You apply 2-means clustering to a set of five observations with two features. You are given the following initial cluster assignments:

Observation	X1	X2	Initial cluster
1	1	3	1
2	0	4	1
3	6	2	1
4	5	2	2
5	1	6	2

Calculate the total within-cluster variation of the initial cluster assignments, based on Euclidean distance measure.

• 32.0
• 70.3
• 77.3
• 118.3
• 141.0

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: C

The means for cluster 1 are (1 + 0 + 6)/3 = 2.3333 for X₁ and (3 + 4 + 2)/3 = 3 for X₂ and the variation is

(1 − 2.3333)2 + (3 − 3)2 + (0 − 2.3333)2 + (4 − 3)2 + (6 − 2.3333)2 + (2 − 3)2 = 22.6667.

The means for cluster 2 are (5 + 1)/2 = 3 for X₁ and (2 + 6)/2 = 4 for X₂ and the variation is

(5 − 3)2 + (2 − 4)2 + (1 − 3)2 + (6 − 4)2 = 16.

The total within-cluster variation is (per equation (10.12) in the first edition of ISLR)

2(22.6667 + 16) = 77.33.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.