From a11ac25e464f3bd98f693c32bb1e2cb67e2c4cc3 Mon Sep 17 00:00:00 2001
From: Raymond Hettinger <raymond.hettinger@gmail.com>
Date: Fri, 14 May 2021 11:40:20 -0700
Subject: [PATCH 1/4] Improve speed and accuracy for correlation()

---
 Lib/statistics.py | 14 ++++++++------
 1 file changed, 8 insertions(+), 6 deletions(-)

diff --git a/Lib/statistics.py b/Lib/statistics.py
index db8c581068b7dd..84b748fc28f9f2 100644
--- a/Lib/statistics.py
+++ b/Lib/statistics.py
@@ -884,8 +884,8 @@ def covariance(x, y, /):
         raise StatisticsError('covariance requires at least two data points')
     xbar = fmean(x)
     ybar = fmean(y)
-    total = fsum((xi - xbar) * (yi - ybar) for xi, yi in zip(x, y))
-    return total / (n - 1)
+    sxy = fsum((xi - xbar) * (yi - ybar) for xi, yi in zip(x, y))
+    return sxy / (n - 1)
 
 
 def correlation(x, y, /):
@@ -910,11 +910,13 @@ def correlation(x, y, /):
         raise StatisticsError('correlation requires that both inputs have same number of data points')
     if n < 2:
         raise StatisticsError('correlation requires at least two data points')
-    cov = covariance(x, y)
-    stdx = stdev(x)
-    stdy = stdev(y)
+    xbar = fmean(x)
+    ybar = fmean(y)
+    sxy = fsum((xi - xbar) * (yi - ybar) for xi, yi in zip(x, y))
+    s2x = fsum((xi - xbar) ** 2.0 for xi in x)
+    s2y = fsum((yi - ybar) ** 2.0 for yi in y)
     try:
-        return cov / (stdx * stdy)
+        return sxy / sqrt(s2x * s2y)
     except ZeroDivisionError:
         raise StatisticsError('at least one of the inputs is constant')
 

From 9d05de138ff7c25e079dffc9453b7e6b023bbdd6 Mon Sep 17 00:00:00 2001
From: Raymond Hettinger <raymond.hettinger@gmail.com>
Date: Fri, 14 May 2021 12:28:28 -0700
Subject: [PATCH 2/4] Consistently use fsum()

---
 Lib/statistics.py | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/Lib/statistics.py b/Lib/statistics.py
index 84b748fc28f9f2..d815f4b9bf56f5 100644
--- a/Lib/statistics.py
+++ b/Lib/statistics.py
@@ -882,8 +882,8 @@ def covariance(x, y, /):
         raise StatisticsError('covariance requires that both inputs have same number of data points')
     if n < 2:
         raise StatisticsError('covariance requires at least two data points')
-    xbar = fmean(x)
-    ybar = fmean(y)
+    xbar = fsum(x) / n
+    ybar = fsum(y) / n
     sxy = fsum((xi - xbar) * (yi - ybar) for xi, yi in zip(x, y))
     return sxy / (n - 1)
 
@@ -910,8 +910,8 @@ def correlation(x, y, /):
         raise StatisticsError('correlation requires that both inputs have same number of data points')
     if n < 2:
         raise StatisticsError('correlation requires at least two data points')
-    xbar = fmean(x)
-    ybar = fmean(y)
+    xbar = fsum(x) / n
+    ybar = fsum(y) / n
     sxy = fsum((xi - xbar) * (yi - ybar) for xi, yi in zip(x, y))
     s2x = fsum((xi - xbar) ** 2.0 for xi in x)
     s2y = fsum((yi - ybar) ** 2.0 for yi in y)

From cf3986ba3d889139b378494dffa1fab7fc668a7a Mon Sep 17 00:00:00 2001
From: Raymond Hettinger <raymond.hettinger@gmail.com>
Date: Sat, 15 May 2021 09:58:26 -0700
Subject: [PATCH 3/4] Restore sort order to __all__

---
 Lib/statistics.py | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/Lib/statistics.py b/Lib/statistics.py
index d815f4b9bf56f5..b2fe37a8308449 100644
--- a/Lib/statistics.py
+++ b/Lib/statistics.py
@@ -107,9 +107,12 @@
 __all__ = [
     'NormalDist',
     'StatisticsError',
+    'correlation',
+    'covariance',
     'fmean',
     'geometric_mean',
     'harmonic_mean',
+    'linear_regression',
     'mean',
     'median',
     'median_grouped',
@@ -122,9 +125,6 @@
     'quantiles',
     'stdev',
     'variance',
-    'correlation',
-    'covariance',
-    'linear_regression',
 ]
 
 import math

From c31b93a04397ea6d901117a7361469d74c838ca1 Mon Sep 17 00:00:00 2001
From: Raymond Hettinger <raymond.hettinger@gmail.com>
Date: Sat, 15 May 2021 10:31:11 -0700
Subject: [PATCH 4/4] Add comment

---
 Lib/statistics.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Lib/statistics.py b/Lib/statistics.py
index b2fe37a8308449..507a5b2d79dce2 100644
--- a/Lib/statistics.py
+++ b/Lib/statistics.py
@@ -960,7 +960,7 @@ def linear_regression(regressor, dependent_variable, /):
     sxy = fsum((xi - xbar) * (yi - ybar) for xi, yi in zip(x, y))
     s2x = fsum((xi - xbar) ** 2.0 for xi in x)
     try:
-        slope = sxy / s2x
+        slope = sxy / s2x   # equivalent to:  covariance(x, y) / variance(x)
     except ZeroDivisionError:
         raise StatisticsError('regressor is constant')
     intercept = ybar - slope * xbar