K-Means groups unlabelled points into K clusters around centres (centroids). The elbow method helps choose K.

Start
Pick K initial centroids (k-means++ spreads them out smartly).
Assign
Each point joins its nearest centroid.
Update
Each centroid moves to the mean of its points.
Repeat
Until assignments stop changing (or max_iter).

Python

from collections import Counter
from sklearn.cluster import KMeans
from sklearn.datasets import make_blobs

X, _ = make_blobs(n_samples=300, centers=4, cluster_std=0.60, random_state=0)
km = KMeans(n_clusters=4, init="k-means++", n_init=10, random_state=0).fit(X)
print(km.cluster_centers_.shape)          # (4, 2)
print(sorted(Counter(km.labels_).values()))   # [75, 75, 75, 75]   ← 4 equal clusters

Choosing K: the elbow method

Run K-Means for K = 1…10 and record the WCSS (within-cluster sum of squares, inertia_). It always falls as K grows; the “elbow” where it stops falling sharply is a good K.

Python

from sklearn.cluster import KMeans
from sklearn.datasets import make_blobs

X, _ = make_blobs(n_samples=300, centers=4, cluster_std=0.60, random_state=0)
wcss = [KMeans(n_clusters=k, n_init=10, random_state=0).fit(X).inertia_ for k in range(1, 11)]
drops = [round(wcss[i] - wcss[i + 1]) for i in range(5)]
print(drops)       # [1621, 644, 335, 23, 18]   ← big drops until K = 4, then small ones

Python

import matplotlib.pyplot as plt
plt.plot(range(1, 11), wcss, marker="o")
plt.xlabel("Number of clusters (K)"); plt.ylabel("WCSS"); plt.title("Elbow Method")
plt.show()

The same with SciPy

Python

import numpy as np
from scipy.cluster.vq import kmeans, vq
from sklearn.datasets import make_blobs

X, _ = make_blobs(n_samples=300, centers=4, cluster_std=0.60, random_state=0)
centroids, _ = kmeans(X, 4, seed=0)       # centroids
labels, _ = vq(X, centroids)              # nearest centroid for each point
print(centroids.shape, len(np.unique(labels)))   # (4, 2) 4