Cheatsheet: The bell curve: the normal distribution

The one idea

The normal distribution is the bell curve defined by a mean and a standard deviation. The 68-95-99.7 rule and the z-score make it usable at a glance.

Continuous distributions

Probability = AREA under a density curve. Total area = 1.
P(value in a range) = area over that range. No single exact value has a probability.

The normal distribution

Symmetric bell, defined by TWO numbers:
  mean               = center (where the peak sits)
  standard deviation = width (bigger = wider/flatter)
Change mean -> slide left/right.   Change sd -> widen/narrow.  Same shape always.

The 68-95-99.7 (empirical) rule

within 1 sd of the mean  -> about 68%
within 2 sd              -> about 95%
within 3 sd              -> about 99.7%

Scores ~ Normal(mean 500, sd 100):
  400 to 600 -> ~68%     300 to 700 -> ~95%     200 to 800 -> ~99.7%
  above 700 (z=+2) -> ~2.5%   below 600 (z=+1) -> ~84%

The z-score

z = (value - mean) / standard deviation     (= standardization from Phase 1)
  600 in N(500,100): z = +1.0      700: z = +2.0      450: z = -0.5
Use: comparability across scales; z=+1 -> ~84% below; |z|>2 or 3 -> unusual (outlier).

In machine learning

• Feature standardization = a z-score per feature value.
• Default noise model and neural-net weight initialization = Gaussian (normal).
• Outlier detection: flag values beyond 2 to 3 standard deviations.
• Why so common: averages/sums of many independent things tend to normal (next phase).

Pitfalls to dodge

• Assuming everything is normal (skewed/bimodal data breaks the rules; check a histogram).
• Confusing curve height with probability (probability is area over a range).
• Forgetting a z-score needs both mean AND standard deviation.
• Reading 99.7% as “all” (the tails extend forever; extremes are rare, not impossible).

Words to use precisely

• Density curve: a curve whose area gives probabilities (total area 1).
• Normal distribution: the symmetric bell set by a mean and standard deviation.
• Empirical rule: 68-95-99.7 within 1-2-3 standard deviations.
• z-score: (value - mean) / standard deviation; standard distance from the mean.