Introduction to the Practice of Statistics
Why do we need statistics? It's not just "applied math"—it's the scientific framework we use to understand a variable world.
The Philosophy of Variability
Statistics is the science of collecting, organizing, summarizing, and analyzing information to draw conclusions. But why does it exist separately from pure math?
"Because of Variability."
If , then . This is always true. 100% certainty.
Measure one person's height: . Measure another: . They are different.
If everyone were identical, we'd only need to measure one person. Statistics exists to quantify the uncertainty created by this variability.
Population vs. Sample (The Inference Loop)
Population
The entire group to be studied. Usually impossible to measure completely.
Sample
A subset of the population. The actual data we possess.
The Statistical Process Loop
The Taxonomy of Variables
Classification determines the valid math operations.
| Variable | Type | Reasoning (Edge Cases) |
|---|---|---|
| Zip Code | Qualitative | Looks numerical (90210), but adding them is meaningless. They are location labels. |
| Shoe Size | Discrete | "8.5" looks like a decimal, but values come in fixed steps. You can't have size 8.72. |
| Temperature | Continuous | Infinite possibilities between any two points. Can be 70.15°F. |
Levels of Measurement
LogicLens Proof: The Failure of "Twice as Hot"
Why implies ?
If "Twice as Hot" is real, the ratio must hold in any valid unit (Physics invariance).
Contradiction. The ratio is not preserved.
Reasoning: is not "No Heat". Without a True Zero, multiplication ratios are invalid.
Common Pitfalls
- Confusing the raw 'Data' list with the calculated summary 'Statistic'.
- Thinking 'Decimal = Continuous'. Money ($1.50) is discrete.
- Trusting Voluntary Response samples (large bias).
Real-World Application
A/B Testing (Netflix/Google)
Companies show Interface A to 99% of users (Pop) and Interface B to 1% (Sample).
If Sample B spends 5% more time (Variable), statistics tells us if that's a real signal or just random noise.
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