📊 Session 4.5 - Normal Distribution

🔧 When does the Normal model make sense?

Engineering — measurement variation (CLT in action)

A micrometer measures a shaft diameter many times. Tiny independent errors (operator, instrument, vibration) add up. By the Central Limit Theorem, the sum of many small effects is well-approximated by a normal:

X ~ N (20.00, {0.02}^{2})

Question: What fraction of parts fall in the tolerance window 19.96–20.04 mm?

P (19.96 < X < 20.04) = Φ (\frac{20.04 - 20.00}{0.02}) - Φ (\frac{19.96 - 20.00}{0.02}) = Φ (2) - Φ (-2) \approx 0.9545

≈ 95.45% of parts pass. This is the “±2σ ≈ 95%” rule.

Student life — exam scores & z-scores

Large classes often have score distributions close to normal. Suppose a midterm has mean μ=72 and SD σ=10.

Q: What fraction scored above 90?

z = \frac{90 - 72}{10} = 1.8, P (X > 90) = 1 - Φ (1.8) \approx 0.0359

About 3.6% of students scored above 90.

Definition: A normal distribution is a continuous probability distribution with a symmetric, bell-shaped curve. Its PDF is

f (x) = \frac{1}{\sqrt{2 π σ^{2}}} e^{- \frac{{(x - μ)}^{2}}{2 σ^{2}}}

Mean (μ): the center of the distribution
Variance (σ²): the spread/dispersion
Notation: $X ~ N (μ, σ^{2})$

🔄 Standardizing (Z-Score)

What is a z-score? It measures how many standard deviations a value is from the mean. Positive z is above μ; negative z is below μ.

Z = \frac{X - μ}{σ}

Why use it? Every normal problem can be converted to the same standard normal (mean 0, SD 1). That means one single CDF $Φ (z)$ handles all normal models. This is convenient for teaching, calculators, Excel, and old textbook tables.

Where do I get Φ(z)?

On this page: use the calculator below (type z, click “Compute Φ(z)”).
Excel: =NORM.S.DIST(z, TRUE) (CDF) and =NORM.S.INV(p) (z for a percentile).
Classic textbook z-table (old way): read row/column for z to 2 decimals. Example: for z=1.50, look at row 1.5 and column 0.00 → Φ(1.50)=0.9332.

📎 Tiny z-table (abridged, for demo)

z \ .00	.00	.01	.02	.03	.04	.05
0.0	0.5000	0.5040	0.5080	0.5120	0.5160	0.5199
0.5	0.6915	0.6950	0.6985	0.7019	0.7054	0.7088
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531
1.5	0.9332	0.9345	0.9357	0.9368	0.9379	0.9390
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798

For negative z, use symmetry: Φ(−z) = 1 − Φ(z).

🧮 Z-Score Cookbook (use this every time)

For $X ~ N (μ, σ^{2})$ , standardize to $Z$ :

z = \frac{x - μ}{σ}

Left tail: $P (X < c) = Φ (\frac{c - μ}{σ})$
Right tail: $P (X > c) = 1 - Φ (\frac{c - μ}{σ})$
Between: $P (a < X < b) = Φ (\frac{b - μ}{σ}) - Φ (\frac{a - μ}{σ})$

Continuous normal ⇒ $P (X \leq c)$ equals $P (X < c)$ , and tails add to 1.

📐 Empirical Rule – 68–95–99.7%

🔢 Standard Normal Calculator

📌 Practice

📌 Where do I get Φ(z)?

On this page: use the built-in standard normal calculator.
Excel / Google Sheets:
- Φ(z): =NORM.S.DIST(z, TRUE)
- P(X > c): =1 - NORM.DIST(c, mu, sigma, TRUE)
- P(a < X < b): =NORM.DIST(b, mu, sigma, TRUE) - NORM.DIST(a, mu, sigma, TRUE)
Old textbooks: use a printed z-table to look up Φ(z).

📊 Excel: Step-by-step (sampling, probabilities, z-scores, histogram)

Compute Φ(z) and one/two-sided probabilities (no sampling)
Let cells hold parameters: B2=μ, B3=σ, B4=a, B5=b, B6=x₀, B7=z.
```
=NORM.S.DIST(B7, TRUE)
```
```
=NORM.DIST(B6, $B$2, $B$3, TRUE)
```
```
=1 - NORM.DIST(B6, $B$2, $B$3, TRUE)
```
```
=NORM.DIST($B$5, $B$2, $B$3, TRUE) - NORM.DIST($B$4, $B$2, $B$3, TRUE)
```
```
=(B6 - $B$2) / $B$3
```
```
=NORM.INV(0.95, $B$2, $B$3)
```
These use the normal CDF. No random numbers needed.
Simulate Normal(μ,σ) data (for intuition & checking)
Put μ in B2, σ in B3. In A2 enter the formula and fill down (e.g., to A5001):
```
=NORM.INV(RAND(), $B$2, $B$3)
```
Estimate a probability from the samples (with bounds L in D2, U in E2):
```
=COUNTIFS(A2:A5001, ">="&$D$2, A2:A5001, "<="&$E$2) / ROWS(A2:A5001)
```
Sample mean/variance to compare with μ, σ:
```
=AVERAGE(A2:A5001)
```
```
=STDEV.S(A2:A5001)
```
Histogram: Excel 365 → Insert → Chart → Histogram. Or create bins and use COUNTIFS.
Worked example (engineering tolerance)
Goal: $P (19.96 < X < 20.04)$ with μ=20.00, σ=0.02.
- Set B2=20, B3=0.02, B4=19.96, B5=20.04.
- Probability: use the “P(a < X < b)” formula above → ≈ 0.9545.
- Optional check: simulate 10,000 values with =NORM.INV(RAND(),$B$2,$B$3), then estimate via COUNTIFS.

📘 Session 4.5 – Normal Distribution