260 -.61624 .95084 -.61945 -.53094 -.48601 -.12913
3 -.23377 .28812 -.00465 -.63814 -.54961 .60212 -.73909 .57522 -.09417 -.83123
3 .04647 .67864 .24538 -.45371 -.75121 .46089 -.12164 .24679 -.96037 -.51968
3 .43037 1.19863 .03038 .01944 -.53914 .57373 -.46845 -.37273 .01472 -.35485
3 -.05359 .58196 .14807 -.16891 -.73624 .49092 .17588 .94871 .67527 -.51166
3 -.37767 .59255 .10369 -1.34807 -.34045 .17858 -.11296 .82858 -.07735 -.37090
3 .06322 .08763 .16581 -.70023 -.07563 .45287 -.00975 .63799 -.28614 -.33280
3 -.51456 .70515 .77411 -.33698 -.30368 .36648 -.31280 .07921 -.25082 -.32052
3 1.14969 .93388 .02509 -.12116 -.07666 .54490 -.09371 .97084 -.30443 .08455
3 .08126 .42957 .18818 -.57343 -.43837 .52381 -.14258 .81012 -.18564 -.39681
3 .00904 .35280 .34457 -.32577 -.20415 1.25479 -.44810 .85328 -.34538 -.56894
3 .57401 .98226 .24527 -.50222 .09948 .73297 -.41718 1.05607 -.78782 -.91479
3 .14213 .83693 -.14598 .11372 .11413 .99507 -.66617 .47991 -.51509 -.75660
3 -.21072 .56046 -.12287 -.09227 -.07915 .65564 -.68789 1.04063 -.82856 -.25795
3 .26696 .46613 .37524 -.30427 -.23610 .84144 -.09927 1.10323 -.29640 -.32335
3 -.35833 .50586 .15598 .27042 -.29747 .84245 .10441 .82107 -.18041 -.32057
3 .38234 .29171 -.07923 -.19716 .05640 .34179 -.69348 .69792 .11555 .00594
3 .39440 1.15490 -.50808 -.38042 .24665 .41070 -.13404 .86015 -.67955 .10854
3 .01013 .05924 -.21301 .25495 -.58790 .72584 -.12632 .82050 -.26645 -.20871
4 .12481 .64539 -.03975 .64673 -.40431 .13063 -.52849 -.07650 -.11618 1.24660
4 -.01452 .00504 -.13760 .66609 1.09446 -.58416 -.53981 .63190 -.96223 1.02409
4 .91232 .48534 -.29704 .18219 .36045 -.79304 .22392 .42187 -.01970 .47275
4 .49115 .46320 -.69861 .69819 .18075 .00179 .49793 .23133 -.56602 .67727
4 .43851 .75341 -.36828 .22170 .55148 -.39889 .17800 -.01949 -.27295 .17505
4 .43239 .08964 .15651 .60433 .58125 -.51438 -.07613 .15524 -.36421 .79300
4 .34400 .25761 -.77976 .16498 .59756 -.34697 -.14232 .04085 -.38930 .48834
4 -.04106 .79610 -1.11204 .44975 .78673 -.06170 -.35702 .36731 -.54020 .49166
4 -.00527 .50502 -.07809 -.18947 .97660 -.60149 -.18532 .15491 .30075 .31105
4 .18186 .55742 -.56654 .47627 .60055 -.69449 .27283 .23622 -.34590 .53413
4 -.50832 .28511 -.04698 .17425 .76304 -.20335 -.03809 -.01743 -.03948 .98610
4 .78833 -.03687 -.62573 .28408 .39957 -.34038 .13137 .29892 -.08960 .52031
4 .95380 .61980 -.52965 .72138 .92326 -.77683 -.34070 -.15212 .14155 .73830
4 .35160 .28312 -.83647 .09864 .83778 -.88948 -.49741 .09432 .34290 .77631
4 .27600 .92787 -.08412 -.20978 .38395 .05653 -.61637 .11731 .33753 -.08716
4 .58476 .62382 -.11519 .50597 .60866 -.47202 -.26130 .21178 -.38129 .77234
4 -.07336 -.40171 -.54984 .17316 .69194 -.32094 -.80006 .54560 -.48985 .46294
4 .25167 .19768 -.56227 .29683 .22327 .10291 -.49781 .57931 .03583 .29051
4 .03777 -.06726 -.28299 .30957 .61156 .24576 -.56313 .70846 -.37586 .46174
4 -.03174 .48932 .03035 .23212 .78673 .22815 -.41416 .00390 -.02809 .25024
4 .47553 .15852 -.52593 .28778 .51391 .03696 -.40845 .06569 -.13843 .78394
4 .66051 -.24317 -.38019 .36846 .63298 -.58473 .06140 .20684 .10806 .90685
4 .78153 .24951 -.04040 1.19602 .19623 -.65166 .22046 .00106 .38641 .83804
4 .43238 .61964 -.38654 .03870 .56794 -.76404 .49067 .22001 -.54198 .40200
4 .81075 .61281 -.60007 .81533 .65307 -.55407 .08571 -.54866 -.51458 .71056
4 -.32060 -.22371 -.59960 .33757 .76813 -.35013 .27732 .19311 -.52868 .27842
4 .43247 .38941 .05445 -.22539 .45748 -.91286 -.26040 .03905 -.29523 .56303
4 .60153 .16959 -1.09957 .82787 .38469 -.41677 -.55122 .29787 -.01731 .85011
4 .09208 -.04104 -.16329 -.32714 .68905 -.16825 -.32004 -.06744 -.34247 .42238
4 .53446 -.02837 -.41285 -.05115 .93666 .09275 .28974 -.43819 -.02035 .78239
4 .52346 .16028 -.30639 .80242 .64151 -.67335 -.27878 .03999 -.17939 .52376
4 -.07272 .86453 -.04191 .51539 1.30292 -.17256 .25074 .13524 -.89523 -.12336
4 .26381 .22033 -.48637 .00789 .07699 -.20436 .30045 -.55430 .28213 .56632
4 -.38775 .18806 -.35541 .16925 1.53684 .20027 -.32236 .29665 .36345 1.10464
4 -.56245 .73046 .07529 -.01363 1.00474 -.58000 -.43443 .49613 .23503 .49014
4 .19397 .60054 -.80352 .06133 .38149 -.01459 -.46508 -.22733 -.02982 .80440
4 -.19880 .54340 -.59345 .37518 .89532 -.57358 -.35951 .04535 .60423 .62265
4 -.11259 .20349 -.61756 .71470 .47077 -.64705 .29557 -.33549 .46786 1.02565
5 .94221 -.76616 .93816 .20651 -.22434 -.64455 .21913 .08471 1.03554 .12535
5 .68932 -.03953 1.31733 .46692 .39142 .55476 -.14973 .06212 .77552 .00833
5 -.78798 -.43622 .70848 .64921 -.12265 -.61095 .16492 .64428 .87688 .66544
5 .10115 -.16724 1.05262 .97713 .66306 -.74898 .12452 -.23221 .60067 .09814
5 .23656 -.37081 .50543 -.30092 .16232 -.40432 .05915 -.08133 -.04027 -.20363
5 -.29437 -.23215 1.13539 .00675 .47423 -.22462 .29859 -.38251 .22398 .11379
5 .20369 -.25498 1.06575 .62940 -.25818 -.47180 .03358 -.38966 .16060 -.27854
5 -.01902 -.15011 .97446 .66675 .21333 -.68202 .58280 -.23840 .65855 -.13566
5 -.59616 -.34664 .48634 .25446 .37241 -.54443 .08664 -.19193 .92828 -.40835
5 .37222 .07618 1.12760 -.50001 .30198 .19466 .07131 -.34506 .45512 .36074
5 .16370 .01577 1.04550 .00619 .28735 -.66035 .61140 .64152 .79330 -.01340
5 -.71456 -.92065 .76481 -.17659 .04013 -.64073 -.03873 -.66962 .53862 -.04873
5 -.14563 -.38052 1.13991 .14477 -.28422 -.80953 -.02142 -.15500 .48110 .02102
5 -.41581 -.09893 .95991 -.69318 .11668 -.42237 -.32851 -.14550 .58129 -.05483
5 -.55230 -.36544 .83642 .48609 -.08735 -.63161 .65751 .18261 .29338 .15508
5 -.08841 -.43305 1.16937 .08712 .54940 -.66578 .47746 -.25139 .51399 -.28219
5 -.70661 -.35850 .42385 -.04263 -.15989 -.17255 -.20838 .46533 1.10884 -.23194
5 -.31880 -.33642 .46561 -.07951 -.30750 -.15734 .42598 -.39341 .63181 -.15968
5 -.02104 .01231 .89536 -.09382 .21141 -.29203 .31900 -.35373 .52825 -.00364
5 .52882 -.19750 .57151 .18083 -.21128 -.39752 -.18213 -.00997 .54937 .20951
5 .18084 -.35750 .98115 .10735 -.15109 .03623 .53613 -.63918 .13051 .41700
5 .13607 -.39431 1.11954 .10425 1.13318 -.38586 -.06870 -.16331 -.02559 -.21759
5 -.24449 -.22553 .72800 .77752 -.16310 -.39704 .20342 .09200 .12099 -.40769
5 .15298 -1.05720 .27149 .21031 -.46929 .23929 .36514 -.08841 .31814 .03095
5 .23835 .22692 .68748 .19415 .30307 -.00560 .13476 -.27850 .37748 -.39022
5 -.46264 -.28041 1.19408 .00246 .92476 -.55617 .28429 -.12166 .52160 .52571
5 .44002 -.41763 1.00519 .37208 -.21578 -.75036 .45686 .03575 .90486 .36819
5 -.21521 -.24764 .77803 .16955 .49566 -.31356 -.16268 -.12568 .94447 -.14197
5 -.01372 -.49895 1.15430 .34893 -.09728 -.00777 .56743 -.10422 .80134 .52607
5 -.01684 -.41250 .64332 .38734 .31766 -.40441 .41563 .38074 .37687 .13039
5 -.34362 -.80602 .89250 .52391 .84036 -.71583 .27336 -.48401 .83208 .40367
5 -.29687 -.81651 1.09081 .72333 .12215 -.18851 .62242 -.34748 1.02239 .26486
5 .15349 .04149 1.01170 -.32899 .18664 -.09604 .25532 -.21718 .31506 -.15446
5 .29247 -1.06331 .95709 .59674 .02793 -.06086 -.15468 .42929 .25093 .15189
5 .32936 .30622 1.19245 .68224 .15637 -.55656 .76485 -.97028 .69600 .15906
5 -.23501 -.50298 .68681 .49305 .00555 -.31013 .62912 -.94264 .82368 .23385
5 -.57203 .00568 .78144 .39737 -.37745 .11250 -.27170 .25716 .31007 -.15374
5 -.07426 -.44952 1.05923 .35666 -.47327 -.61224 .19806 -.10865 .38987 -.23939
5 -.37858 .07292 .63529 .37387 -.07960 -.41198 .31239 -.52773 .77381 -.47123
5 -.09884 -.28480 .62785 .06343 -.45201 -1.05897 .35904 .29022 .41229 .20190
5 .46010 -.69470 1.57011 .59279 .53526 -.09801 .22513 -.02678 1.05208 .53716
5 .09403 -.18182 1.24049 .48869 .43755 .17045 .30166 -.04614 .60966 .06173
5 -.01407 -.19142 .98127 .67015 .11047 .15014 -.54993 .13654 .09494 -.22172
The SSwithin (pooled within-cluster sum of squares of deviations from the cluster centres) is 217.342.
I performed a series of K-means cluster analyses in SPSS on these data. I used default settings to seed initial cluster centres (In SPSS, that is k-utmost points in the data cloud, found by a special procedure) each time. The 14 analyses formed k=2 to k=15 clusters. SSwithin for the obtained cluster solutions were:
k
2 430.350
3 352.252
4 281.481
5 217.012
6 210.398
7 203.786
8 197.991
9 194.528
10 190.398
11 182.382
12 183.242
13 179.123
14 179.375
15 171.383
and on the plot
Clearly, for such clear clusters as ours had been generated there is a nice elbow, and at k=5 clusters (the clusters of that solution are very close to be what we generated).
But besides that we observe two minor "inversions" of SSwithin: for k=12 it is higher than for k=11, and for k=14 it is higher than for k=13.
Final centres in a K-means clustering solution are multivariate means (centroids) and as such they minimize SSwithin to the limit attainable with that specific assignment of points to k clusters. K-means algorithm tries to find the assignment which will yield the minimal SSw but does not guarantee to find it, the global optimum. The global optimum at k=i is lower SSw than at k=i-1 for any i but it may happen that the procedure succeeds to find the global (or near global) optimum at i-1 while badly fails to find it at i; then SSw observed in the obtained cluster solution may be lower at i-1 than at i.
Good initial cluster centres fed to the program play the crucial role. If those centres are already close to the real centres of the clusters hidden in the data or the data is such that they can easily "slide down" to them during the iterations then the solution will be about the global optimum: SSw utmost minimized. It is possible that in your example you somehow had had very good initial centres for k=5 and rather poor initial centres for most of other solutions. But SPSS's default gimmick to set initial centres to the utmost juts of the data cloud often provide easy "slide" to final centres which are not far from optimal. Hence the inversions of the SSw I've found in my example are slight, while yours was pronounced.
SSwithin elbow method isn't particularly reliable or sensitive as a criterion to select the "best" number of clusters. Its various standardized versions which compare SSw relative SSbetween or SStotal are far better. Two most well-known are Calinski-Harabasz and Davies-Bouldin clustering criterions. Shown applied to my example:
They unequivocally indicate 5-cluster solution. The two criterions are rather similar and are ideally suited to validate spherical clusters having compact centre (exactly the shape that K-means is inclined to). Calinski-Harabasz is somewhat in favour with similar-sized (in respect to the number of objects) clusters, while Davies-Bouldin a bit tends to prefer equally-distanced from each other clusters.
