diff --git a/x-pack/plugin/ml/src/main/java/org/elasticsearch/xpack/ml/aggs/changepoint/KDE.java b/x-pack/plugin/ml/src/main/java/org/elasticsearch/xpack/ml/aggs/changepoint/KDE.java
index 1a0342981c1d3..551d2db2e9580 100644
--- a/x-pack/plugin/ml/src/main/java/org/elasticsearch/xpack/ml/aggs/changepoint/KDE.java
+++ b/x-pack/plugin/ml/src/main/java/org/elasticsearch/xpack/ml/aggs/changepoint/KDE.java
@@ -38,9 +38,13 @@ private interface KernelFunction {
     }
 
     private ValueAndMagnitude evaluate(IntervalComplementFunction complement, KernelFunction kernel, double x) {
-        if (bandwidth == 0.0 || orderedValues.length == 0) {
+        if (orderedValues.length == 0) {
             return new ValueAndMagnitude(1.0, 0.0);
         }
+        if (bandwidth == 0) {
+            return new ValueAndMagnitude(x != orderedValues[0] ? 0.0 : 1.0, Math.abs(orderedValues[0] - x));
+        }
+
         int a = Math.min(lowerBound(orderedValues, x - 3.0 * bandwidth), orderedValues.length - 1);
         int b = Math.max(lowerBound(orderedValues, x + 3.0 * bandwidth), a + 1);
         // Account for all the values outside the interval [a, b) using the kernel complement.
diff --git a/x-pack/plugin/ml/src/test/java/org/elasticsearch/xpack/ml/aggs/changepoint/ChangePointDetectorTests.java b/x-pack/plugin/ml/src/test/java/org/elasticsearch/xpack/ml/aggs/changepoint/ChangePointDetectorTests.java
new file mode 100644
index 0000000000000..6fe1108a3f1df
--- /dev/null
+++ b/x-pack/plugin/ml/src/test/java/org/elasticsearch/xpack/ml/aggs/changepoint/ChangePointDetectorTests.java
@@ -0,0 +1,49 @@
+/*
+ * Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
+ * or more contributor license agreements. Licensed under the Elastic License
+ * 2.0; you may not use this file except in compliance with the Elastic License
+ * 2.0.
+ */
+
+package org.elasticsearch.xpack.ml.aggs.changepoint;
+
+import org.elasticsearch.test.ESTestCase;
+import org.elasticsearch.xpack.ml.aggs.MlAggsHelper;
+
+import java.util.Arrays;
+
+import static org.hamcrest.Matchers.instanceOf;
+
+public class ChangePointDetectorTests extends ESTestCase {
+
+    /**
+     * Testing the special case where values without the spike have a variance of 0.
+     */
+    public void testConstantWithSpike() {
+        double[] values = new double[25];
+        Arrays.fill(values, 51);
+        values[12] = 500001;
+
+        MlAggsHelper.DoubleBucketValues bucketValues = new MlAggsHelper.DoubleBucketValues(null, values);
+
+        ChangeType change = ChangePointDetector.getChangeType(bucketValues);
+
+        assertEquals(12, change.changePoint());
+        assertThat(change, instanceOf(ChangeType.Spike.class));
+    }
+
+    public void testRandomWithSpike() {
+        double[] values = new double[25];
+        for (int i = 0; i < values.length; i++) {
+            values[i] = randomLongBetween(50, 53);
+        }
+        values[12] = 500001;
+
+        MlAggsHelper.DoubleBucketValues bucketValues = new MlAggsHelper.DoubleBucketValues(null, values);
+
+        ChangeType change = ChangePointDetector.getChangeType(bucketValues);
+
+        assertEquals(12, change.changePoint());
+        assertThat(change, instanceOf(ChangeType.Spike.class));
+    }
+}