diff --git a/commons-math-legacy/src/main/java/org/apache/commons/math4/legacy/stat/descriptive/moment/Variance.java b/commons-math-legacy/src/main/java/org/apache/commons/math4/legacy/stat/descriptive/moment/Variance.java
index 9c235da70..415bbb1f4 100644
--- a/commons-math-legacy/src/main/java/org/apache/commons/math4/legacy/stat/descriptive/moment/Variance.java
+++ b/commons-math-legacy/src/main/java/org/apache/commons/math4/legacy/stat/descriptive/moment/Variance.java
@@ -297,6 +297,7 @@ public class Variance extends AbstractStorelessUnivariateStatistic implements Se
      *     <li>the weights array contains one or more infinite values</li>
      *     <li>the weights array contains one or more NaN values</li>
      *     <li>the weights array contains negative values</li>
+     *     <li>the weights array does not contain at least one non-zero value (applies when length is non zero)</li>
      *     <li>the start and length arguments do not determine a valid array</li>
      * </ul>
      * <p>
@@ -318,7 +319,7 @@ public class Variance extends AbstractStorelessUnivariateStatistic implements Se
 
         double var = Double.NaN;
 
-        if (MathArrays.verifyValues(values, weights,begin, length)) {
+        if (MathArrays.verifyValues(values, weights, begin, length)) {
             if (length == 1) {
                 var = 0.0;
             } else if (length > 1) {
@@ -356,6 +357,7 @@ public class Variance extends AbstractStorelessUnivariateStatistic implements Se
      *     <li>the weights array contains one or more infinite values</li>
      *     <li>the weights array contains one or more NaN values</li>
      *     <li>the weights array contains negative values</li>
+     *     <li>the weights array does not contain at least one non-zero value (applies when length is non zero)</li>
      * </ul>
      * <p>
      * Does not change the internal state of the statistic.</p>
@@ -488,6 +490,7 @@ public class Variance extends AbstractStorelessUnivariateStatistic implements Se
      *     <li>the weights array contains one or more infinite values</li>
      *     <li>the weights array contains one or more NaN values</li>
      *     <li>the weights array contains negative values</li>
+     *     <li>the weights array does not contain at least one non-zero value (applies when length is non zero)</li>
      *     <li>the start and length arguments do not determine a valid array</li>
      * </ul>
      * <p>
@@ -527,6 +530,10 @@ public class Variance extends AbstractStorelessUnivariateStatistic implements Se
                 }
 
                 if (isBiasCorrected) {
+                    // Note: For this to be valid the weights should correspond to counts
+                    // of each observation where the weights are positive integers; the
+                    // sum of the weights is the total number of observations and should
+                    // be at least 2.
                     var = (accum - (accum2 * accum2 / sumWts)) / (sumWts - 1.0);
                 } else {
                     var = (accum - (accum2 * accum2 / sumWts)) / sumWts;
@@ -566,6 +573,7 @@ public class Variance extends AbstractStorelessUnivariateStatistic implements Se
      *     <li>the weights array contains one or more infinite values</li>
      *     <li>the weights array contains one or more NaN values</li>
      *     <li>the weights array contains negative values</li>
+     *     <li>the weights array does not contain at least one non-zero value (applies when length is non zero)</li>
      * </ul>
      * <p>
      * Does not change the internal state of the statistic.</p>
diff --git a/commons-math-legacy/src/test/java/org/apache/commons/math4/legacy/stat/descriptive/moment/VarianceTest.java b/commons-math-legacy/src/test/java/org/apache/commons/math4/legacy/stat/descriptive/moment/VarianceTest.java
index a3261af32..bd7228d0f 100644
--- a/commons-math-legacy/src/test/java/org/apache/commons/math4/legacy/stat/descriptive/moment/VarianceTest.java
+++ b/commons-math-legacy/src/test/java/org/apache/commons/math4/legacy/stat/descriptive/moment/VarianceTest.java
@@ -19,8 +19,10 @@ package org.apache.commons.math4.legacy.stat.descriptive.moment;
 import org.apache.commons.math4.legacy.stat.descriptive.StorelessUnivariateStatisticAbstractTest;
 import org.apache.commons.math4.legacy.stat.descriptive.UnivariateStatistic;
 import org.apache.commons.math4.legacy.core.MathArrays;
+import org.apache.commons.math4.legacy.exception.MathIllegalArgumentException;
 import org.junit.Assert;
 import org.junit.Test;
+import org.junit.jupiter.api.Assertions;
 
 /**
  * Test cases for the {@link UnivariateStatistic} class.
@@ -114,4 +116,29 @@ public class VarianceTest extends StorelessUnivariateStatisticAbstractTest{
 
     }
 
+    @Test
+    public void testZeroWeights() {
+        Variance variance = new Variance();
+        final double[] values = {1, 2, 3, 4};
+        final double[] weights = new double[values.length];
+
+        // No weights
+        Assertions.assertThrows(MathIllegalArgumentException.class, () -> {
+            variance.evaluate(values, weights);
+        });
+
+        // No length
+        final int begin = 1;
+        final int zeroLength = 0;
+        Assertions.assertEquals(Double.NaN, variance.evaluate(values, weights, begin, zeroLength));
+
+        // One weight (must be non-zero)
+        Assertions.assertThrows(MathIllegalArgumentException.class, () -> {
+            variance.evaluate(values, weights, begin, zeroLength + 1);
+        });
+
+        weights[begin] = Double.MIN_VALUE;
+        Assertions.assertEquals(0.0, variance.evaluate(values, weights, begin, zeroLength + 1));
+    }
+
 }