Central Tendency

Arithmetic Mean

Compute the arithmetic mean as the sum of all values divided by the count. The trace builds the running total step by step, then shows a second pass where each deviation v - mean accumulates into dev_sum, ending at 0 — demonstrating that the mean is the balance point of the distribution.

By hand

Accumulate total and count elements in a first loop, then divide. A second loop sums up every deviation v - mean to confirm they cancel out exactly.

naive.py
Replay: real traced execution (multi-file project)
values = [4, 8, 6, 13, 10, 6, 8, 7]
total = 0
n = 0
for v in values:
    total = total + v
    n = n + 1
mean = total / n
dev_sum = 0
for v in values:
    dev_sum = dev_sum + (v - mean)
print('RESULT:', round(mean, 10))

  1. values ← [4, 8, 6, 13, 10, 6, 8, 7]

    1values = [4, 8, 6, 13, 10, 6, 8, 7]2total = 0
    values this step[4, 8, 6, 13, 10, 6, 8, 7]values

  2. total ← 0

    1values = [4, 8, 6, 13, 10, 6, 8, 7]2total = 03n = 0
    values this step0total

  3. n ← 0

    2total = 03n = 04for v in values:
    values this step0n

  4. v ← 4, total ← 4, n ← 1

    pass 1 of 8
    3n = 04for v in values:5    total = total + v6    n = n + 17mean = total / n
    values this step4v0 → 4total0 → 1n
    All 8 passes — pass 1 is the card above
    passvtotaln
    140 → 40 → 1
    24 → 84 → 121 → 2
    38 → 612 → 182 → 3
    46 → 1318 → 313 → 4
    513 → 1031 → 414 → 5
    610 → 641 → 475 → 6
    76 → 847 → 556 → 7
    88 → 755 → 627 → 8

  5. for v in values:

    3n = 04for v in values:5    total = total + v

  6. mean ← 7.75

    6    n = n + 17mean = total / n8dev_sum = 0
    values this step7.75mean

  7. dev_sum ← 0

    7mean = total / n8dev_sum = 09for v in values:
    values this step0dev_sum

  8. v ← 4, dev_sum ← -3.75

    pass 1 of 8
    8dev_sum = 09for v in values:10    dev_sum = dev_sum + (v - mean)11print('RESULT:', round(mean, 10))
    values this step7 → 4v0 → -3.75dev_sum
    All 8 passes — pass 1 is the card above
    passvdev_sum
    17 → 40 → -3.75
    24 → 8-3.75 → -3.5
    38 → 6-3.5 → -5.25
    46 → 13-5.25 → 0.0
    513 → 100.0 → 2.25
    610 → 62.25 → 0.5
    76 → 80.5 → 0.75
    88 → 70.75 → 0.0

  9. for v in values:

    8dev_sum = 09for v in values:10    dev_sum = dev_sum + (v - mean)

  10. stdout ← RESULT: 7.75

    10    dev_sum = dev_sum + (v - mean)11print('RESULT:', round(mean, 10))
    values this stepRESULT: 7.75stdout

With the library

statistics.mean and numpy.mean both reduce the list to a single value. statistics.mean operates on pure Python sequences; np.mean works on any array-like and is vectorised for large datasets.

library.py 
import statistics
import numpy as np
from dalib.display import set_display
set_display()

values = [4, 8, 6, 13, 10, 6, 8, 7]
mean_stdlib = float(statistics.mean(values))
mean_numpy = float(np.mean(values))
print('statistics.mean:', mean_stdlib)
print('np.mean:        ', mean_numpy)
print('RESULT:', round(mean_stdlib, 10))

statistics.mean: 7.75
np.mean:         7.75
RESULT: 7.75

Implementation notes

  • The deviations summing to zero is the defining property of the mean as a balance point: sum(v - mean for v in values) == 0 (up to float rounding). No other value has this property.
  • This lesson shares mechanics with python-data-basics/list-sum-mean; the framing there is algorithmic (how to compute it), while here it is statistical (what the result means).
  • statistics.mean returns int when the mean divides evenly for an all-integer input; float() normalises before printing.
  • For large arrays, prefer np.mean — it is vectorised and avoids Python loop overhead. Its internal summation order also tends to produce smaller rounding error than a scalar running total, though the exact behaviour depends on array shape and NumPy version.