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Monte

The normal curve and z

  1. Read it, not yet
  2. Practice it, not yet
  3. Master it, not yet

The normal is the bell. Measure in SDs from the mean, z = (y − μ) / σ, and any normal becomes the standard one. Area under it is probability.

Why it matters

Add up many small effects and you get the bell.

Know the mean and SD, and the bell tells you how rare any value is. Every cutoff and quality limit uses it.

  • InventoryStock enough for 95% of days: the mean plus 1.645 SDs.
  • QualityParts more than 3 SDs from target get flagged.
  • PercentilesYour percentile is the share of the bell's area below you.

Everyday example

A cafe sells 400 coffees a day on average, SD 50. More than 475 is z = 1.5, an area of 6.7%. Stock 464 cups and you run out on about 10% of days (z = 1.28).

Any normal: set μ, σ, and a cutoff

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z = (475 − 400) / 50 = 1.5. P(Y < 475) = 0.9332 (shaded). P(Y > 475) = 0.0668.

Check yourself

  1. Y ~ N(400, 50). P(Y ≤ 525)?

  2. P(Y ≥ 475)?

  3. P(400 ≤ Y ≤ 475)?

The trap

The mistake people make most on this one.

Some z tables give the area from 0 to z, not from the far left. For less than a positive z, add 0.5; Excel's NORM.S.DIST already starts from the left.

Formulas

5 formulas and a way to remember them.
z score
z = (y − μ) / σ, and back again y = μ + zσ
Areas
P(Y < y) = Φ(z), P(Y > y) = 1 − Φ(z), P(a < Y < b) = Φ(z of b) − Φ(z of a)
Excel
NORM.DIST(y, μ, σ, TRUE), NORM.S.DIST(z, TRUE), NORM.INV(p, μ, σ)
Key z values
1.28 (90%), 1.645 (95% one-sided), 1.96 (95% two-sided), 2.576 (99%)
Continuous uniform
Flat from a to b: P(c < Y < d) = (d − c) / (b − a). Mean (a + b) / 2, SD (b − a) / √12
Remember it
z is how many SDs from average. Memorize 1.28, 1.645, 1.96, 2.576 and every bell question is arithmetic.

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