Discrete & Normal Distributions

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Skill: distributions

Discrete & Normal Distributions

Compute binomial probabilities, standardize with z-scores, and use the empirical rule and normal tables.

1 Binomial: counting successes

A binomial setting has n independent trials, each success/failure with the same probability p. X = number of successes.

Binomial formulas
P(X = k) = C(n, k) pk (1−p)n−k   |   μ = np   |   σ = √(np(1−p))

Check BINS: Binary outcomes, Independent trials, fixed Number of trials, same Success probability.

Example 5 free throws, 70% shooter
  1. P(exactly 4) = C(5,4)(0.7)4(0.3)1 = 5 × 0.2401 × 0.3 = 0.360.
  2. Mean μ = 5(0.7) = 3.5; SD σ = √(5×0.7×0.3) ≈ 1.02.

2 z-scores: standardize everything

A z-score counts how many standard deviations a value sits from the mean. Negative z means below the mean — the sign carries information.

z-score
z = (x − μ) / σ   |   x = μ + zσ

z-scores let you compare values from different distributions (SAT vs ACT, heights vs weights).

Example Test score 82, mean 74, SD 4
  1. z = (82 − 74)/4 = 2.0 — two SDs above the mean.
  2. A score of 70 gives z = −1.0 — one SD below.

3 The normal model and the empirical rule

The normal (bell) curve is symmetric and fully described by μ and σ. The empirical rule (68–95–99.7) applies only to approximately normal data.

Empirical rule (normal data only)
≈68% within 1σ   |   ≈95% within 2σ   |   ≈99.7% within 3σ
μ μ−σ μ+σ μ−2σ μ+2σ 68% 95% within 2σ · 99.7% within 3σ
Example IQ: μ = 100, σ = 15
  1. About 95% of IQs fall within 2σ: 100 ± 30, i.e. 70 to 130.
  2. About 2.5% score above 130 (half of the 5% outside ±2σ).

4 Reading a z-table

A z-table gives Φ(z) = P(Z < z), the area under the standard normal curve to the left of z. For “greater than”, use 1 − Φ(z); for “between”, subtract two lookups.

Mini z-table (Φ(z) = area left of z)
z: 1.00 → 0.8413  |  1.28 → 0.8997  |  1.65 → 0.9505  |  1.96 → 0.9750  |  2.33 → 0.9901
Example P(Z < 1.65)
  1. Look up 1.65: Φ = 0.9505.
  2. P(Z > 1.65) = 1 − 0.9505 = 0.0495.

5 Common mistakes

Empirical rule on non-normal data. 68–95–99.7 describes the normal curve only. For skewed data (incomes, waiting times), it can be wildly wrong.
Ignoring the z-sign. z = −1.5 is a full 1.5 SDs below the mean. Dropping the negative sign flips the interpretation.
Reading “area right” from the table. Tables give area to the left. For P(Z > z), compute 1 − Φ(z).

6 Key vocabulary

Binomial distribution
Counts successes in n independent trials: P(X=k) = C(n,k)pk(1−p)n−k.
z-score
(x − μ)/σ: SDs from the mean; sign matters.
Normal distribution
Symmetric bell curve set by μ and σ.
Empirical rule
68–95–99.7% within 1–2–3 SD (normal data only).
z-table
Φ(z) = P(Z < z), area left of z under N(0,1).

7 Quick checks

1. Binomial n = 10, p = 0.4. What are μ and σ?
μ = 4, σ = √(2.4) ≈ 1.55. μ = np; σ = √(np(1−p)).
2. x = 90, μ = 100, σ = 5. Find z and say what it means.
z = −2. Two SDs below the mean.
3. Normal data: about what percent lies more than 2σ above μ?
≈2.5%. 5% falls outside ±2σ; half of that is above.
Practice Normal calculator Curve lab
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