diff --git a/src/morphobricks_normalization/normalization.py b/src/morphobricks_normalization/normalization.py
index cfc0df3..747ee76 100644
--- a/src/morphobricks_normalization/normalization.py
+++ b/src/morphobricks_normalization/normalization.py
@@ -99,7 +99,14 @@ def robust_z_score(
 
 
 def _parameters(method: str, offset: float, scale: float) -> NormalizationParameters:
-    if not np.isfinite(scale) or scale <= 0:
+    """Make the parameters, refusing a scale that is not finite or not positive, by cause."""
+    if not np.isfinite(scale):
+        raise ValueError(
+            f"The {method} normalization cannot be used for this group: its intensity spread "
+            f"is {scale!r}, not a finite number, because the pixel values are too large for "
+            "float64 sums of squares (overflow). Choose another method or a different range."
+        )
+    if scale <= 0:
         raise ValueError(
             f"The {method} normalization cannot be used for this group: its intensity spread "
             "is zero (all pixels have the same value). Choose a larger group or another method."
diff --git a/src/morphobricks_normalization/pooling.py b/src/morphobricks_normalization/pooling.py
index 5ee29a1..32d3323 100644
--- a/src/morphobricks_normalization/pooling.py
+++ b/src/morphobricks_normalization/pooling.py
@@ -361,7 +361,29 @@ def pooled_statistics(
 def _mean_with_tails(
     values: np.ndarray, weights: np.ndarray, tails: list[tuple[float, float, float]], total: float
 ) -> float:
-    return (weighted_sum(weights, values) + math.fsum(s for _, s, _ in tails)) / total
+    sums = [s for _, s, _ in tails]
+    if not all(math.isfinite(s) for s in sums):
+        # math.fsum would raise a bare ValueError or OverflowError from the standard library.
+        raise ValueError(_non_finite_tail_message(sums))
+    return (weighted_sum(weights, values) + math.fsum(sums)) / total
+
+
+def _non_finite_tail_message(sums: list[float]) -> str:
+    """Refusal text for tail sums that are infinite or NaN, naming each such tail."""
+    bad = ", ".join(
+        f"tail {number} of {len(sums)} has sum {s!r}"
+        for number, s in enumerate(sums, 1)
+        if not math.isfinite(s)
+    )
+    opposing = any(s == math.inf for s in sums) and any(s == -math.inf for s in sums)
+    reason = (
+        "infinite sums of opposite sign have no mean" if opposing else "a sum that is not finite"
+    )
+    return (
+        f"The mean cannot be pooled: {bad} ({reason}). The pixels outside the robust range "
+        "are too large for float64 sums. Use a method that does not need the mean, or "
+        "another range."
+    )
 
 
 def _std_with_tails(
diff --git a/tests/test_normalization.py b/tests/test_normalization.py
index 5a2eb52..4c3d616 100644
--- a/tests/test_normalization.py
+++ b/tests/test_normalization.py
@@ -56,3 +56,11 @@ def test_zero_spread_is_rejected_with_a_clear_message():
     for method in (min_max, z_score, robust_z_score):
         with pytest.raises(ValueError, match="spread is zero"):
             method(constant)
+
+
+def test_non_finite_scale_is_refused_as_overflow_not_zero_spread():
+    plane = np.array([[-1.7e308, 1.7e308], [-1.7e308, 1.7e308]])
+    pooled = PooledHistogram([compute_plane_histogram(plane)])
+    with pytest.raises(ValueError, match="not a finite number") as caught:
+        z_score(pooled)
+    assert "spread is zero" not in str(caught.value)
diff --git a/tests/test_pooling.py b/tests/test_pooling.py
index a2555a5..b825989 100644
--- a/tests/test_pooling.py
+++ b/tests/test_pooling.py
@@ -9,8 +9,11 @@
     pooled_statistics,
 )
 from morphobricks_normalization.pooling import DEFAULT_PERCENTILES
+from morphobricks_normalization.ranges import HistogramSettings
 from simulation import direct_stats, make_records, simulate_experiment
 
+ROBUST = HistogramSettings(histogram_range="robust", robust_quantiles=(1.0, 99.0))
+
 LEVELS = ("image", "technical_repeat", "biological_repeat", "condition")
 
 
@@ -83,3 +86,17 @@ def test_percentile_limits():
     assert pooled.percentile(0) == 0 and pooled.percentile(100) == 9
     with pytest.raises(ValueError):
         pooled.percentile(101)
+
+
+def test_opposing_infinite_tail_sums_are_a_named_refusal():
+    # Float64 pixels at both ends overflow the tails' sums to +inf and -inf; the mean then
+    # needs a refusal that says why, not a bare error from math.fsum.
+    plane = np.full((80, 80), 1.0)
+    plane.ravel()[:5] = -1.7e308
+    plane.ravel()[5:10] = 1.7e308
+    plane.ravel()[10:] = np.linspace(0, 10, 6390)
+    histogram = compute_plane_histogram(plane, settings=ROBUST)
+    assert histogram.underflow.sum == -np.inf and histogram.overflow.sum == np.inf
+    with pytest.raises(ValueError, match="opposite sign") as caught:
+        _ = PooledHistogram([histogram]).mean
+    assert "tail 1 of 2" in str(caught.value) and "tail 2 of 2" in str(caught.value)