The mean and the standard deviation of an array

mean(a): sum(a) / numel(a), over every entry whatever the rank.
std(a): the population standard deviation, which divides by n (numpy's np.std default; torch's std and Julia's std divide by n - 1): sqrt(sum(d * d) / numel(a)) with d = a - mean(a).

Both add in sum's order, NumPy's pairwise order, which is left to right below 8 entries, so tools/random_replay.py reproduces the six-sample test in Python.

Used with import stats.

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[1]lines 17–45

#| The mean of every entry of `a`.
def mean(a: Arr) {
    return sum(a) / numel(a)
}

#| The population standard deviation of every entry of `a` (divides by n).
def std(a: Arr) {
    d = a - mean(a)
    return sqrt(sum(d * d) / numel(a))
}

test "mean and std of six normals are the ones Python computes" {
    # ---- replayed by tools/random_replay.py ----
    SAMPLE = [4.16423517651071, -1.2113924334265904, 0.017206071253852796, -1.2683694313171443, -1.2785648787291637, 1.2992678807294153]
    REF_MEAN = 0.28706373083684655
    REF_STD = 1.968323100081351
    # ---- end of the replayed block ----
    assert_near(mean(SAMPLE), REF_MEAN, 1.0e-15)
    assert_near(std(SAMPLE), REF_STD, 1.0e-15)
}

test "every rank counts every entry" {
    assert_near(mean([1.0, 2.0, 3.0, 6.0]), 3.0, 0.0)
    assert_near(std([[1.0, 3.0], [1.0, 3.0]]), 1.0, 0.0)
    assert_near(std([5.0, 5.0, 5.0]), 0.0, 0.0)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors