Using the elbow method to determine the optimal number of clusters for k-means clustering

K-means is a simple unsupervised machine learning algorithm that groups a dataset into a user-specified number (k) of clusters. The algorithm is somewhat naive–it clusters the data into k clusters, even if k is not the right number of clusters to use. Therefore, when using k-means clustering, users need some way to determine whether they are using the right number of clusters.

One method to validate the number of clusters is the elbow method. The idea of the elbow method is to run k-means clustering on the dataset for a range of values of k (say, k from 1 to 10 in the examples above), and for each value of k calculate the sum of squared errors (SSE). Like this:

var sse = {};
for (var k = 1; k <= maxK; ++k) {
    sse[k] = 0;
    clusters = kmeans(dataset, k);
    clusters.forEach(function(cluster) {
        mean = clusterMean(cluster);
        cluster.forEach(function(datapoint) {
            sse[k] += Math.pow(datapoint - mean, 2);
        });
    });
}

Then, plot a line chart of the SSE for each value of k. If the line chart looks like an arm, then the “elbow” on the arm is the value of k that is the best. The idea is that we want a small SSE, but that the SSE tends to decrease toward 0 as we increase k (the SSE is 0 when k is equal to the number of data points in the dataset, because then each data point is its own cluster, and there is no error between it and the center of its cluster). So our goal is to choose a small value of k that still has a low SSE, and the elbow usually represents where we start to have diminishing returns by increasing k.

Note Dataset A on the left. At the top we see a number line plotting each point in the dataset, and below we see an elbow chart showing the SSE after running k-means clustering for k going from 1 to 10. We see a pretty clear elbow at k = 3, indicating that 3 is the best number of clusters.

However, the elbow method doesn’t always work well; especially if the data is not very clustered. Notice how the elbow chart for Dataset B does not have a clear elbow. Instead, we see a fairly smooth curve, and it’s unclear what is the best value of k to choose. In cases like this, we might try a different method for determining the optimal k, such as computing silhouette scores, or we might reevaluate whether clustering is the right thing to do on our data.

This has practical applications for other methods too, such as determining the best number of classes to use in the Jenks natural breaks algorithm for building color scales in choropleth maps.

You can paste in your own comma-separated list of numbers in the input fields above and click “Parse datasets” to see your numbers plotted and their corresponding elbow charts.