Question 1

What is the 68-95-99.7 empirical rule?

Accepted Answer

For any normal distribution N(μ, σ²): 68.27% of values fall within μ ± 1σ. 95.45% fall within μ ± 2σ. 99.73% fall within μ ± 3σ. Only 0.27% of values (about 1 in 370) fall more than 3 standard deviations from the mean. This rule lets you quickly estimate probabilities without a calculator: e.g., IQ scores (μ=100, σ=15) — 95% of people have IQ between 70 and 130.

Question 2

How do I compute P(x₁ ≤ X ≤ x₂) for a normal distribution?

Accepted Answer

P(x₁ ≤ X ≤ x₂) = Φ(z₂) − Φ(z₁), where z₁ = (x₁ − μ)/σ, z₂ = (x₂ − μ)/σ, and Φ is the standard normal CDF. Example: Heights ~ N(170, 10²). P(160 ≤ X ≤ 180) = P(−1 ≤ Z ≤ 1) = Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6827 (≈ 68% by the empirical rule).

Question 3

What does the standard deviation σ control in the bell curve?

Accepted Answer

σ controls the width (spread) of the curve. A larger σ produces a flatter, wider curve with lower peak. A smaller σ gives a taller, narrower curve. The peak height is always 1/(σ√2π). The total area under the curve is always 1.0 regardless of σ. The curve is symmetric around μ, with inflection points at μ ± σ (the points where the curve changes from concave down to concave up).

Question 4

How do I use z-scores with the bell curve?

Accepted Answer

A z-score converts any normal distribution value to the standard normal: z = (x − μ)/σ. Then use z-score tables or this calculator to find P(X ≤ x) = Φ(z). Positive z: x is above the mean. z = 1: x is 1 standard deviation above. Negative z: x is below the mean. P(Z > 1.96) ≈ 0.025 and P(Z > 2.576) ≈ 0.005 — the critical values commonly used in hypothesis testing.

Question 5

Why is the normal distribution so common in statistics?

Accepted Answer

The Central Limit Theorem (CLT) states that the sample mean of n independent identically distributed variables converges to a normal distribution as n → ∞, regardless of the original distribution. This is why sample means of almost anything — heights, test scores, measurement errors — are normally distributed for large enough n. The normal distribution is also the maximum entropy distribution for a given mean and variance, making it the "least informative" prior.

Question 6

What are common real-world examples of normal distributions?

Accepted Answer

Examples: IQ scores (μ = 100, σ = 15). Adult male heights in the US (μ ≈ 178cm, σ ≈ 7cm). Blood pressure readings in a healthy population. Measurement errors in scientific instruments. Annual rainfall over many years in a stable climate. Exam scores for a well-designed test. Stock returns over short periods (approximately). The normal distribution is the "default assumption" in many fields, though it should always be verified for your specific data.

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