diff --git a/src/ert/gui/tools/plot/plot_widget.py b/src/ert/gui/tools/plot/plot_widget.py
index 00792acd0ab..76a85e986ac 100644
--- a/src/ert/gui/tools/plot/plot_widget.py
+++ b/src/ert/gui/tools/plot/plot_widget.py
@@ -203,7 +203,7 @@ def __init__(
         vbox.addSpacing(8)
         self.setLayout(vbox)
 
-        self._negative_values_in_data = False
+        self._negative_or_non_unique_values_in_data = False
         self.resetPlot()
 
     @property
@@ -221,7 +221,7 @@ def _sync_log_checkbox(self, key_def: PlotApiKeyDefinition | None = None) -> Non
                 "DistributionPlot",
                 "GaussianKDEPlot",
             }
-            and self._negative_values_in_data is False
+            and self._negative_or_non_unique_values_in_data is False
         ):
             if key_def is not None:
                 a = key_def.parameter
@@ -271,7 +271,7 @@ def updatePlot(
             plot_context.log_scale = (
                 self._log_checkbox.isVisible()
                 and self._log_checkbox.isChecked()
-                and self._negative_values_in_data is False
+                and self._negative_or_non_unique_values_in_data is False
             )
             plot_context.extended_plot_information = (
                 self._extended_plot_information_checkbox.isVisible()
diff --git a/src/ert/gui/tools/plot/plot_window.py b/src/ert/gui/tools/plot/plot_window.py
index 1936bbd9a86..75a60a15cc5 100644
--- a/src/ert/gui/tools/plot/plot_window.py
+++ b/src/ert/gui/tools/plot/plot_window.py
@@ -415,15 +415,22 @@ def fetch_data(
                     elif result is not None:
                         ensemble_to_data_map[ensemble] = result
 
-            negative_values_in_data = False
+            negative_or_non_unique_values_in_data = False
             if key_def.parameter is not None and key_def.parameter.type == "gen_kw":
                 for data in ensemble_to_data_map.values():
                     numeric = data.select_dtypes(include=["number"])
-                    if not numeric.empty and numeric.le(0).any().any():
-                        negative_values_in_data = True
+                    # Need non-unique check to disable log scale for
+                    # single realization runs, even though the
+                    # distribution is not set as CONSTANT
+                    if not numeric.empty and (
+                        numeric.le(0).any().any() or numeric.nunique().le(1).all()
+                    ):
+                        negative_or_non_unique_values_in_data = True
                         break
 
-            plot_widget._negative_values_in_data = negative_values_in_data
+            plot_widget._negative_or_non_unique_values_in_data = (
+                negative_or_non_unique_values_in_data
+            )
             observations = pd.DataFrame()
             if key_def.observations and selected_ensembles:
                 try:
diff --git a/src/ert/gui/tools/plot/plottery/plots/histogram.py b/src/ert/gui/tools/plot/plottery/plots/histogram.py
index 5bd0e9c1a3a..5e91a88c8f3 100644
--- a/src/ert/gui/tools/plot/plottery/plots/histogram.py
+++ b/src/ert/gui/tools/plot/plottery/plots/histogram.py
@@ -188,9 +188,6 @@ def _plotHistogram(
     if minimum is not None and maximum is not None:
         # Ensure we have at least 2 bin edges to create 1 bin
         effective_bin_count = max(bin_count + 1, 2)
-        if minimum == maximum:
-            minimum -= 0.5
-            maximum += 0.5
         if use_log_scale:
             bins = _histogramLogBins(effective_bin_count, minimum, maximum)  # type: ignore
             axes.set_xscale("log")