The big idea: Many measurements follow a bell-shaped pattern: symmetric, with values clustered near center and fewer at extremes. This is the normal distribution.
Examples: heights, test scores, measurement errors. Not all data is normal (income is skewed), but natural phenomena often are.
Notation: X ~ N(μ, σ²) where μ is mean and σ² is variance. Sometimes the question gives σ (SD) instead — always check.
Why this matters
Key characteristics
- Most data clusters near mean.
- Fewer values far from mean.
- Curve is symmetric left-right.
- Mean = median = mode.
The 68-95-99.7 rule
One rule for all normal distributions: No matter what μ and σ are, the percentage of data within fixed distances from the mean is always the same.
The golden percentages
- 68% within 1 SD: between μ - σ and μ + σ
- 95% within 2 SDs: between μ - 2σ and μ + 2σ
- 99.7% within 3 SDs: between μ - 3σ and μ + 3σ
Worked example
Heights: N(170, 10²). What % between 160 and 180 cm?
Step by step
- 160 = 170 - 10 = μ - σ. 180 = 170 + 10 = μ + σ
- Asking for percent within 1 SD of mean
- By the rule: 68%
Final answer
68% of people are 160-180 cm tall.
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Converting to the standard normal
To use z-tables and calculate exact probabilities, convert any normal distribution to standard normal: Z ~ N(0, 1) with mean 0 and SD 1.
Interpreting z-scores: z = 0 means X = μ (at mean). z = 1 means 1 SD above. z = -2 means 2 SDs below. Positive z is above mean, negative z is below.
Worked example
X ~ N(100, 15²). Find z for x = 115 and x = 85.
Step by step
- For x = 115:
- For x = 85:
- Both exactly 1 SD away from mean.
Final answer
z(115) = 1, z(85) = -1.
Standard normal probabilities
The z-table gives Φ(z): Φ(z) = P(Z ≤ z) = area left of z. This is what z-tables show. Total area under curve = 1.
Using the z-table
- Φ(z) = area to left of z
- P(Z ≥ z) = 1 - Φ(z) (area to right)
- P(Z ≤ -z) = 1 - Φ(z) by symmetry
Worked example: Find probability
Find P(Z ≤ 1.23) and P(Z > 1.23).
Step by step
- Look up z = 1.23 in table: Φ(1.23) = 0.8907
- P(Z > 1.23) = 1 - 0.8907 = 0.1093
Final answer
P(Z ≤ 1.23) = 89.07%. P(Z > 1.23) = 10.93%.