Signal vs Noise Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
You flip a coin 10 times and get 6 heads. Is this signal (the coin is biased) or noise (random variation)? Calculate the expected range of heads for a fair coin.

Solution

  1. 1
    Expected heads: np=10×0.5=5np = 10 \times 0.5 = 5
  2. 2
    Standard deviation: np(1−p)=10×0.5×0.5=2.5≈1.58\sqrt{np(1-p)} = \sqrt{10 \times 0.5 \times 0.5} = \sqrt{2.5} \approx 1.58
  3. 3
    Expected range: 5±2(1.58)=[1.84,8.16]5 \pm 2(1.58) = [1.84, 8.16] — so 6 is well within normal random variation

Answer

6 heads is noise — well within the expected range [2,8][2, 8] for a fair coin with 10 flips.
Six heads out of 10 is not unusual for a fair coin — it's within 1 standard deviation of the mean. To detect bias (signal), more flips are needed so that random noise becomes small relative to any true bias.

About Signal vs Noise

Signal versus noise describes the fundamental challenge of separating meaningful patterns (signal) from random, unpredictable variation (noise) in data — the central task of all statistical analysis.

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