Normalization Arithmetic

Normalization Arithmetic

Ada re-derives this chapter’s own numbers step by step, at full precision

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Ada ADA · CALCULATION AUDIT

Normalization Arithmetic

The chapter normalizes five temperature readings — 18, 22, 25, 19, 21 C — two ways: min-max scaling puts 22 C at 0.571, while z-score places it at 0.408 standard deviations. The two methods disagree on purpose, and a wrong scale silently reweights every downstream model. This audit re-derives the normalization arithmetic at full precision to show why an outlier that pushes the max to 85 collapses that same 0.571 reading.

Companion to the chapter Lab: Feature Scaling — every number here comes from that chapter.

See the relationship before changing it

The figure reads from left to right. The blue card is observed maximum. The middle card applies the page rule. The green card is scaled 22 c reading. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

Observed maximum changes scaled 22 c reading An input card leads through the rule scaled value = (22 - 18) / (maximum - 18) to the scaled 22 c reading result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. One extreme maximum squeezes every ordinary reading toward zero.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 85 C.

  2. 2

    Name the relationship. scaled value = (22 - 18) / (maximum - 18)

  3. 3

    Substitute with units. (22 - 18) / (85 - 18) = 4 / 67 = 0.0597

  4. 4

    Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change observed maximum

Try Predict the direction of scaled value = (22 - 18) / (maximum - 18). Test another observed maximum, then compare scaled 22 c reading.

85 C
Chapter baseline
Scaled 22 C reading

Observe One extreme maximum squeezes every ordinary reading toward zero. Reset observed maximum to 85 and compare scaled 22 c reading.

Explain One extreme maximum squeezes every ordinary reading toward zero.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only observed maximum moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only observed maximum. It is not a field prediction.

Try

The chapter normalizes five temperature readings — 18, 22, 25, 19, 21 C — two ways: min-max scaling puts 22 C at 0.571 , while z-score places it at 0.408 standard deviations. Calculate this case.

Observe

This audit re-derives the normalization arithmetic at full precision to show why an outlier that pushes the max to 85 collapses that same 0.571 reading. Check shows this.

Explain

The two methods disagree on purpose. Min-max forces 25 C to exactly 1.000, its meaning tied to the observed max; z-score places 25 C at 1.633 standard deviations, its meaning tied to the spread. The audit conclusion is narrow: if a later outlier pushes x_max to 85, the same 22 C reading that scaled to 0.571 here collapses toward (22 - 18) / (85 - 18) = 4 / 67 = 0.059701, about0.060, exposing min-max sensitivity to one extreme. Check confirms it.

Technical boundaries

Excluded from the “Normalization Arithmetic” calculation are future distribution shift, outliers, missing values, leakage from fitting on test data, or whether scaling improves the downstream model; “Normalization Arithmetic” therefore reports only its named fixtures.

Ada: Normalization decides how much each sensor channel is allowed to say, so a wrong scale silently reweights every downstream model. Let me re-derive the worked example above at full precision using only its five readings: 18, 22, 25, 19, 21 (degrees C).

Min-max scaling uses x' = (x - x_min) / (x_max - x_min) with x_min = 18 and x_max = 25, so the range is 25 - 18 = 7:

  • 18 C: (18 - 18) / 7 = 0 / 7 = 0.000000
  • 19 C: (19 - 18) / 7 = 1 / 7 = 0.142857
  • 21 C: (21 - 18) / 7 = 3 / 7 = 0.428571
  • 22 C: (22 - 18) / 7 = 4 / 7 = 0.571429, rounded to 0.571.
  • 25 C: (25 - 18) / 7 = 7 / 7 = 1.000000.

Z-score uses z = (x - mu) / sigma. The mean is mu = (18 + 22 + 25 + 19 + 21) / 5 = 105 / 5 = 21. The population standard deviation:

  • Squared deviations: (18-21)^2 + (22-21)^2 + (25-21)^2 + (19-21)^2 + (21-21)^2 = 9 + 1 + 16 + 4 + 0 = 30
  • Variance: 30 / 5 = 6
  • sigma = sqrt(6) = 2.449490, rounded to 2.449.
  • For 22 C: (22 - 21) / 2.449490 = 0.408248, about 0.408.
  • For 25 C: (25 - 21) / 2.449490 = 1.632993, about 1.633.

The two methods disagree on purpose. Min-max forces 25 C to exactly 1.000, its meaning tied to the observed max; z-score places 25 C at 1.633 standard deviations, its meaning tied to the spread. The audit conclusion is narrow: if a later outlier pushes x_max to 85, the same 22 C reading that scaled to 0.571 here collapses toward (22 - 18) / (85 - 18) = 4 / 67 = 0.059701, about 0.060, which is why the chapter reaches for robust scaling when spikes are expected.

Every number above is taken from the chapter’s own material and re-derived step by step.