Issue #18606: Add the new "statistics" module (PEP 450). Contributed

by Steven D'Aprano.

1 files changed, 1534 insertions, 0 deletions

diff --git a/Lib/test/test_statistics.py b/Lib/test/test_statistics.py
new file mode 100644
index 0000000000..71b3654526
--- /dev/null
+++ b/Lib/test/test_statistics.py
@@ -0,0 +1,1534 @@
+"""Test suite for statistics module, including helper NumericTestCase and
+approx_equal function.
+
+"""
+
+import collections
+import decimal
+import doctest
+import math
+import random
+import types
+import unittest
+
+from decimal import Decimal
+from fractions import Fraction
+
+
+# Module to be tested.
+import statistics
+
+
+# === Helper functions and class ===
+
+def _calc_errors(actual, expected):
+    """Return the absolute and relative errors between two numbers.
+
+    >>> _calc_errors(100, 75)
+    (25, 0.25)
+    >>> _calc_errors(100, 100)
+    (0, 0.0)
+
+    Returns the (absolute error, relative error) between the two arguments.
+    """
+    base = max(abs(actual), abs(expected))
+    abs_err = abs(actual - expected)
+    rel_err = abs_err/base if base else float('inf')
+    return (abs_err, rel_err)
+
+
+def approx_equal(x, y, tol=1e-12, rel=1e-7):
+    """approx_equal(x, y [, tol [, rel]]) => True|False
+
+    Return True if numbers x and y are approximately equal, to within some
+    margin of error, otherwise return False. Numbers which compare equal
+    will also compare approximately equal.
+
+    x is approximately equal to y if the difference between them is less than
+    an absolute error tol or a relative error rel, whichever is bigger.
+
+    If given, both tol and rel must be finite, non-negative numbers. If not
+    given, default values are tol=1e-12 and rel=1e-7.
+
+    >>> approx_equal(1.2589, 1.2587, tol=0.0003, rel=0)
+    True
+    >>> approx_equal(1.2589, 1.2587, tol=0.0001, rel=0)
+    False
+
+    Absolute error is defined as abs(x-y); if that is less than or equal to
+    tol, x and y are considered approximately equal.
+
+    Relative error is defined as abs((x-y)/x) or abs((x-y)/y), whichever is
+    smaller, provided x or y are not zero. If that figure is less than or
+    equal to rel, x and y are considered approximately equal.
+
+    Complex numbers are not directly supported. If you wish to compare to
+    complex numbers, extract their real and imaginary parts and compare them
+    individually.
+
+    NANs always compare unequal, even with themselves. Infinities compare
+    approximately equal if they have the same sign (both positive or both
+    negative). Infinities with different signs compare unequal; so do
+    comparisons of infinities with finite numbers.
+    """
+    if tol < 0 or rel < 0:
+        raise ValueError('error tolerances must be non-negative')
+    # NANs are never equal to anything, approximately or otherwise.
+    if math.isnan(x) or math.isnan(y):
+        return False
+    # Numbers which compare equal also compare approximately equal.
+    if x == y:
+        # This includes the case of two infinities with the same sign.
+        return True
+    if math.isinf(x) or math.isinf(y):
+        # This includes the case of two infinities of opposite sign, or
+        # one infinity and one finite number.
+        return False
+    # Two finite numbers.
+    actual_error = abs(x - y)
+    allowed_error = max(tol, rel*max(abs(x), abs(y)))
+    return actual_error <= allowed_error
+
+
+# This class exists only as somewhere to stick a docstring containing
+# doctests. The following docstring and tests were originally in a separate
+# module. Now that it has been merged in here, I need somewhere to hang the.
+# docstring. Ultimately, this class will die, and the information below will
+# either become redundant, or be moved into more appropriate places.
+class _DoNothing:
+    """
+    When doing numeric work, especially with floats, exact equality is often
+    not what you want. Due to round-off error, it is often a bad idea to try
+    to compare floats with equality. Instead the usual procedure is to test
+    them with some (hopefully small!) allowance for error.
+
+    The ``approx_equal`` function allows you to specify either an absolute
+    error tolerance, or a relative error, or both.
+
+    Absolute error tolerances are simple, but you need to know the magnitude
+    of the quantities being compared:
+
+    >>> approx_equal(12.345, 12.346, tol=1e-3)
+    True
+    >>> approx_equal(12.345e6, 12.346e6, tol=1e-3)  # tol is too small.
+    False
+
+    Relative errors are more suitable when the values you are comparing can
+    vary in magnitude:
+
+    >>> approx_equal(12.345, 12.346, rel=1e-4)
+    True
+    >>> approx_equal(12.345e6, 12.346e6, rel=1e-4)
+    True
+
+    but a naive implementation of relative error testing can run into trouble
+    around zero.
+
+    If you supply both an absolute tolerance and a relative error, the
+    comparison succeeds if either individual test succeeds:
+
+    >>> approx_equal(12.345e6, 12.346e6, tol=1e-3, rel=1e-4)
+    True
+
+    """
+    pass
+
+
+
+# We prefer this for testing numeric values that may not be exactly equal,
+# and avoid using TestCase.assertAlmostEqual, because it sucks :-)
+
+class NumericTestCase(unittest.TestCase):
+    """Unit test class for numeric work.
+
+    This subclasses TestCase. In addition to the standard method
+    ``TestCase.assertAlmostEqual``,  ``assertApproxEqual`` is provided.
+    """
+    # By default, we expect exact equality, unless overridden.
+    tol = rel = 0
+
+    def assertApproxEqual(
+            self, first, second, tol=None, rel=None, msg=None
+            ):
+        """Test passes if ``first`` and ``second`` are approximately equal.
+
+        This test passes if ``first`` and ``second`` are equal to
+        within ``tol``, an absolute error, or ``rel``, a relative error.
+
+        If either ``tol`` or ``rel`` are None or not given, they default to
+        test attributes of the same name (by default, 0).
+
+        The objects may be either numbers, or sequences of numbers. Sequences
+        are tested element-by-element.
+
+        >>> class MyTest(NumericTestCase):
+        ...     def test_number(self):
+        ...         x = 1.0/6
+        ...         y = sum([x]*6)
+        ...         self.assertApproxEqual(y, 1.0, tol=1e-15)
+        ...     def test_sequence(self):
+        ...         a = [1.001, 1.001e-10, 1.001e10]
+        ...         b = [1.0, 1e-10, 1e10]
+        ...         self.assertApproxEqual(a, b, rel=1e-3)
+        ...
+        >>> import unittest
+        >>> from io import StringIO  # Suppress test runner output.
+        >>> suite = unittest.TestLoader().loadTestsFromTestCase(MyTest)
+        >>> unittest.TextTestRunner(stream=StringIO()).run(suite)
+        <unittest.runner.TextTestResult run=2 errors=0 failures=0>
+
+        """
+        if tol is None:
+            tol = self.tol
+        if rel is None:
+            rel = self.rel
+        if (
+                isinstance(first, collections.Sequence) and
+                isinstance(second, collections.Sequence)
+            ):
+            check = self._check_approx_seq
+        else:
+            check = self._check_approx_num
+        check(first, second, tol, rel, msg)
+
+    def _check_approx_seq(self, first, second, tol, rel, msg):
+        if len(first) != len(second):
+            standardMsg = (
+                "sequences differ in length: %d items != %d items"
+                % (len(first), len(second))
+                )
+            msg = self._formatMessage(msg, standardMsg)
+            raise self.failureException(msg)
+        for i, (a,e) in enumerate(zip(first, second)):
+            self._check_approx_num(a, e, tol, rel, msg, i)
+
+    def _check_approx_num(self, first, second, tol, rel, msg, idx=None):
+        if approx_equal(first, second, tol, rel):
+            # Test passes. Return early, we are done.
+            return None
+        # Otherwise we failed.
+        standardMsg = self._make_std_err_msg(first, second, tol, rel, idx)
+        msg = self._formatMessage(msg, standardMsg)
+        raise self.failureException(msg)
+
+    @staticmethod
+    def _make_std_err_msg(first, second, tol, rel, idx):
+        # Create the standard error message for approx_equal failures.
+        assert first != second
+        template = (
+            '  %r != %r\n'
+            '  values differ by more than tol=%r and rel=%r\n'
+            '  -> absolute error = %r\n'
+            '  -> relative error = %r'
+            )
+        if idx is not None:
+            header = 'numeric sequences first differ at index %d.\n' % idx
+            template = header + template
+        # Calculate actual errors:
+        abs_err, rel_err = _calc_errors(first, second)
+        return template % (first, second, tol, rel, abs_err, rel_err)
+
+
+# ========================
+# === Test the helpers ===
+# ========================
+
+
+# --- Tests for approx_equal ---
+
+class ApproxEqualSymmetryTest(unittest.TestCase):
+    # Test symmetry of approx_equal.
+
+    def test_relative_symmetry(self):
+        # Check that approx_equal treats relative error symmetrically.
+        # (a-b)/a is usually not equal to (a-b)/b. Ensure that this
+        # doesn't matter.
+        #
+        #   Note: the reason for this test is that an early version
+        #   of approx_equal was not symmetric. A relative error test
+        #   would pass, or fail, depending on which value was passed
+        #   as the first argument.
+        #
+        args1 = [2456, 37.8, -12.45, Decimal('2.54'), Fraction(17, 54)]
+        args2 = [2459, 37.2, -12.41, Decimal('2.59'), Fraction(15, 54)]
+        assert len(args1) == len(args2)
+        for a, b in zip(args1, args2):
+            self.do_relative_symmetry(a, b)
+
+    def do_relative_symmetry(self, a, b):
+        a, b = min(a, b), max(a, b)
+        assert a < b
+        delta = b - a  # The absolute difference between the values.
+        rel_err1, rel_err2 = abs(delta/a), abs(delta/b)
+        # Choose an error margin halfway between the two.
+        rel = (rel_err1 + rel_err2)/2
+        # Now see that values a and b compare approx equal regardless of
+        # which is given first.
+        self.assertTrue(approx_equal(a, b, tol=0, rel=rel))
+        self.assertTrue(approx_equal(b, a, tol=0, rel=rel))
+
+    def test_symmetry(self):
+        # Test that approx_equal(a, b) == approx_equal(b, a)
+        args = [-23, -2, 5, 107, 93568]
+        delta = 2
+        for x in args:
+            for type_ in (int, float, Decimal, Fraction):
+                x = type_(x)*100
+                y = x + delta
+                r = abs(delta/max(x, y))
+                # There are five cases to check:
+                # 1) actual error <= tol, <= rel
+                self.do_symmetry_test(x, y, tol=delta, rel=r)
+                self.do_symmetry_test(x, y, tol=delta+1, rel=2*r)
+                # 2) actual error > tol, > rel
+                self.do_symmetry_test(x, y, tol=delta-1, rel=r/2)
+                # 3) actual error <= tol, > rel
+                self.do_symmetry_test(x, y, tol=delta, rel=r/2)
+                # 4) actual error > tol, <= rel
+                self.do_symmetry_test(x, y, tol=delta-1, rel=r)
+                self.do_symmetry_test(x, y, tol=delta-1, rel=2*r)
+                # 5) exact equality test
+                self.do_symmetry_test(x, x, tol=0, rel=0)
+                self.do_symmetry_test(x, y, tol=0, rel=0)
+
+    def do_symmetry_test(self, a, b, tol, rel):
+        template = "approx_equal comparisons don't match for %r"
+        flag1 = approx_equal(a, b, tol, rel)
+        flag2 = approx_equal(b, a, tol, rel)
+        self.assertEqual(flag1, flag2, template.format((a, b, tol, rel)))
+
+
+class ApproxEqualExactTest(unittest.TestCase):
+    # Test the approx_equal function with exactly equal values.
+    # Equal values should compare as approximately equal.
+    # Test cases for exactly equal values, which should compare approx
+    # equal regardless of the error tolerances given.
+
+    def do_exactly_equal_test(self, x, tol, rel):
+        result = approx_equal(x, x, tol=tol, rel=rel)
+        self.assertTrue(result, 'equality failure for x=%r' % x)
+        result = approx_equal(-x, -x, tol=tol, rel=rel)
+        self.assertTrue(result, 'equality failure for x=%r' % -x)
+
+    def test_exactly_equal_ints(self):
+        # Test that equal int values are exactly equal.
+        for n in [42, 19740, 14974, 230, 1795, 700245, 36587]:
+            self.do_exactly_equal_test(n, 0, 0)
+
+    def test_exactly_equal_floats(self):
+        # Test that equal float values are exactly equal.
+        for x in [0.42, 1.9740, 1497.4, 23.0, 179.5, 70.0245, 36.587]:
+            self.do_exactly_equal_test(x, 0, 0)
+
+    def test_exactly_equal_fractions(self):
+        # Test that equal Fraction values are exactly equal.
+        F = Fraction
+        for f in [F(1, 2), F(0), F(5, 3), F(9, 7), F(35, 36), F(3, 7)]:
+            self.do_exactly_equal_test(f, 0, 0)
+
+    def test_exactly_equal_decimals(self):
+        # Test that equal Decimal values are exactly equal.
+        D = Decimal
+        for d in map(D, "8.2 31.274 912.04 16.745 1.2047".split()):
+            self.do_exactly_equal_test(d, 0, 0)
+
+    def test_exactly_equal_absolute(self):
+        # Test that equal values are exactly equal with an absolute error.
+        for n in [16, 1013, 1372, 1198, 971, 4]:
+            # Test as ints.
+            self.do_exactly_equal_test(n, 0.01, 0)
+            # Test as floats.
+            self.do_exactly_equal_test(n/10, 0.01, 0)
+            # Test as Fractions.
+            f = Fraction(n, 1234)
+            self.do_exactly_equal_test(f, 0.01, 0)
+
+    def test_exactly_equal_absolute_decimals(self):
+        # Test equal Decimal values are exactly equal with an absolute error.
+        self.do_exactly_equal_test(Decimal("3.571"), Decimal("0.01"), 0)
+        self.do_exactly_equal_test(-Decimal("81.3971"), Decimal("0.01"), 0)
+
+    def test_exactly_equal_relative(self):
+        # Test that equal values are exactly equal with a relative error.
+        for x in [8347, 101.3, -7910.28, Fraction(5, 21)]:
+            self.do_exactly_equal_test(x, 0, 0.01)
+        self.do_exactly_equal_test(Decimal("11.68"), 0, Decimal("0.01"))
+
+    def test_exactly_equal_both(self):
+        # Test that equal values are equal when both tol and rel are given.
+        for x in [41017, 16.742, -813.02, Fraction(3, 8)]:
+            self.do_exactly_equal_test(x, 0.1, 0.01)
+        D = Decimal
+        self.do_exactly_equal_test(D("7.2"), D("0.1"), D("0.01"))
+
+
+class ApproxEqualUnequalTest(unittest.TestCase):
+    # Unequal values should compare unequal with zero error tolerances.
+    # Test cases for unequal values, with exact equality test.
+
+    def do_exactly_unequal_test(self, x):
+        for a in (x, -x):
+            result = approx_equal(a, a+1, tol=0, rel=0)
+            self.assertFalse(result, 'inequality failure for x=%r' % a)
+
+    def test_exactly_unequal_ints(self):
+        # Test unequal int values are unequal with zero error tolerance.
+        for n in [951, 572305, 478, 917, 17240]:
+            self.do_exactly_unequal_test(n)
+
+    def test_exactly_unequal_floats(self):
+        # Test unequal float values are unequal with zero error tolerance.
+        for x in [9.51, 5723.05, 47.8, 9.17, 17.24]:
+            self.do_exactly_unequal_test(x)
+
+    def test_exactly_unequal_fractions(self):
+        # Test that unequal Fractions are unequal with zero error tolerance.
+        F = Fraction
+        for f in [F(1, 5), F(7, 9), F(12, 11), F(101, 99023)]:
+            self.do_exactly_unequal_test(f)
+
+    def test_exactly_unequal_decimals(self):
+        # Test that unequal Decimals are unequal with zero error tolerance.
+        for d in map(Decimal, "3.1415 298.12 3.47 18.996 0.00245".split()):
+            self.do_exactly_unequal_test(d)
+
+
+class ApproxEqualInexactTest(unittest.TestCase):
+    # Inexact test cases for approx_error.
+    # Test cases when comparing two values that are not exactly equal.
+
+    # === Absolute error tests ===
+
+    def do_approx_equal_abs_test(self, x, delta):
+        template = "Test failure for x={!r}, y={!r}"
+        for y in (x + delta, x - delta):
+            msg = template.format(x, y)
+            self.assertTrue(approx_equal(x, y, tol=2*delta, rel=0), msg)
+            self.assertFalse(approx_equal(x, y, tol=delta/2, rel=0), msg)
+
+    def test_approx_equal_absolute_ints(self):
+        # Test approximate equality of ints with an absolute error.
+        for n in [-10737, -1975, -7, -2, 0, 1, 9, 37, 423, 9874, 23789110]:
+            self.do_approx_equal_abs_test(n, 10)
+            self.do_approx_equal_abs_test(n, 2)
+
+    def test_approx_equal_absolute_floats(self):
+        # Test approximate equality of floats with an absolute error.
+        for x in [-284.126, -97.1, -3.4, -2.15, 0.5, 1.0, 7.8, 4.23, 3817.4]:
+            self.do_approx_equal_abs_test(x, 1.5)
+            self.do_approx_equal_abs_test(x, 0.01)
+            self.do_approx_equal_abs_test(x, 0.0001)
+
+    def test_approx_equal_absolute_fractions(self):
+        # Test approximate equality of Fractions with an absolute error.
+        delta = Fraction(1, 29)
+        numerators = [-84, -15, -2, -1, 0, 1, 5, 17, 23, 34, 71]
+        for f in (Fraction(n, 29) for n in numerators):
+            self.do_approx_equal_abs_test(f, delta)
+            self.do_approx_equal_abs_test(f, float(delta))
+
+    def test_approx_equal_absolute_decimals(self):
+        # Test approximate equality of Decimals with an absolute error.
+        delta = Decimal("0.01")
+        for d in map(Decimal, "1.0 3.5 36.08 61.79 7912.3648".split()):
+            self.do_approx_equal_abs_test(d, delta)
+            self.do_approx_equal_abs_test(-d, delta)
+
+    def test_cross_zero(self):
+        # Test for the case of the two values having opposite signs.
+        self.assertTrue(approx_equal(1e-5, -1e-5, tol=1e-4, rel=0))
+
+    # === Relative error tests ===
+
+    def do_approx_equal_rel_test(self, x, delta):
+        template = "Test failure for x={!r}, y={!r}"
+        for y in (x*(1+delta), x*(1-delta)):
+            msg = template.format(x, y)
+            self.assertTrue(approx_equal(x, y, tol=0, rel=2*delta), msg)
+            self.assertFalse(approx_equal(x, y, tol=0, rel=delta/2), msg)
+
+    def test_approx_equal_relative_ints(self):
+        # Test approximate equality of ints with a relative error.
+        self.assertTrue(approx_equal(64, 47, tol=0, rel=0.36))
+        self.assertTrue(approx_equal(64, 47, tol=0, rel=0.37))
+        # ---
+        self.assertTrue(approx_equal(449, 512, tol=0, rel=0.125))
+        self.assertTrue(approx_equal(448, 512, tol=0, rel=0.125))
+        self.assertFalse(approx_equal(447, 512, tol=0, rel=0.125))
+
+    def test_approx_equal_relative_floats(self):
+        # Test approximate equality of floats with a relative error.
+        for x in [-178.34, -0.1, 0.1, 1.0, 36.97, 2847.136, 9145.074]:
+            self.do_approx_equal_rel_test(x, 0.02)
+            self.do_approx_equal_rel_test(x, 0.0001)
+
+    def test_approx_equal_relative_fractions(self):
+        # Test approximate equality of Fractions with a relative error.
+        F = Fraction
+        delta = Fraction(3, 8)
+        for f in [F(3, 84), F(17, 30), F(49, 50), F(92, 85)]:
+            for d in (delta, float(delta)):
+                self.do_approx_equal_rel_test(f, d)
+                self.do_approx_equal_rel_test(-f, d)
+
+    def test_approx_equal_relative_decimals(self):
+        # Test approximate equality of Decimals with a relative error.
+        for d in map(Decimal, "0.02 1.0 5.7 13.67 94.138 91027.9321".split()):
+            self.do_approx_equal_rel_test(d, Decimal("0.001"))
+            self.do_approx_equal_rel_test(-d, Decimal("0.05"))
+
+    # === Both absolute and relative error tests ===
+
+    # There are four cases to consider:
+    #   1) actual error <= both absolute and relative error
+    #   2) actual error <= absolute error but > relative error
+    #   3) actual error <= relative error but > absolute error
+    #   4) actual error > both absolute and relative error
+
+    def do_check_both(self, a, b, tol, rel, tol_flag, rel_flag):
+        check = self.assertTrue if tol_flag else self.assertFalse
+        check(approx_equal(a, b, tol=tol, rel=0))
+        check = self.assertTrue if rel_flag else self.assertFalse
+        check(approx_equal(a, b, tol=0, rel=rel))
+        check = self.assertTrue if (tol_flag or rel_flag) else self.assertFalse
+        check(approx_equal(a, b, tol=tol, rel=rel))
+
+    def test_approx_equal_both1(self):
+        # Test actual error <= both absolute and relative error.
+        self.do_check_both(7.955, 7.952, 0.004, 3.8e-4, True, True)
+        self.do_check_both(-7.387, -7.386, 0.002, 0.0002, True, True)
+
+    def test_approx_equal_both2(self):
+        # Test actual error <= absolute error but > relative error.
+        self.do_check_both(7.955, 7.952, 0.004, 3.7e-4, True, False)
+
+    def test_approx_equal_both3(self):
+        # Test actual error <= relative error but > absolute error.
+        self.do_check_both(7.955, 7.952, 0.001, 3.8e-4, False, True)
+
+    def test_approx_equal_both4(self):
+        # Test actual error > both absolute and relative error.
+        self.do_check_both(2.78, 2.75, 0.01, 0.001, False, False)
+        self.do_check_both(971.44, 971.47, 0.02, 3e-5, False, False)
+
+
+class ApproxEqualSpecialsTest(unittest.TestCase):
+    # Test approx_equal with NANs and INFs and zeroes.
+
+    def test_inf(self):
+        for type_ in (float, Decimal):
+            inf = type_('inf')
+            self.assertTrue(approx_equal(inf, inf))
+            self.assertTrue(approx_equal(inf, inf, 0, 0))
+            self.assertTrue(approx_equal(inf, inf, 1, 0.01))
+            self.assertTrue(approx_equal(-inf, -inf))
+            self.assertFalse(approx_equal(inf, -inf))
+            self.assertFalse(approx_equal(inf, 1000))
+
+    def test_nan(self):
+        for type_ in (float, Decimal):
+            nan = type_('nan')
+            for other in (nan, type_('inf'), 1000):
+                self.assertFalse(approx_equal(nan, other))
+
+    def test_float_zeroes(self):
+        nzero = math.copysign(0.0, -1)
+        self.assertTrue(approx_equal(nzero, 0.0, tol=0.1, rel=0.1))
+
+    def test_decimal_zeroes(self):
+        nzero = Decimal("-0.0")
+        self.assertTrue(approx_equal(nzero, Decimal(0), tol=0.1, rel=0.1))
+
+
+class TestApproxEqualErrors(unittest.TestCase):
+    # Test error conditions of approx_equal.
+
+    def test_bad_tol(self):
+        # Test negative tol raises.
+        self.assertRaises(ValueError, approx_equal, 100, 100, -1, 0.1)
+
+    def test_bad_rel(self):
+        # Test negative rel raises.
+        self.assertRaises(ValueError, approx_equal, 100, 100, 1, -0.1)
+
+
+# --- Tests for NumericTestCase ---
+
+# The formatting routine that generates the error messages is complex enough
+# that it too needs testing.
+
+class TestNumericTestCase(unittest.TestCase):
+    # The exact wording of NumericTestCase error messages is *not* guaranteed,
+    # but we need to give them some sort of test to ensure that they are
+    # generated correctly. As a compromise, we look for specific substrings
+    # that are expected to be found even if the overall error message changes.
+
+    def do_test(self, args):
+        actual_msg = NumericTestCase._make_std_err_msg(*args)
+        expected = self.generate_substrings(*args)
+        for substring in expected:
+            self.assertIn(substring, actual_msg)
+
+    def test_numerictestcase_is_testcase(self):
+        # Ensure that NumericTestCase actually is a TestCase.
+        self.assertTrue(issubclass(NumericTestCase, unittest.TestCase))
+
+    def test_error_msg_numeric(self):
+        # Test the error message generated for numeric comparisons.
+        args = (2.5, 4.0, 0.5, 0.25, None)
+        self.do_test(args)
+
+    def test_error_msg_sequence(self):
+        # Test the error message generated for sequence comparisons.
+        args = (3.75, 8.25, 1.25, 0.5, 7)
+        self.do_test(args)
+
+    def generate_substrings(self, first, second, tol, rel, idx):
+        """Return substrings we expect to see in error messages."""
+        abs_err, rel_err = _calc_errors(first, second)
+        substrings = [
+                'tol=%r' % tol,
+                'rel=%r' % rel,
+                'absolute error = %r' % abs_err,
+                'relative error = %r' % rel_err,
+                ]
+        if idx is not None:
+            substrings.append('differ at index %d' % idx)
+        return substrings
+
+
+# =======================================
+# === Tests for the statistics module ===
+# =======================================
+
+
+class GlobalsTest(unittest.TestCase):
+    module = statistics
+    expected_metadata = ["__doc__", "__all__"]
+
+    def test_meta(self):
+        # Test for the existence of metadata.
+        for meta in self.expected_metadata:
+            self.assertTrue(hasattr(self.module, meta),
+                            "%s not present" % meta)
+
+    def test_check_all(self):
+        # Check everything in __all__ exists and is public.
+        module = self.module
+        for name in module.__all__:
+            # No private names in __all__:
+            self.assertFalse(name.startswith("_"),
+                             'private name "%s" in __all__' % name)
+            # And anything in __all__ must exist:
+            self.assertTrue(hasattr(module, name),
+                            'missing name "%s" in __all__' % name)
+
+
+class DocTests(unittest.TestCase):
+    def test_doc_tests(self):
+        failed, tried = doctest.testmod(statistics)
+        self.assertGreater(tried, 0)
+        self.assertEqual(failed, 0)
+
+class StatisticsErrorTest(unittest.TestCase):
+    def test_has_exception(self):
+        errmsg = (
+                "Expected StatisticsError to be a ValueError, but got a"
+                " subclass of %r instead."
+                )
+        self.assertTrue(hasattr(statistics, 'StatisticsError'))
+        self.assertTrue(
+                issubclass(statistics.StatisticsError, ValueError),
+                errmsg % statistics.StatisticsError.__base__
+                )
+
+
+# === Tests for private utility functions ===
+
+class ExactRatioTest(unittest.TestCase):
+    # Test _exact_ratio utility.
+
+    def test_int(self):
+        for i in (-20, -3, 0, 5, 99, 10**20):
+            self.assertEqual(statistics._exact_ratio(i), (i, 1))
+
+    def test_fraction(self):
+        numerators = (-5, 1, 12, 38)
+        for n in numerators:
+            f = Fraction(n, 37)
+            self.assertEqual(statistics._exact_ratio(f), (n, 37))
+
+    def test_float(self):
+        self.assertEqual(statistics._exact_ratio(0.125), (1, 8))
+        self.assertEqual(statistics._exact_ratio(1.125), (9, 8))
+        data = [random.uniform(-100, 100) for _ in range(100)]
+        for x in data:
+            num, den = statistics._exact_ratio(x)
+            self.assertEqual(x, num/den)
+
+    def test_decimal(self):
+        D = Decimal
+        _exact_ratio = statistics._exact_ratio
+        self.assertEqual(_exact_ratio(D("0.125")), (125, 1000))
+        self.assertEqual(_exact_ratio(D("12.345")), (12345, 1000))
+        self.assertEqual(_exact_ratio(D("-1.98")), (-198, 100))
+
+
+class DecimalToRatioTest(unittest.TestCase):
+    # Test _decimal_to_ratio private function.
+
+    def testSpecialsRaise(self):
+        # Test that NANs and INFs raise ValueError.
+        # Non-special values are covered by _exact_ratio above.
+        for d in (Decimal('NAN'), Decimal('sNAN'), Decimal('INF')):
+            self.assertRaises(ValueError, statistics._decimal_to_ratio, d)
+
+
+
+# === Tests for public functions ===
+
+class UnivariateCommonMixin:
+    # Common tests for most univariate functions that take a data argument.
+
+    def test_no_args(self):
+        # Fail if given no arguments.
+        self.assertRaises(TypeError, self.func)
+
+    def test_empty_data(self):
+        # Fail when the data argument (first argument) is empty.
+        for empty in ([], (), iter([])):
+            self.assertRaises(statistics.StatisticsError, self.func, empty)
+
+    def prepare_data(self):
+        """Return int data for various tests."""
+        data = list(range(10))
+        while data == sorted(data):
+            random.shuffle(data)
+        return data
+
+    def test_no_inplace_modifications(self):
+        # Test that the function does not modify its input data.
+        data = self.prepare_data()
+        assert len(data) != 1  # Necessary to avoid infinite loop.
+        assert data != sorted(data)
+        saved = data[:]
+        assert data is not saved
+        _ = self.func(data)
+        self.assertListEqual(data, saved, "data has been modified")
+
+    def test_order_doesnt_matter(self):
+        # Test that the order of data points doesn't change the result.
+
+        # CAUTION: due to floating point rounding errors, the result actually
+        # may depend on the order. Consider this test representing an ideal.
+        # To avoid this test failing, only test with exact values such as ints
+        # or Fractions.
+        data = [1, 2, 3, 3, 3, 4, 5, 6]*100
+        expected = self.func(data)
+        random.shuffle(data)
+        actual = self.func(data)
+        self.assertEqual(expected, actual)
+
+    def test_type_of_data_collection(self):
+        # Test that the type of iterable data doesn't effect the result.
+        class MyList(list):
+            pass
+        class MyTuple(tuple):
+            pass
+        def generator(data):
+            return (obj for obj in data)
+        data = self.prepare_data()
+        expected = self.func(data)
+        for kind in (list, tuple, iter, MyList, MyTuple, generator):
+            result = self.func(kind(data))
+            self.assertEqual(result, expected)
+
+    def test_range_data(self):
+        # Test that functions work with range objects.
+        data = range(20, 50, 3)
+        expected = self.func(list(data))
+        self.assertEqual(self.func(data), expected)
+
+    def test_bad_arg_types(self):
+        # Test that function raises when given data of the wrong type.
+
+        # Don't roll the following into a loop like this:
+        #   for bad in list_of_bad:
+        #       self.check_for_type_error(bad)
+        #
+        # Since assertRaises doesn't show the arguments that caused the test
+        # failure, it is very difficult to debug these test failures when the
+        # following are in a loop.
+        self.check_for_type_error(None)
+        self.check_for_type_error(23)
+        self.check_for_type_error(42.0)
+        self.check_for_type_error(object())
+
+    def check_for_type_error(self, *args):
+        self.assertRaises(TypeError, self.func, *args)
+
+    def test_type_of_data_element(self):
+        # Check the type of data elements doesn't affect the numeric result.
+        # This is a weaker test than UnivariateTypeMixin.testTypesConserved,
+        # because it checks the numeric result by equality, but not by type.
+        class MyFloat(float):
+            def __truediv__(self, other):
+                return type(self)(super().__truediv__(other))
+            def __add__(self, other):
+                return type(self)(super().__add__(other))
+            __radd__ = __add__
+
+        raw = self.prepare_data()
+        expected = self.func(raw)
+        for kind in (float, MyFloat, Decimal, Fraction):
+            data = [kind(x) for x in raw]
+            result = type(expected)(self.func(data))
+            self.assertEqual(result, expected)
+
+
+class UnivariateTypeMixin:
+    """Mixin class for type-conserving functions.
+
+    This mixin class holds test(s) for functions which conserve the type of
+    individual data points. E.g. the mean of a list of Fractions should itself
+    be a Fraction.
+
+    Not all tests to do with types need go in this class. Only those that
+    rely on the function returning the same type as its input data.
+    """
+    def test_types_conserved(self):
+        # Test that functions keeps the same type as their data points.
+        # (Excludes mixed data types.) This only tests the type of the return
+        # result, not the value.
+        class MyFloat(float):
+            def __truediv__(self, other):
+                return type(self)(super().__truediv__(other))
+            def __sub__(self, other):
+                return type(self)(super().__sub__(other))
+            def __rsub__(self, other):
+                return type(self)(super().__rsub__(other))
+            def __pow__(self, other):
+                return type(self)(super().__pow__(other))
+            def __add__(self, other):
+                return type(self)(super().__add__(other))
+            __radd__ = __add__
+
+        data = self.prepare_data()
+        for kind in (float, Decimal, Fraction, MyFloat):
+            d = [kind(x) for x in data]
+            result = self.func(d)
+            self.assertIs(type(result), kind)
+
+
+class TestSum(NumericTestCase, UnivariateCommonMixin, UnivariateTypeMixin):
+    # Test cases for statistics._sum() function.
+
+    def setUp(self):
+        self.func = statistics._sum
+
+    def test_empty_data(self):
+        # Override test for empty data.
+        for data in ([], (), iter([])):
+            self.assertEqual(self.func(data), 0)
+            self.assertEqual(self.func(data, 23), 23)
+            self.assertEqual(self.func(data, 2.3), 2.3)
+
+    def test_ints(self):
+        self.assertEqual(self.func([1, 5, 3, -4, -8, 20, 42, 1]), 60)
+        self.assertEqual(self.func([4, 2, 3, -8, 7], 1000), 1008)
+
+    def test_floats(self):
+        self.assertEqual(self.func([0.25]*20), 5.0)
+        self.assertEqual(self.func([0.125, 0.25, 0.5, 0.75], 1.5), 3.125)
+
+    def test_fractions(self):
+        F = Fraction
+        self.assertEqual(self.func([Fraction(1, 1000)]*500), Fraction(1, 2))
+
+    def test_decimals(self):
+        D = Decimal
+        data = [D("0.001"), D("5.246"), D("1.702"), D("-0.025"),
+                D("3.974"), D("2.328"), D("4.617"), D("2.843"),
+                ]
+        self.assertEqual(self.func(data), Decimal("20.686"))
+
+    def test_compare_with_math_fsum(self):
+        # Compare with the math.fsum function.
+        # Ideally we ought to get the exact same result, but sometimes
+        # we differ by a very slight amount :-(
+        data = [random.uniform(-100, 1000) for _ in range(1000)]
+        self.assertApproxEqual(self.func(data), math.fsum(data), rel=2e-16)
+
+    def test_start_argument(self):
+        # Test that the optional start argument works correctly.
+        data = [random.uniform(1, 1000) for _ in range(100)]
+        t = self.func(data)
+        self.assertEqual(t+42, self.func(data, 42))
+        self.assertEqual(t-23, self.func(data, -23))
+        self.assertEqual(t+1e20, self.func(data, 1e20))
+
+    def test_strings_fail(self):
+        # Sum of strings should fail.
+        self.assertRaises(TypeError, self.func, [1, 2, 3], '999')
+        self.assertRaises(TypeError, self.func, [1, 2, 3, '999'])
+
+    def test_bytes_fail(self):
+        # Sum of bytes should fail.
+        self.assertRaises(TypeError, self.func, [1, 2, 3], b'999')
+        self.assertRaises(TypeError, self.func, [1, 2, 3, b'999'])
+
+    def test_mixed_sum(self):
+        # Mixed sums are allowed.
+
+        # Careful here: order matters. Can't mix Fraction and Decimal directly,
+        # only after they're converted to float.
+        data = [1, 2, Fraction(1, 2), 3.0, Decimal("0.25")]
+        self.assertEqual(self.func(data), 6.75)
+
+
+class SumInternalsTest(NumericTestCase):
+    # Test internals of the sum function.
+
+    def test_ignore_instance_float_method(self):
+        # Test that __float__ methods on data instances are ignored.
+
+        # Python typically calls __dunder__ methods on the class, not the
+        # instance. The ``sum`` implementation calls __float__ directly. To
+        # better match the behaviour of Python, we call it only on the class,
+        # not the instance. This test will fail if somebody "fixes" that code.
+
+        # Create a fake __float__ method.
+        def __float__(self):
+            raise AssertionError('test fails')
+
+        # Inject it into an instance.
+        class MyNumber(Fraction):
+            pass
+        x = MyNumber(3)
+        x.__float__ = types.MethodType(__float__, x)
+
+        # Check it works as expected.
+        self.assertRaises(AssertionError, x.__float__)
+        self.assertEqual(float(x), 3.0)
+        # And now test the function.
+        self.assertEqual(statistics._sum([1.0, 2.0, x, 4.0]), 10.0)
+
+
+class SumTortureTest(NumericTestCase):
+    def test_torture(self):
+        # Tim Peters' torture test for sum, and variants of same.
+        self.assertEqual(statistics._sum([1, 1e100, 1, -1e100]*10000), 20000.0)
+        self.assertEqual(statistics._sum([1e100, 1, 1, -1e100]*10000), 20000.0)
+        self.assertApproxEqual(
+            statistics._sum([1e-100, 1, 1e-100, -1]*10000), 2.0e-96, rel=5e-16
+            )
+
+
+class SumSpecialValues(NumericTestCase):
+    # Test that sum works correctly with IEEE-754 special values.
+
+    def test_nan(self):
+        for type_ in (float, Decimal):
+            nan = type_('nan')
+            result = statistics._sum([1, nan, 2])
+            self.assertIs(type(result), type_)
+            self.assertTrue(math.isnan(result))
+
+    def check_infinity(self, x, inf):
+        """Check x is an infinity of the same type and sign as inf."""
+        self.assertTrue(math.isinf(x))
+        self.assertIs(type(x), type(inf))
+        self.assertEqual(x > 0, inf > 0)
+        assert x == inf
+
+    def do_test_inf(self, inf):
+        # Adding a single infinity gives infinity.
+        result = statistics._sum([1, 2, inf, 3])
+        self.check_infinity(result, inf)
+        # Adding two infinities of the same sign also gives infinity.
+        result = statistics._sum([1, 2, inf, 3, inf, 4])
+        self.check_infinity(result, inf)
+
+    def test_float_inf(self):
+        inf = float('inf')
+        for sign in (+1, -1):
+            self.do_test_inf(sign*inf)
+
+    def test_decimal_inf(self):
+        inf = Decimal('inf')
+        for sign in (+1, -1):
+            self.do_test_inf(sign*inf)
+
+    def test_float_mismatched_infs(self):
+        # Test that adding two infinities of opposite sign gives a NAN.
+        inf = float('inf')
+        result = statistics._sum([1, 2, inf, 3, -inf, 4])
+        self.assertTrue(math.isnan(result))
+
+    def test_decimal_mismatched_infs_to_nan(self):
+        # Test adding Decimal INFs with opposite sign returns NAN.
+        inf = Decimal('inf')
+        data = [1, 2, inf, 3, -inf, 4]
+        with decimal.localcontext(decimal.ExtendedContext):
+            self.assertTrue(math.isnan(statistics._sum(data)))
+
+    def test_decimal_mismatched_infs_to_nan(self):
+        # Test adding Decimal INFs with opposite sign raises InvalidOperation.
+        inf = Decimal('inf')
+        data = [1, 2, inf, 3, -inf, 4]
+        with decimal.localcontext(decimal.BasicContext):
+            self.assertRaises(decimal.InvalidOperation, statistics._sum, data)
+
+    def test_decimal_snan_raises(self):
+        # Adding sNAN should raise InvalidOperation.
+        sNAN = Decimal('sNAN')
+        data = [1, sNAN, 2]
+        self.assertRaises(decimal.InvalidOperation, statistics._sum, data)
+
+
+# === Tests for averages ===
+
+class AverageMixin(UnivariateCommonMixin):
+    # Mixin class holding common tests for averages.
+
+    def test_single_value(self):
+        # Average of a single value is the value itself.
+        for x in (23, 42.5, 1.3e15, Fraction(15, 19), Decimal('0.28')):
+            self.assertEqual(self.func([x]), x)
+
+    def test_repeated_single_value(self):
+        # The average of a single repeated value is the value itself.
+        for x in (3.5, 17, 2.5e15, Fraction(61, 67), Decimal('4.9712')):
+            for count in (2, 5, 10, 20):
+                data = [x]*count
+                self.assertEqual(self.func(data), x)
+
+
+class TestMean(NumericTestCase, AverageMixin, UnivariateTypeMixin):
+    def setUp(self):
+        self.func = statistics.mean
+
+    def test_torture_pep(self):
+        # "Torture Test" from PEP-450.
+        self.assertEqual(self.func([1e100, 1, 3, -1e100]), 1)
+
+    def test_ints(self):
+        # Test mean with ints.
+        data = [0, 1, 2, 3, 3, 3, 4, 5, 5, 6, 7, 7, 7, 7, 8, 9]
+        random.shuffle(data)
+        self.assertEqual(self.func(data), 4.8125)
+
+    def test_floats(self):
+        # Test mean with floats.
+        data = [17.25, 19.75, 20.0, 21.5, 21.75, 23.25, 25.125, 27.5]
+        random.shuffle(data)
+        self.assertEqual(self.func(data), 22.015625)
+
+    def test_decimals(self):
+        # Test mean with ints.
+        D = Decimal
+        data = [D("1.634"), D("2.517"), D("3.912"), D("4.072"), D("5.813")]
+        random.shuffle(data)
+        self.assertEqual(self.func(data), D("3.5896"))
+
+    def test_fractions(self):
+        # Test mean with Fractions.
+        F = Fraction
+        data = [F(1, 2), F(2, 3), F(3, 4), F(4, 5), F(5, 6), F(6, 7), F(7, 8)]
+        random.shuffle(data)
+        self.assertEqual(self.func(data), F(1479, 1960))
+
+    def test_inf(self):
+        # Test mean with infinities.
+        raw = [1, 3, 5, 7, 9]  # Use only ints, to avoid TypeError later.
+        for kind in (float, Decimal):
+            for sign in (1, -1):
+                inf = kind("inf")*sign
+                data = raw + [inf]
+                result = self.func(data)
+                self.assertTrue(math.isinf(result))
+                self.assertEqual(result, inf)
+
+    def test_mismatched_infs(self):
+        # Test mean with infinities of opposite sign.
+        data = [2, 4, 6, float('inf'), 1, 3, 5, float('-inf')]
+        result = self.func(data)
+        self.assertTrue(math.isnan(result))
+
+    def test_nan(self):
+        # Test mean with NANs.
+        raw = [1, 3, 5, 7, 9]  # Use only ints, to avoid TypeError later.
+        for kind in (float, Decimal):
+            inf = kind("nan")
+            data = raw + [inf]
+            result = self.func(data)
+            self.assertTrue(math.isnan(result))
+
+    def test_big_data(self):
+        # Test adding a large constant to every data point.
+        c = 1e9
+        data = [3.4, 4.5, 4.9, 6.7, 6.8, 7.2, 8.0, 8.1, 9.4]
+        expected = self.func(data) + c
+        assert expected != c
+        result = self.func([x+c for x in data])
+        self.assertEqual(result, expected)
+
+    def test_doubled_data(self):
+        # Mean of [a,b,c...z] should be same as for [a,a,b,b,c,c...z,z].
+        data = [random.uniform(-3, 5) for _ in range(1000)]
+        expected = self.func(data)
+        actual = self.func(data*2)
+        self.assertApproxEqual(actual, expected)
+
+
+class TestMedian(NumericTestCase, AverageMixin):
+    # Common tests for median and all median.* functions.
+    def setUp(self):
+        self.func = statistics.median
+
+    def prepare_data(self):
+        """Overload method from UnivariateCommonMixin."""
+        data = super().prepare_data()
+        if len(data)%2 != 1:
+            data.append(2)
+        return data
+
+    def test_even_ints(self):
+        # Test median with an even number of int data points.
+        data = [1, 2, 3, 4, 5, 6]
+        assert len(data)%2 == 0
+        self.assertEqual(self.func(data), 3.5)
+
+    def test_odd_ints(self):
+        # Test median with an odd number of int data points.
+        data = [1, 2, 3, 4, 5, 6, 9]
+        assert len(data)%2 == 1
+        self.assertEqual(self.func(data), 4)
+
+    def test_odd_fractions(self):
+        # Test median works with an odd number of Fractions.
+        F = Fraction
+        data = [F(1, 7), F(2, 7), F(3, 7), F(4, 7), F(5, 7)]
+        assert len(data)%2 == 1
+        random.shuffle(data)
+        self.assertEqual(self.func(data), F(3, 7))
+
+    def test_even_fractions(self):
+        # Test median works with an even number of Fractions.
+        F = Fraction
+        data = [F(1, 7), F(2, 7), F(3, 7), F(4, 7), F(5, 7), F(6, 7)]
+        assert len(data)%2 == 0
+        random.shuffle(data)
+        self.assertEqual(self.func(data), F(1, 2))
+
+    def test_odd_decimals(self):
+        # Test median works with an odd number of Decimals.
+        D = Decimal
+        data = [D('2.5'), D('3.1'), D('4.2'), D('5.7'), D('5.8')]
+        assert len(data)%2 == 1
+        random.shuffle(data)
+        self.assertEqual(self.func(data), D('4.2'))
+
+    def test_even_decimals(self):
+        # Test median works with an even number of Decimals.
+        D = Decimal
+        data = [D('1.2'), D('2.5'), D('3.1'), D('4.2'), D('5.7'), D('5.8')]
+        assert len(data)%2 == 0
+        random.shuffle(data)
+        self.assertEqual(self.func(data), D('3.65'))
+
+
+class TestMedianDataType(NumericTestCase, UnivariateTypeMixin):
+    # Test conservation of data element type for median.
+    def setUp(self):
+        self.func = statistics.median
+
+    def prepare_data(self):
+        data = list(range(15))
+        assert len(data)%2 == 1
+        while data == sorted(data):
+            random.shuffle(data)
+        return data
+
+
+class TestMedianLow(TestMedian, UnivariateTypeMixin):
+    def setUp(self):
+        self.func = statistics.median_low
+
+    def test_even_ints(self):
+        # Test median_low with an even number of ints.
+        data = [1, 2, 3, 4, 5, 6]
+        assert len(data)%2 == 0
+        self.assertEqual(self.func(data), 3)
+
+    def test_even_fractions(self):
+        # Test median_low works with an even number of Fractions.
+        F = Fraction
+        data = [F(1, 7), F(2, 7), F(3, 7), F(4, 7), F(5, 7), F(6, 7)]
+        assert len(data)%2 == 0
+        random.shuffle(data)
+        self.assertEqual(self.func(data), F(3, 7))
+
+    def test_even_decimals(self):
+        # Test median_low works with an even number of Decimals.
+        D = Decimal
+        data = [D('1.1'), D('2.2'), D('3.3'), D('4.4'), D('5.5'), D('6.6')]
+        assert len(data)%2 == 0
+        random.shuffle(data)
+        self.assertEqual(self.func(data), D('3.3'))
+
+
+class TestMedianHigh(TestMedian, UnivariateTypeMixin):
+    def setUp(self):
+        self.func = statistics.median_high
+
+    def test_even_ints(self):
+        # Test median_high with an even number of ints.
+        data = [1, 2, 3, 4, 5, 6]
+        assert len(data)%2 == 0
+        self.assertEqual(self.func(data), 4)
+
+    def test_even_fractions(self):
+        # Test median_high works with an even number of Fractions.
+        F = Fraction
+        data = [F(1, 7), F(2, 7), F(3, 7), F(4, 7), F(5, 7), F(6, 7)]
+        assert len(data)%2 == 0
+        random.shuffle(data)
+        self.assertEqual(self.func(data), F(4, 7))
+
+    def test_even_decimals(self):
+        # Test median_high works with an even number of Decimals.
+        D = Decimal
+        data = [D('1.1'), D('2.2'), D('3.3'), D('4.4'), D('5.5'), D('6.6')]
+        assert len(data)%2 == 0
+        random.shuffle(data)
+        self.assertEqual(self.func(data), D('4.4'))
+
+
+class TestMedianGrouped(TestMedian):
+    # Test median_grouped.
+    # Doesn't conserve data element types, so don't use TestMedianType.
+    def setUp(self):
+        self.func = statistics.median_grouped
+
+    def test_odd_number_repeated(self):
+        # Test median.grouped with repeated median values.
+        data = [12, 13, 14, 14, 14, 15, 15]
+        assert len(data)%2 == 1
+        self.assertEqual(self.func(data), 14)
+        #---
+        data = [12, 13, 14, 14, 14, 14, 15]
+        assert len(data)%2 == 1
+        self.assertEqual(self.func(data), 13.875)
+        #---
+        data = [5, 10, 10, 15, 20, 20, 20, 20, 25, 25, 30]
+        assert len(data)%2 == 1
+        self.assertEqual(self.func(data, 5), 19.375)
+        #---
+        data = [16, 18, 18, 18, 18, 20, 20, 20, 22, 22, 22, 24, 24, 26, 28]
+        assert len(data)%2 == 1
+        self.assertApproxEqual(self.func(data, 2), 20.66666667, tol=1e-8)
+
+    def test_even_number_repeated(self):
+        # Test median.grouped with repeated median values.
+        data = [5, 10, 10, 15, 20, 20, 20, 25, 25, 30]
+        assert len(data)%2 == 0
+        self.assertApproxEqual(self.func(data, 5), 19.16666667, tol=1e-8)
+        #---
+        data = [2, 3, 4, 4, 4, 5]
+        assert len(data)%2 == 0
+        self.assertApproxEqual(self.func(data), 3.83333333, tol=1e-8)
+        #---
+        data = [2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6]
+        assert len(data)%2 == 0
+        self.assertEqual(self.func(data), 4.5)
+        #---
+        data = [3, 4, 4, 4, 5, 5, 5, 5, 6, 6]
+        assert len(data)%2 == 0
+        self.assertEqual(self.func(data), 4.75)
+
+    def test_repeated_single_value(self):
+        # Override method from AverageMixin.
+        # Yet again, failure of median_grouped to conserve the data type
+        # causes me headaches :-(
+        for x in (5.3, 68, 4.3e17, Fraction(29, 101), Decimal('32.9714')):
+            for count in (2, 5, 10, 20):
+                data = [x]*count
+                self.assertEqual(self.func(data), float(x))
+
+    def test_odd_fractions(self):
+        # Test median_grouped works with an odd number of Fractions.
+        F = Fraction
+        data = [F(5, 4), F(9, 4), F(13, 4), F(13, 4), F(17, 4)]
+        assert len(data)%2 == 1
+        random.shuffle(data)
+        self.assertEqual(self.func(data), 3.0)
+
+    def test_even_fractions(self):
+        # Test median_grouped works with an even number of Fractions.
+        F = Fraction
+        data = [F(5, 4), F(9, 4), F(13, 4), F(13, 4), F(17, 4), F(17, 4)]
+        assert len(data)%2 == 0
+        random.shuffle(data)
+        self.assertEqual(self.func(data), 3.25)
+
+    def test_odd_decimals(self):
+        # Test median_grouped works with an odd number of Decimals.
+        D = Decimal
+        data = [D('5.5'), D('6.5'), D('6.5'), D('7.5'), D('8.5')]
+        assert len(data)%2 == 1
+        random.shuffle(data)
+        self.assertEqual(self.func(data), 6.75)
+
+    def test_even_decimals(self):
+        # Test median_grouped works with an even number of Decimals.
+        D = Decimal
+        data = [D('5.5'), D('5.5'), D('6.5'), D('6.5'), D('7.5'), D('8.5')]
+        assert len(data)%2 == 0
+        random.shuffle(data)
+        self.assertEqual(self.func(data), 6.5)
+        #---
+        data = [D('5.5'), D('5.5'), D('6.5'), D('7.5'), D('7.5'), D('8.5')]
+        assert len(data)%2 == 0
+        random.shuffle(data)
+        self.assertEqual(self.func(data), 7.0)
+
+    def test_interval(self):
+        # Test median_grouped with interval argument.
+        data = [2.25, 2.5, 2.5, 2.75, 2.75, 3.0, 3.0, 3.25, 3.5, 3.75]
+        self.assertEqual(self.func(data, 0.25), 2.875)
+        data = [2.25, 2.5, 2.5, 2.75, 2.75, 2.75, 3.0, 3.0, 3.25, 3.5, 3.75]
+        self.assertApproxEqual(self.func(data, 0.25), 2.83333333, tol=1e-8)
+        data = [220, 220, 240, 260, 260, 260, 260, 280, 280, 300, 320, 340]
+        self.assertEqual(self.func(data, 20), 265.0)
+
+
+class TestMode(NumericTestCase, AverageMixin, UnivariateTypeMixin):
+    # Test cases for the discrete version of mode.
+    def setUp(self):
+        self.func = statistics.mode
+
+    def prepare_data(self):
+        """Overload method from UnivariateCommonMixin."""
+        # Make sure test data has exactly one mode.
+        return [1, 1, 1, 1, 3, 4, 7, 9, 0, 8, 2]
+
+    def test_range_data(self):
+        # Override test from UnivariateCommonMixin.
+        data = range(20, 50, 3)
+        self.assertRaises(statistics.StatisticsError, self.func, data)
+
+    def test_nominal_data(self):
+        # Test mode with nominal data.
+        data = 'abcbdb'
+        self.assertEqual(self.func(data), 'b')
+        data = 'fe fi fo fum fi fi'.split()
+        self.assertEqual(self.func(data), 'fi')
+
+    def test_discrete_data(self):
+        # Test mode with discrete numeric data.
+        data = list(range(10))
+        for i in range(10):
+            d = data + [i]
+            random.shuffle(d)
+            self.assertEqual(self.func(d), i)
+
+    def test_bimodal_data(self):
+        # Test mode with bimodal data.
+        data = [1, 1, 2, 2, 2, 2, 3, 4, 5, 6, 6, 6, 6, 7, 8, 9, 9]
+        assert data.count(2) == data.count(6) == 4
+        # Check for an exception.
+        self.assertRaises(statistics.StatisticsError, self.func, data)
+
+    def test_unique_data_failure(self):
+        # Test mode exception when data points are all unique.
+        data = list(range(10))
+        self.assertRaises(statistics.StatisticsError, self.func, data)
+
+    def test_none_data(self):
+        # Test that mode raises TypeError if given None as data.
+
+        # This test is necessary because the implementation of mode uses
+        # collections.Counter, which accepts None and returns an empty dict.
+        self.assertRaises(TypeError, self.func, None)
+
+
+# === Tests for variances and standard deviations ===
+
+class VarianceStdevMixin(UnivariateCommonMixin):
+    # Mixin class holding common tests for variance and std dev.
+
+    # Subclasses should inherit from this before NumericTestClass, in order
+    # to see the rel attribute below. See testShiftData for an explanation.
+
+    rel = 1e-12
+
+    def test_single_value(self):
+        # Deviation of a single value is zero.
+        for x in (11, 19.8, 4.6e14, Fraction(21, 34), Decimal('8.392')):
+            self.assertEqual(self.func([x]), 0)
+
+    def test_repeated_single_value(self):
+        # The deviation of a single repeated value is zero.
+        for x in (7.2, 49, 8.1e15, Fraction(3, 7), Decimal('62.4802')):
+            for count in (2, 3, 5, 15):
+                data = [x]*count
+                self.assertEqual(self.func(data), 0)
+
+    def test_domain_error_regression(self):
+        # Regression test for a domain error exception.
+        # (Thanks to Geremy Condra.)
+        data = [0.123456789012345]*10000
+        # All the items are identical, so variance should be exactly zero.
+        # We allow some small round-off error, but not much.
+        result = self.func(data)
+        self.assertApproxEqual(result, 0.0, tol=5e-17)
+        self.assertGreaterEqual(result, 0)  # A negative result must fail.
+
+    def test_shift_data(self):
+        # Test that shifting the data by a constant amount does not affect
+        # the variance or stdev. Or at least not much.
+
+        # Due to rounding, this test should be considered an ideal. We allow
+        # some tolerance away from "no change at all" by setting tol and/or rel
+        # attributes. Subclasses may set tighter or looser error tolerances.
+        raw = [1.03, 1.27, 1.94, 2.04, 2.58, 3.14, 4.75, 4.98, 5.42, 6.78]
+        expected = self.func(raw)
+        # Don't set shift too high, the bigger it is, the more rounding error.
+        shift = 1e5
+        data = [x + shift for x in raw]
+        self.assertApproxEqual(self.func(data), expected)
+
+    def test_shift_data_exact(self):
+        # Like test_shift_data, but result is always exact.
+        raw = [1, 3, 3, 4, 5, 7, 9, 10, 11, 16]
+        assert all(x==int(x) for x in raw)
+        expected = self.func(raw)
+        shift = 10**9
+        data = [x + shift for x in raw]
+        self.assertEqual(self.func(data), expected)
+
+    def test_iter_list_same(self):
+        # Test that iter data and list data give the same result.
+
+        # This is an explicit test that iterators and lists are treated the
+        # same; justification for this test over and above the similar test
+        # in UnivariateCommonMixin is that an earlier design had variance and
+        # friends swap between one- and two-pass algorithms, which would
+        # sometimes give different results.
+        data = [random.uniform(-3, 8) for _ in range(1000)]
+        expected = self.func(data)
+        self.assertEqual(self.func(iter(data)), expected)
+
+
+class TestPVariance(VarianceStdevMixin, NumericTestCase, UnivariateTypeMixin):
+    # Tests for population variance.
+    def setUp(self):
+        self.func = statistics.pvariance
+
+    def test_exact_uniform(self):
+        # Test the variance against an exact result for uniform data.
+        data = list(range(10000))
+        random.shuffle(data)
+        expected = (10000**2 - 1)/12  # Exact value.
+        self.assertEqual(self.func(data), expected)
+
+    def test_ints(self):
+        # Test population variance with int data.
+        data = [4, 7, 13, 16]
+        exact = 22.5
+        self.assertEqual(self.func(data), exact)
+
+    def test_fractions(self):
+        # Test population variance with Fraction data.
+        F = Fraction
+        data = [F(1, 4), F(1, 4), F(3, 4), F(7, 4)]
+        exact = F(3, 8)
+        result = self.func(data)
+        self.assertEqual(result, exact)
+        self.assertIsInstance(result, Fraction)
+
+    def test_decimals(self):
+        # Test population variance with Decimal data.
+        D = Decimal
+        data = [D("12.1"), D("12.2"), D("12.5"), D("12.9")]
+        exact = D('0.096875')
+        result = self.func(data)
+        self.assertEqual(result, exact)
+        self.assertIsInstance(result, Decimal)
+
+
+class TestVariance(VarianceStdevMixin, NumericTestCase, UnivariateTypeMixin):
+    # Tests for sample variance.
+    def setUp(self):
+        self.func = statistics.variance
+
+    def test_single_value(self):
+        # Override method from VarianceStdevMixin.
+        for x in (35, 24.7, 8.2e15, Fraction(19, 30), Decimal('4.2084')):
+            self.assertRaises(statistics.StatisticsError, self.func, [x])
+
+    def test_ints(self):
+        # Test sample variance with int data.
+        data = [4, 7, 13, 16]
+        exact = 30
+        self.assertEqual(self.func(data), exact)
+
+    def test_fractions(self):
+        # Test sample variance with Fraction data.
+        F = Fraction
+        data = [F(1, 4), F(1, 4), F(3, 4), F(7, 4)]
+        exact = F(1, 2)
+        result = self.func(data)
+        self.assertEqual(result, exact)
+        self.assertIsInstance(result, Fraction)
+
+    def test_decimals(self):
+        # Test sample variance with Decimal data.
+        D = Decimal
+        data = [D(2), D(2), D(7), D(9)]
+        exact = 4*D('9.5')/D(3)
+        result = self.func(data)
+        self.assertEqual(result, exact)
+        self.assertIsInstance(result, Decimal)
+
+
+class TestPStdev(VarianceStdevMixin, NumericTestCase):
+    # Tests for population standard deviation.
+    def setUp(self):
+        self.func = statistics.pstdev
+
+    def test_compare_to_variance(self):
+        # Test that stdev is, in fact, the square root of variance.
+        data = [random.uniform(-17, 24) for _ in range(1000)]
+        expected = math.sqrt(statistics.pvariance(data))
+        self.assertEqual(self.func(data), expected)
+
+
+class TestStdev(VarianceStdevMixin, NumericTestCase):
+    # Tests for sample standard deviation.
+    def setUp(self):
+        self.func = statistics.stdev
+
+    def test_single_value(self):
+        # Override method from VarianceStdevMixin.
+        for x in (81, 203.74, 3.9e14, Fraction(5, 21), Decimal('35.719')):
+            self.assertRaises(statistics.StatisticsError, self.func, [x])
+
+    def test_compare_to_variance(self):
+        # Test that stdev is, in fact, the square root of variance.
+        data = [random.uniform(-2, 9) for _ in range(1000)]
+        expected = math.sqrt(statistics.variance(data))
+        self.assertEqual(self.func(data), expected)
+
+
+# === Run tests ===
+
+def load_tests(loader, tests, ignore):
+    """Used for doctest/unittest integration."""
+    tests.addTests(doctest.DocTestSuite())
+    return tests
+
+
+if __name__ == "__main__":
+    unittest.main()