SAS Output

Breakfast Cereals - Johnson & Wichern Table 11.9

Raw Data

Obs	Brand	Label	Calories	Protein	Fat	Na	Fiber	Carbs	Sugar	K
1	ACCheerios	ACC	110	2	2	180	1.5	10.5	10	70
2	Cheerios	Chee	110	6	2	290	2.0	17.0	1	105
3	CocoaPuffs	Coco	110	1	1	180	0.0	12.0	13	55
4	CountChocula	CC	110	1	1	180	0.0	12.0	13	65
5	GoldenGrahams	GG	110	1	1	280	0.0	15.0	9	45
6	HoneyNutCheerios	HNC	110	3	1	250	1.5	11.5	10	90
7	Kix	Kix	110	2	1	260	0.0	21.0	3	40
8	LuckyCharms	LC	110	2	1	180	0.0	12.0	12	55
9	MultiGrainCheerios	MGC	100	2	1	220	2.0	15.0	6	90
10	OatmealRaisinCrisp	ORC	130	3	2	170	1.5	13.5	10	120
11	RaisinNutBran	RNB	100	3	2	140	2.5	10.5	8	140
12	TotalCornFlakes	TCF	110	2	1	200	0.0	21.0	3	35
13	TotalRaisinBran	TRB	140	3	1	190	4.0	15.0	14	230
14	TotalWholeGrain	TWG	100	3	1	200	3.0	16.0	3	110
15	Trix	Trix	110	1	1	140	0.0	13.0	12	25
16	Cheaties	Chea	100	3	1	200	3.0	17.0	3	110
17	WheatiesHoneyGold	WHG	110	2	1	200	1.0	16.0	8	60
18	AllBran	AB	70	4	1	260	9.0	7.0	5	320
19	AppleJacks	AJ	110	2	0	125	1.0	11.0	14	30
20	CornFlakes	CF	100	2	0	290	1.0	21.0	2	35
21	CornPops	CP	110	1	0	90	1.0	13.0	12	20
22	CracklinOatBran	COB	110	3	3	140	4.0	10.0	7	160
23	Crispix	Cris	110	2	0	220	1.0	21.0	3	30
24	FrootLoops	FL	110	2	1	125	1.0	11.0	13	30
25	FrostedFlakes	FF	110	1	0	200	1.0	14.0	11	25
26	FrostedMiniWheats	FMW	100	3	0	0	3.0	14.0	7	100
27	FruitfulBran	FB	120	3	0	240	5.0	14.0	12	190
28	JustRightCrunchyNuggets	JRCN	110	2	1	170	1.0	17.0	6	60
29	MueslixCrispyBlend	MCB	160	3	2	150	3.0	17.0	13	160
30	NutNHoneyCrunch	NNHC	120	2	1	190	0.0	15.0	9	40
31	NutriGrainAlmondRaisin	NGAR	140	3	2	220	3.0	21.0	7	130
32	NutriGrainWheat	NGW	90	3	0	170	3.0	18.0	2	90
33	Product19	P19	100	3	0	320	1.0	20.0	3	45
34	RaisinBran	RB	120	3	1	210	5.0	14.0	12	240
35	RiceKrispies	RK	110	2	0	290	0.0	22.0	3	35
36	Smacks	Smac	110	2	1	70	1.0	9.0	15	40
37	SpecialK	SK	110	6	0	230	1.0	16.0	3	55
38	CapNCrunch	CNC	120	1	2	220	0.0	12.0	12	35
39	HoneyGrahamOhs	HGO	120	1	2	220	1.0	12.0	11	45
40	Life	Life	100	4	2	150	2.0	12.0	6	95
41	PuffedRice	PR	50	1	0	0	0.0	13.0	0	15
42	PuffedWheat	PW	50	2	0	0	1.0	10.0	0	50
43	QuakerOatmeal	QO	100	5	2	0	2.7	1.0	1	110

Breakfast Cereals - Johnson & Wichern Table 11.9

Raw Data

Summary Statistics

The MEANS Procedure

Calories
Protein
Fat
Sodium
Fiber
Carbohydrates
Sugar
Potassium

107.9069767
2.4651163
0.9767442
180.4651163
1.7139535
14.2558140
7.6046512
84.4186047

18.9684123
1.2218027
0.8014384
79.2134996
1.7992831
4.2571678
4.5363061
66.1096609

50.0000000
1.0000000
0
0
0
1.0000000
0
15.0000000

160.0000000
6.0000000
3.0000000
320.0000000
9.0000000
22.0000000
15.0000000
320.0000000

Breakfast Cereals - Johnson & Wichern Table 11.9

Standardized Data

Summary Statistics

The MEANS Procedure

Calories
Protein
Fat
Sodium
Fiber
Carbohydrates
Sugar
Potassium

-3.65341E-16

7.229359E-17

6.196594E-17

-1.03277E-16

2.581914E-18

-1.03277E-16

-6.19659E-17

-7.22936E-17

1.0000000
1.0000000
1.0000000
1.0000000
1.0000000
1.0000000
1.0000000
1.0000000

-3.0528110
-1.1991431
-1.2187390
-2.2782116
-0.9525758
-3.1137636
-1.6763973
-1.0500523

2.7463038
2.8931707
2.5245308
1.7615038
4.0494164
1.8190935
1.6302579
3.5634942

Breakfast Cereals - Johnson & Wichern Table 11.9

Single Linkage Cluster Analysis

The CLUSTER Procedure

Single Linkage Cluster Analysis

Eigenvalues of the Covariance Matrix
	Eigenvalue	Difference	Proportion	Cumulative
1	2.54756944	0.69280765	0.3184	0.3184
2	1.85476179	0.08593184	0.2318	0.5503
3	1.76882996	0.90054660	0.2211	0.7714
4	0.86828336	0.37098255	0.1085	0.8799
5	0.49730081	0.13987236	0.0622	0.9421
6	0.35742845	0.29692509	0.0447	0.9868
7	0.06050337	0.01518055	0.0076	0.9943
8	0.04532281		0.0057	1.0000

Root-Mean-Square Total-Sample Standard Deviation = 1

Mean Distance Between Observations = 3.711894

Cluster History
NCL	Clusters Joined		FREQ	Norm Min Dist	T i e
42	Coco	CC	2	0.0408
41	TWG	Chea	2	0.0633
40	CNC	HGO	2	0.1662
39	WHG	JRCN	2	0.1689
38	CL42	Trix	3	0.2024
37	Kix	TCF	2	0.2051
36	CF	RK	2	0.2239
35	CL38	LC	4	0.2284
34	CL39	NNHC	3	0.2407
33	FL	Smac	2	0.2584
32	CL36	P19	3	0.2612
31	CL35	CL33	6	0.2744
30	MGC	CL34	4	0.2833
29	CL32	Cris	4	0.2842
28	CL31	CL30	10	0.3047
27	AJ	CP	2	0.3075
26	RNB	Life	2	0.3349
25	ACC	CL40	3	0.3403
24	CL28	CL27	12	0.3414
23	CL24	CL41	14	0.3436
22	PR	PW	2	0.357
21	TRB	RB	2	0.3571
20	CL37	CL29	6	0.3575
19	CL23	CL20	20	0.3729
18	CL19	FF	21	0.3846
17	CL18	GG	22	0.3954
16	CL17	NGW	23	0.3972
15	CL25	CL16	26	0.4023
14	CL21	FB	3	0.4061
13	CL15	HNC	27	0.4362
12	CL26	COB	3	0.4413
11	CL13	CL12	30	0.453
10	CL11	ORC	31	0.4561
9	CL14	MCB	4	0.5792
8	CL9	NGAR	5	0.586
7	CL10	CL8	36	0.5865
6	CL7	FMW	37	0.713
5	Chee	SK	2	0.7587
4	CL6	CL5	39	0.7856
3	CL4	QO	40	0.9467
2	CL3	CL22	42	0.9597
1	CL2	AB	43	1.1899

The TREE Procedure

Single Linkage Cluster Analysis

The TREE Procedure

Average Linkage Cluster Analysis

The TREE Procedure

Complete Linkage Cluster Analysis

The TREE Procedure

Ward's Minimum Variance Cluster Analysis

Breakfast Cereals - Johnson & Wichern Table 11.9

Ward's Method Cluster Analysis

Cereals Assigned To One of Three Clusters

Obs	Cl#	Brand	Label	Cal	Protein	Fat	Na	Fiber	Carbs	Sugar	K
1	1	ACCheerios	ACC	110	2	2	180	1.5	10.5	10	70
2	1	AppleJacks	AJ	110	2	0	125	1.0	11.0	14	30
3	1	CapNCrunch	CNC	120	1	2	220	0.0	12.0	12	35
4	1	CocoaPuffs	Coco	110	1	1	180	0.0	12.0	13	55
5	1	CornPops	CP	110	1	0	90	1.0	13.0	12	20
6	1	CountChocula	CC	110	1	1	180	0.0	12.0	13	65
7	1	FrootLoops	FL	110	2	1	125	1.0	11.0	13	30
8	1	FrostedFlakes	FF	110	1	0	200	1.0	14.0	11	25
9	1	GoldenGrahams	GG	110	1	1	280	0.0	15.0	9	45
10	1	HoneyGrahamOhs	HGO	120	1	2	220	1.0	12.0	11	45
11	1	HoneyNutCheerios	HNC	110	3	1	250	1.5	11.5	10	90
12	1	JustRightCrunchyNuggets	JRCN	110	2	1	170	1.0	17.0	6	60
13	1	LuckyCharms	LC	110	2	1	180	0.0	12.0	12	55
14	1	NutNHoneyCrunch	NNHC	120	2	1	190	0.0	15.0	9	40
15	1	OatmealRaisinCrisp	ORC	130	3	2	170	1.5	13.5	10	120
16	1	Smacks	Smac	110	2	1	70	1.0	9.0	15	40
17	1	Trix	Trix	110	1	1	140	0.0	13.0	12	25
18	1	WheatiesHoneyGold	WHG	110	2	1	200	1.0	16.0	8	60
19	2	Cheaties	Chea	100	3	1	200	3.0	17.0	3	110
20	2	Cheerios	Chee	110	6	2	290	2.0	17.0	1	105
21	2	CornFlakes	CF	100	2	0	290	1.0	21.0	2	35
22	2	Crispix	Cris	110	2	0	220	1.0	21.0	3	30
23	2	FrostedMiniWheats	FMW	100	3	0	0	3.0	14.0	7	100
24	2	Kix	Kix	110	2	1	260	0.0	21.0	3	40
25	2	MultiGrainCheerios	MGC	100	2	1	220	2.0	15.0	6	90
26	2	NutriGrainWheat	NGW	90	3	0	170	3.0	18.0	2	90
27	2	Product19	P19	100	3	0	320	1.0	20.0	3	45
28	2	PuffedRice	PR	50	1	0	0	0.0	13.0	0	15
29	2	PuffedWheat	PW	50	2	0	0	1.0	10.0	0	50
30	2	RiceKrispies	RK	110	2	0	290	0.0	22.0	3	35
31	2	SpecialK	SK	110	6	0	230	1.0	16.0	3	55
32	2	TotalCornFlakes	TCF	110	2	1	200	0.0	21.0	3	35
33	2	TotalWholeGrain	TWG	100	3	1	200	3.0	16.0	3	110
34	3	AllBran	AB	70	4	1	260	9.0	7.0	5	320
35	3	CracklinOatBran	COB	110	3	3	140	4.0	10.0	7	160
36	3	FruitfulBran	FB	120	3	0	240	5.0	14.0	12	190
37	3	Life	Life	100	4	2	150	2.0	12.0	6	95
38	3	MueslixCrispyBlend	MCB	160	3	2	150	3.0	17.0	13	160
39	3	NutriGrainAlmondRaisin	NGAR	140	3	2	220	3.0	21.0	7	130
40	3	QuakerOatmeal	QO	100	5	2	0	2.7	1.0	1	110
41	3	RaisinBran	RB	120	3	1	210	5.0	14.0	12	240
42	3	RaisinNutBran	RNB	100	3	2	140	2.5	10.5	8	140
43	3	TotalRaisinBran	TRB	140	3	1	190	4.0	15.0	14	230

Breakfast Cereals - Johnson & Wichern Table 11.9

K-Means Clustering: Least-squares criterion

The FASTCLUS Procedure

Replace=FULL Radius=0 Maxclusters=3 Maxiter=10 Converge=0.02

Initial Seeds
Cluster	Calories	Protein	Fat	Sodium	Fiber	Carbohydrates	Sugar	Potassium
1	-1.998426446	1.256245182	0.029017596	1.004057189	4.049416386	-1.704375860	-0.574178881	3.563494233
2	0.110342565	2.893170721	1.276774203	1.382780515	0.158978045	0.644603691	-1.455953590	0.311322053
3	-3.052810952	-1.199143128	-1.218739012	-2.278211634	-0.952575767	-0.294988130	-1.676397268	-1.050052348

Minimum Distance Between Initial Seeds =	6.390159

Iteration History
Iteration	Criterion	Relative Change in Cluster Seeds
Iteration	Criterion	1	2	3
1	1.4707	0.4753	0.5214	0.5829
2	0.8291	0.1081	0.0224	0.0203
3	0.8244	0	0	0

Convergence criterion is satisfied.

Criterion Based on Final Seeds =	0.8244

Cluster Summary
Cluster	Frequency	RMS Std Deviation	Maximum Distance from Seed to Observation	Radius Exceeded	Nearest Cluster	Distance Between Cluster Centroids
1	4	0.9286	3.6763		2	3.7323
2	27	0.8752	4.9961		3	2.3859
3	12	0.7819	3.8213		2	2.3859

Statistics for Variables
Variable	Total STD	Within STD	R-Square	RSQ/(1-RSQ)
Calories	1.00000	0.98040	0.084590	0.092407
Protein	1.00000	0.90450	0.220835	0.283425
Fat	1.00000	0.93408	0.169037	0.203424
Sodium	1.00000	0.83191	0.340881	0.517177
Fiber	1.00000	0.67434	0.566919	1.309039
Carbohydrates	1.00000	0.94307	0.152979	0.180608
Sugar	1.00000	0.90651	0.217363	0.277732
Potassium	1.00000	0.57902	0.680704	2.131887
OVER-ALL	1.00000	0.85477	0.304164	0.437119

Pseudo F Statistic =	8.74

Approximate Expected Over-All R-Squared =	0.24233

Cubic Clustering Criterion =	3.044

WARNING: The two values above are invalid for correlated variables.

Cluster Means
Cluster	Calories	Protein	Fat	Sodium	Fiber	Carbohydrates	Sugar	Potassium
1	0.242140628	0.642398104	-0.282921556	0.562213309	2.243141442	-0.412437107	0.693372264	2.429015565
2	0.168919481	0.225588360	0.306296842	0.326095680	-0.094209212	0.296606720	-0.353735203	-0.078042230
3	-0.460782376	-0.721706512	-0.594860708	-0.921119717	-0.535743087	-0.529886085	0.564780119	-0.634076837

Cluster Standard Deviations
Cluster	Calories	Protein	Fat	Sodium	Fiber	Carbohydrates	Sugar	Potassium
1	1.574237615	0.409231385	0.623878304	0.392499557	1.232355136	0.868381449	0.870217530	0.823890408
2	0.750326459	1.054823139	1.057247314	0.815244941	0.652950687	1.107284088	0.797169055	0.617545164
3	1.220089700	0.547189770	0.651619636	0.950626105	0.481316919	0.361134989	1.131229092	0.364148749

Breakfast Cereals - Johnson & Wichern Table 11.9

K-Means Clustering: Least absolute values criterion

The FASTCLUS Procedure

Replace=FULL Radius=0 Maxclusters=3 Maxiter=20 Converge=0.0001 Least=1

Initial Seeds
Cluster	Calories	Protein	Fat	Sodium	Fiber	Carbohydrates	Sugar	Potassium
1	-1.998426446	1.256245182	0.029017596	1.004057189	4.049416386	-1.704375860	-0.574178881	3.563494233
2	0.110342565	2.893170721	1.276774203	1.382780515	0.158978045	0.644603691	-1.455953590	0.311322053
3	-3.052810952	-1.199143128	-1.218739012	-2.278211634	-0.952575767	-0.294988130	-1.676397268	-1.050052348

Minimum Distance Between Initial Seeds =	15.74554

Preliminary L(1) Scale Estimate =	8.475935

Number of Bins =	100

Iteration History
Iteration	Criterion	Maximum Bin Size	Relative Change in Cluster Seeds
Iteration	Criterion	Maximum Bin Size	1	2	3
1	1.0595	0.0498	0.6316	0.7829	1.1225
2	0.6024	0.0266	1.0579	0.0503	0.2902
3	0.5582	0.0281	0.1280	0.0731	0.1557
4	0.5529	0.0121	0.0107	0.0314	0.00431
5	0.5523	0.0289	0.0789	0.1455	0.00488
6	0.5477	0.0121	0.00560	0.1217	0.0635
7	0.5471	0.0303	0.0858	0.0301	0.000645
8	0.5470	0.00635	0.00343	0.000984	0.000445
9	0.5469	0.00610	6.831E-8	2.541E-6	0.000032
10	0.5469	0.00610	7.184E-9	1.918E-7	6.452E-7
11	0.5469	0.0373	6.99E-10	0.00426	5.724E-7
12	0.5469	0.00610	5.44E-11	0.00128	3.184E-8
13	0.5469	0.00610	5.44E-12	1.92E-10	6.46E-10

Convergence criterion is satisfied.

Criterion Based on Final Seeds =	0.5469

Cluster Summary
Cluster	Frequency	Mean Absolute Deviation	Maximum Distance from Seed to Observation	Radius Exceeded	Nearest Cluster	Distance Between Cluster Medians
1	6	0.7638	11.5599		2	6.8494
2	22	0.6667	10.7255		3	6.7846
3	15	0.4069	9.9416		2	6.7846

Cluster Medians
Cluster	Calories	Protein	Fat	Sodium	Fiber	Carbohydrates	Sugar	Potassium
1	0.637534817	0.437782521	0.029017778	0.247133354	1.550865728	-0.060090175	0.968926861	1.900903140
2	-0.152987303	0.437782108	0.029017459	0.246610811	-0.118910408	0.527976322	-1.015065962	0.084426137
3	0.110342664	-1.199142763	0.029017527	-0.005871816	-0.952575288	-0.529885866	0.968926725	-0.671893066

Mean Absolute Deviations from Final Seeds
Cluster	Calories	Protein	Fat	Sodium	Fiber	Carbohydrates	Sugar	Potassium
1	1.265261406	0.163692554	0.998205286	0.580709099	0.889243049	0.704693865	0.661331032	0.847077405
2	0.552296646	0.701539517	0.831837738	0.835595909	0.508138885	0.900442161	0.524865899	0.479002104
3	0.338909305	0.350769759	0.623878304	0.667274431	0.297737628	0.293622444	0.472379309	0.210688895

The ACECLUS Procedure

Approximate Covariance Estimation for Cluster Analysis

Observations	43	Proportion	0.0200
Variables	8	Converge	0.00100

Means and Standard Deviations
Variable	Mean	Standard Deviation
Calories	0	1.0000
Protein	0	1.0000
Fat	0	1.0000
Sodium	0	1.0000
Fiber	0	1.0000
Carbohydrates	0	1.0000
Sugar	0	1.0000
Potassium	0	1.0000

COV: Total Sample Covariances
	Calories	Protein	Fat	Sodium	Fiber	Carbohydrates	Sugar	Potassium
Calories	1.000000000	0.032731866	0.388274042	0.336599201	-0.019354957	0.255935380	0.579536593	0.141408489
Protein	0.032731866	1.000000000	0.205831787	0.093654934	0.512511257	-0.076061109	-0.399911986	0.500116652
Fat	0.388274042	0.205831787	1.000000000	0.009550521	0.163692191	-0.322713374	0.187333260	0.312059211
Sodium	0.336599201	0.093654934	0.009550521	1.000000000	0.043386933	0.567296810	-0.048839528	0.114399813
Fiber	-0.019354957	0.512511257	0.163692191	0.043386933	1.000000000	-0.240752843	-0.034313042	0.928831438
Carbohydrates	0.255935380	-0.076061109	-0.322713374	0.567296810	-0.240752843	1.000000000	-0.315192130	-0.223223205
Sugar	0.579536593	-0.399911986	0.187333260	-0.048839528	-0.034313042	-0.315192130	1.000000000	0.081387284
Potassium	0.141408489	0.500116652	0.312059211	0.114399813	0.928831438	-0.223223205	0.081387284	1.000000000

Initial Within-Cluster Covariance Estimate = Full Covariance Matrix

Threshold =	0.552617

Iteration History
Iteration	RMS Distance	Distance Cutoff	Pairs Within Cutoff	Convergence Measure
1	4.000	2.210	84.0000	0.284963
2	14.678	8.111	403.0	0.152804
3	13.910	7.687	511.0	0.074106
4	13.432	7.423	529.0	0.033408
5	13.452	7.434	535.0	0.014791
6	13.399	7.405	535.0	0.000000

Algorithm converged.

ACE: Approximate Covariance Estimate Within Clusters
	Calories	Protein	Fat	Sodium	Fiber	Carbohydrates	Sugar	Potassium
Calories	0.1607847706	-.1072669192	0.1450865398	0.0095165048	-.1233617534	-.0841971663	0.1796463486	-.0549272429
Protein	-.1072669192	0.7443821237	0.2166560829	0.0691399360	0.3250073164	0.0987329434	-.4444861208	0.3096830742
Fat	0.1450865398	0.2166560829	0.9661488882	-.0290746168	0.1370745943	-.4197839404	0.1290468670	0.3027784622
Sodium	0.0095165048	0.0691399360	-.0290746168	0.7255049985	-.1543561640	0.4428389930	-.3956017094	-.0707612154
Fiber	-.1233617534	0.3250073164	0.1370745943	-.1543561640	0.3778825715	-.0827838856	-.2018676273	0.2750704261
Carbohydrates	-.0841971663	0.0987329434	-.4197839404	0.4428389930	-.0827838856	0.7673735237	-.6612580299	-.1149794722
Sugar	0.1796463486	-.4444861208	0.1290468670	-.3956017094	-.2018676273	-.6612580299	0.8512372540	-.1429634491
Potassium	-.0549272429	0.3096830742	0.3027784622	-.0707612154	0.2750704261	-.1149794722	-.1429634491	0.3070724140

Eigenvalues of Inv(ACE)*(COV-ACE)
	Eigenvalue	Difference	Proportion	Cumulative
1	71.8123	66.3345	0.8783	0.8783
2	5.4778	1.5173	0.0670	0.9452
3	3.9606	3.2757	0.0484	0.9937
4	0.6849	0.3489	0.00838	1.0021
5	0.3360	0.4370	0.00411	1.0062
6	-0.1009	0.0783	-0.00123	1.0049
7	-0.1792	0.0451	-0.00219	1.0027
8	-0.2243		-0.00274	1.0000

Eigenvectors (Raw Canonical Coefficients)
	Can1	Can2	Can3	Can4	Can5	Can6	Can7	Can8
Calories	-2.95286	4.60795	1.28743	-0.46250	-0.73439	-0.61697	-0.68900	0.85072
Protein	1.28958	-0.40249	0.68388	0.79341	1.22080	0.70055	0.34409	-0.73407
Fat	2.08206	-1.96164	0.67740	0.08682	-0.76870	0.58606	0.64977	-0.51082
Sodium	0.68934	0.26105	-0.62878	1.29344	-0.19473	-0.46657	0.30988	0.46676
Fiber	4.22198	-0.46922	0.25869	1.74952	-2.07466	0.96160	-1.54820	-0.68199
Carbohydrates	8.66311	-3.69952	0.73498	-0.48069	0.52716	1.09485	0.10350	-0.52198
Sugar	9.13032	-3.49735	0.02093	0.67211	0.98335	0.33202	0.28976	-1.40260
Potassium	0.16191	1.24269	-2.13168	-2.49584	1.61989	-0.78975	1.32810	0.89566

Standardized Canonical Coefficients
	Can1	Can2	Can3	Can4	Can5	Can6	Can7	Can8
Calories	-2.95286	4.60795	1.28743	-0.46250	-0.73439	-0.61697	-0.68900	0.85072
Protein	1.28958	-0.40249	0.68388	0.79341	1.22080	0.70055	0.34409	-0.73407
Fat	2.08206	-1.96164	0.67740	0.08682	-0.76870	0.58606	0.64977	-0.51082
Sodium	0.68934	0.26105	-0.62878	1.29344	-0.19473	-0.46657	0.30988	0.46676
Fiber	4.22198	-0.46922	0.25869	1.74952	-2.07466	0.96160	-1.54820	-0.68199
Carbohydrates	8.66311	-3.69952	0.73498	-0.48069	0.52716	1.09485	0.10350	-0.52198
Sugar	9.13032	-3.49735	0.02093	0.67211	0.98335	0.33202	0.28976	-1.40260
Potassium	0.16191	1.24269	-2.13168	-2.49584	1.61989	-0.78975	1.32810	0.89566

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=2 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	31	0.0003805	0	.
2	3	0.00006715	0	.
3	2	0.00004476	0	.
4	1	0.00002238	0	.
5	1	0.00002238	0	.
6	1	0.00002238	0	.
7	1	0.00002238	0	.
8	1	0.00002238	0	.
9	1	0.00002238	0	.
10	1	0.00002238	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=2.1 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	31	0.00027269	0	.
2	3	0.00004545	0	.
3	2	0.0000303	0	.
4	1	0.00001515	0	.
5	1	0.00001515	0	.
6	1	0.00001515	0	.
7	1	0.00001515	0	.
8	1	0.00001515	0	.
9	1	0.00001515	0	.
10	1	0.00001515	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=2.2 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	36	0.00020883	0	.
2	2	0.00002088	0	.
3	1	0.00001044	0	.
4	1	0.00001044	0	.
5	1	0.00001044	0	.
6	1	0.00001044	0	.
7	1	0.00001044	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=2.3 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	36	0.00015365	0	.
2	2	0.00001463	0	.
3	1	7.31679E-6	0	.
4	1	7.31679E-6	0	.
5	1	7.31679E-6	0	.
6	1	7.31679E-6	0	.
7	1	7.31679E-6	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=2.4 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	36	0.00010931	0	.
2	2	0.00001041	0	.
3	1	5.20539E-6	0	.
4	1	5.20539E-6	0	.
5	1	5.20539E-6	0	.
6	1	5.20539E-6	0	.
7	1	5.20539E-6	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=2.5 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	36	0.00009012	0	.
2	2	7.51023E-6	0	.
3	1	3.75511E-6	0	.
4	1	3.75511E-6	0	.
5	1	3.75511E-6	0	.
6	1	3.75511E-6	0	.
7	1	3.75511E-6	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=2.6 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	36	0.00007134	0	.
2	2	5.48765E-6	0	.
3	1	2.74383E-6	0	.
4	1	2.74383E-6	0	.
5	1	2.74383E-6	0	.
6	1	2.74383E-6	0	.
7	1	2.74383E-6	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=2.7 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	37	0.00005883	0	.
2	2	4.05754E-6	0	.
3	1	2.02877E-6	0	.
4	1	2.02877E-6	0	.
5	1	2.02877E-6	0	.
6	1	2.02877E-6	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=2.8 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	37	0.00004398	0	.
2	2	3.03326E-6	0	.
3	1	1.51663E-6	0	.
4	1	1.51663E-6	0	.
5	1	1.51663E-6	0	.
6	1	1.51663E-6	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=2.9 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	37	0.00003436	0	.
2	2	2.29081E-6	0	.
3	2	2.29081E-6	0	.
4	1	1.14541E-6	0	.
5	1	1.14541E-6	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=3 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	39	0.00002707	0	.
2	2	1.74664E-6	0	.
3	1	8.7332E-7	0	.
4	1	8.7332E-7	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=3.1 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	39	0.0000215	0	.
2	2	1.34363E-6	0	.
3	1	6.71815E-7	0	.
4	1	6.71815E-7	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=3.2 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	39	0.00001668	0	.
2	2	1.04225E-6	0	.
3	1	5.21127E-7	0	.
4	1	5.21127E-7	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=3.3 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	39	0.00001344	0	.
2	2	8.1482E-7	0	.
3	1	4.0741E-7	0	.
4	1	4.0741E-7	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=3.4 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	39	0.00001091	0	.
2	2	6.41715E-7	0	.
3	1	3.20857E-7	0	.
4	1	3.20857E-7	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=3.5 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	39	8.90568E-6	0	.
2	2	5.08896E-7	0	.
3	1	2.54448E-7	0	.
4	1	2.54448E-7	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

R=3.6 METHOD=1

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	42	7.31183E-6	0	.
2	1	2.03106E-7	0	.

Breakfast Cereals - Johnson & Wichern Table 11.9

Non-parametric Density Clustering - Bump Hunting

The MODECLUS Procedure

Cluster Summary
R	Number of Clusters	Frequency of Unclassified Objects
2	10	0
2.1	10	0
2.2	7	0
2.3	7	0
2.4	7	0
2.5	7	0
2.6	7	0
2.7	6	0
2.8	6	0
2.9	5	0
3	4	0
3.1	4	0
3.2	4	0
3.3	4	0
3.4	4	0
3.5	4	0
3.6	2	0