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Monte

How spread out is it

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

Range is biggest minus smallest. Standard deviation is the typical distance from the average; in finance, risk. Empirical rule: on hill-shaped data, 68% fall within 1 SD, 95% within 2.

Why it matters

Two routes average 30 minutes. One always takes 30; the other, 15 to 60. SD is why you're late.

Risk is spread. Two things with the same average can be completely different bets.

  • FinanceVolatility is the SD of returns.
  • QualityA machine right on average but swinging a lot still makes bad parts.
  • ComparingA z score shows how unusual a value is, whatever its units.

Tennis example

Your first serve averages 115 mph with an SD of 5. About two thirds land between 110 and 120, 95% between 105 and 125. A 132 mph serve is over 3 SD out: a radar glitch, or something changed.

Drag the cutoff

A cafe sells an average of 400 coffees a day, SD 50.

250300350400450500550

450 is 1.0 SD above the average. Share above it: 15.9%. Share below: 84.1%.

2.5%below 300
13.5%300 to 350
34%350 to 400
34%400 to 450
13.5%450 to 500
2.5%above 500

Control chart: drag the process SD and watch points break the limits

A filling machine targets 500 grams per bag.

ULCLLL

In control. Limits were set for an SD of 4 g (500 ± 12). At SD 3, 0 of 40 points broke a limit.

Check yourself

  1. Range of 42, 57, 61, 38, 75, 12?

  2. Range of −40, −25, 8, 15?

  3. Problem with range?

The trap

The mistake people make most on this one.

Same range, different spread. 10, 50, 50, 50, 90 and 10, 10, 50, 90, 90 both span 80, but SDs of about 28 and 40. Range sees only the ends.

Formulas

7 formulas and a way to remember them.
Range
max − min
Population variance
σ² = Σ(y − μ)² / N
Sample variance
s² = Σ(y − ȳ)² / (n − 1)
Shortcut
s² = [Σy² − (Σy)² / n] / (n − 1)
Standard deviation
s = √s²
Coefficient of variation
CV = s / x bar
Empirical rule
x bar ± 1s holds 68%, x bar ± 2s holds 95%, x bar ± 3s holds 99.7%
Remember it
My Desk Shows Some Daily Risk: mean, distance, square, sum, divide, root. SD is how bumpy the ride is.

Watch and go deeper

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