🔮 The normal curve and sampling distributions

Turn any value into a z-score and one curve answers everything: area = probability = percentile.

⭐ The one idea

Lots of measurements (birth weight, BP, lab values, test scores) pile up in a bell curve. Once you turn any value into a z-score, one curve answers every question: area under the curve = probability = percentile.

🔮 The normal curve

Standard normal: μ = 0, σ = 1−3−2−10123
Properties: symmetric about the mean, mean = median = mode, total area 1, tails never touch the axis. The inflection points sit exactly 1 SD from the mean.
68 – 95 – 99.7−3σ−2σ−1σμ+1σ+2σ+3σ68% within 1 SD95% within 2 SD99.7% within 3 SD
The Empirical Rule: about 68% of values within 1 SD, 95% within 2, 99.7% within 3. Works only for bell-shaped data.

🧪 Empirical rule with real numbers

Adult systolic BP in a clinic: mean 120, SD 15.

📏 z-scores

z = (x − μ) / σ
z on birth weightμ = 3,400 g σ = 500 g−31,900 g−2−103,400 g+1+2+34,900 g2,500 g4,000 gz = (x − μ) / σ. Negative isbelow the mean. Past ±2 isunusual, past ±3 is rare.
A z-score is a value translated into “SDs from the mean”. It lets you compare a birth weight with a blood pressure or a test score.

🔢 Compute one

Birth weights: μ = 3,400 g, σ = 500 g. A 2,500 g newborn: z = (2500 − 3400)/500 = −1.8. Below the mean by 1.8 SDs.

🔄 Go backwards

x = μ + zσ

z = +1.2 → x = 3400 + 1.2(500) = 4,000 g.

⚖️ Compare apples to oranges

Student A scored 82 on an exam with mean 70, SD 8 (z = 1.5). Student B scored 90 where mean 85, SD 5 (z = 1.0). A did relatively better.

⚠️ Unusual

|z| > 2 is unusual (outside the middle 95%). |z| > 3 is rare. A lab value at z = 3.5 gets a phone call.

📊 Finding areas (probabilities)

1 Drawthe curve, mark μ and the value
2 Shadethe side the question asks about
3 z-scorez = (x − μ)/σ
4 Areatable or Desmos cdf
5 Answeras a probability or a %
Birth weight: P(X < 2,500 g)2,4002,9003,4003,9004,4003.6%
P(X < 2,500) → z = −1.8 → area to the left = 0.0359. About 3.6% of newborns are low birth weight (< 2,500 g) in this population.
Birth weight: P(3,000 < X < 4,000)3,0003,4004,00067.3%
P(3,000 < X < 4,000) → z from −0.8 to +1.2 → 0.8849 − 0.2119 = 0.673. Between = big area minus small area.

⬅️ Less than

Area to the left. Table value directly, or normaldist(μ,σ).cdf(−∞, x). P(X < 2500) = 0.0359

➡️ Greater than

Area to the right = 1 − left area. P(X > 2500) = 1 − 0.0359 = 0.9641. Desmos: .cdf(x, ∞).

↔️ Between

Left area of the bigger value minus left area of the smaller. Desmos: .cdf(a, b). P(3000<X<4000) = 0.673

💡 Table habit

The z-table gives area to the left of z. Row = first two digits (−1.8), column = hundredths (.00). For “greater than” subtract from 1.

🔄 Backwards: from a percentile to a value

The 90th percentile3,4004,041 g10%90% below
“Which weight is at the 90th percentile?” → find z with 0.90 to its left (z = 1.28) → x = 3400 + 1.28(500) = 4,041 g. Desmos: normaldist(3400,500).inversecdf(0.90).

⭐ Memorize these z cut-offs

Middle areaz*Used for
90%1.64590% CI
95%1.9695% CI, α = 0.05 two-tailed
99%2.57699% CI

One-tailed 5% cut-off is 1.645; two-tailed 5% is ±1.96.

🎲 Sampling distributions and the Central Limit Theorem (week 9)

⭐ The CLT in one breath

Take many samples of size n, average each one. Those sample means form their own distribution that is approximately normal with mean μ and SD σ/√n — as long as n ≥ 30 or the population was normal to begin with.

Why bigger samples settle downPopulation: skewedmean μ, SD σMeans of samples, n = 5spread σ/2.2 — narrowerMeans of samples, n = 30spread σ/5.5 — narrower stillSame population every time.Averages of bigger samples pileup closer to μ.
Left: a skewed population (hospital stays). Middle and right: distributions of sample MEANS. They tighten and turn bell-shaped as n grows.

📏 Standard error of the mean

σx̄ = σ / √n

Birth weight σ = 500. For samples of 25 babies: 500/√25 = 100 g. For 100 babies: 50 g. Averages wobble less than individuals.

🔢 z for a sample mean

z = (x̄ − μ) / (σ/√n)

Same z idea, but divide by the standard error, not σ. Forgetting the √n is the #1 exam error.

Sample means, n = 25: P(x̄ < 3,200)3,1003,2003,4003,6003,7002.3%
25 newborns from a unit: P(mean weight < 3,200 g)? z = (3200 − 3400)/(500/√25) = −2.0 → 0.0228. For ONE baby, P(X < 3200) would be z = −0.4 → 0.345. The mean of 25 is far less likely to be that low.

📊 Sample proportions too

μp̂ = p    σp̂ = √( p(1−p) / n )

Normal enough when np ≥ 10 and n(1−p) ≥ 10 (some books use 5). 30% of patients are readmitted. In a sample of 100, P(p̂ ≥ 0.36)? SE = √(0.3·0.7/100) = 0.0458, z = 0.06/0.0458 = 1.31, P = 0.095.

🧮 In Desmos

1W = normaldist(3400, 500)birth weights
2W.cdf(−∞, 2500)= 0.0359
3W.cdf(3000, 4000)= 0.6731
4W.cdf(2500, ∞)= 0.9641
5W.inversecdf(0.90)= 4040.8
6M = normaldist(3400, 500/sqrt(25))sample means
7M.cdf(−∞, 3200)= 0.0228
8normaldist().cdf(−∞, −1.8)= plain z-table lookup
⌨️ Type infty for ∞. normaldist() with nothing inside is the standard normal (μ = 0, σ = 1). Click the graph of the distribution and Desmos shades the area for you.

✅ Quick self-check

❓ Temperatures: μ = 98.6 °F, σ = 0.6. What percent of healthy adults read above 100.0?
z = (100 − 98.6)/0.6 = 2.33 → right-tail area 0.0098 ≈ 1%. A 100 °F reading is unusual.
❓ Adult BP μ = 120, σ = 15. P(100 < X < 140)?
z from −1.33 to +1.33 → 0.9088 − 0.0912 = 0.818.
❓ Length of stay: μ = 4.2 days, σ = 2.1 (skewed). For 36 patients, P(mean stay > 5 days)?
CLT (n ≥ 30) → normal. SE = 2.1/6 = 0.35. z = 0.8/0.35 = 2.29 → 0.0111. The skew does not matter because n = 36.
❓ Which is more likely: one newborn under 3,000 g, or a sample of 16 newborns averaging under 3,000 g?
One newborn. Individual: z = −0.8 (21%). Mean of 16: z = −200/125 = −1.6 (5.5%). Averages are less spread out.
MAT 300 · built from your own course files (the statistics study guide, the Desmos guide, the formula sheet and the final-exam study questions) with nursing examples. Not a substitute for the textbook — check any number against your own notes before an exam.