The Normal Distribution
The bell curve — why it appears everywhere and what it tells you
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Explanation
The normal distribution (bell curve) is the most important distribution in statistics. It appears naturally wherever many small independent factors add up — heights, exam scores, measurement errors.
Defined by two parameters:
- μ (mu) — the mean (center of the curve)
- σ (sigma) — the standard deviation (width of the curve)
The 68-95-99.7 rule (empirical rule):
- 68% of data falls within 1σ of the mean
- 95% falls within 2σ
- 99.7% falls within 3σ
mean=100, std=15 (IQ scores):
68% score between 85 and 115
95% score between 70 and 130
99.7% score between 55 and 145Z-score — how many standard deviations a value is from the mean: `` z = (x - μ) / σ `` A z-score of 2 means the value is 2 standard deviations above the mean.
Examples
Z-score calculation
Z-scores let you compare values across different scales
# IQ: mean=100, std=15
mean, std = 100, 15
score = 130
z = (score - mean) / std
print(f"z-score: {z}") # z-score: 2.0
# Score is 2 std above mean → top ~2.5%How well did you understand this?
Next in Mathematics for Data Science
Correlation vs Causation