A marketing researcher uses the k$-$means algorithm to cluster data concerning

a sample of customers of a large apparel retailer. The researcher does not

have any prior information about the number of clusters K$.$ The clustering

allocation for different values of K is compared to an external classification

of the subjects into different types of purchasing behaviors. The table below

reports the adjusted Rand index $($ARI$)$ values for each value of K considered:

K ARI

$20\mathrm{.}43$

$30\mathrm{.}51$

$40\mathrm{.}63$

$50\mathrm{.}59$

$60\mathrm{.}39$

Do you think that K $=4$ corresponds to the optimal number of clusters for

these data? Justify your answer.