1.1. Uncertainty

Randomness vs. Uncertainty: The Core of Probability

Randomness is about processes — the world generates outcomes unpredictably.
Uncertainty is about knowledge — we lack full information.

Think: a die roll (random process) vs. not knowing if the die landed on 3 or 4 (uncertainty).

Probability as a Consistent Language for Uncertainty

A probability model assigns a number to how plausible each outcome is, following three axioms:

$$ 0 \le P(A) \le 1, \quad P(\Omega)=1, \quad P(A \cup B)=P(A)+P(B)-P(A\cap B) $$

Common interpretations:

Perspective	Meaning	Example
Frequentist	Long-run frequency	30% defective parts in many samples
Bayesian	Degree of belief	Prior belief defect rate = 30%, updated with new data
Geometric/Measure	Proportion of total “space”	Dart hits area covering 0.3 of total

• Q1: Is “I don’t know tomorrow’s temperature” randomness or uncertainty?
  • Uncertainty (temperature is determined; our knowledge isn’t).
• Q2: If (P(A)=0.4) and (P(B)=0.5), estimate (P(A∩B)).
  • ≈ 0.2 if independent. (P(A∩B) ≤ 0.4
• Q3: Why is $(P(A|B)\neq P(B|A))$ generally?
  • Conditioning direction matters—what’s known vs. what’s inferred.

Descriptive Statistics by Measurement Scale

Not all averages are meaningful. Match your summary to the data scale.

Scale	Typical statistics	Meaningful central tendency	Meaningful spread	Notes
Nominal	Counts, proportions, mode	Mode	None (just frequencies)	Never take means or SDs of labels.
Ordinal	Median, percentiles, IQR, rank correlation	Median	IQR	Preserve order; spacing not uniform.
Interval	Mean, SD, z-score	Mean	SD, variance	Ratios (×2) meaningless (no true zero).
Ratio	Mean, geometric mean, SD, CV	Mean or geometric mean	SD, CV	Ratios and proportions valid.

Descriptive Statistics by Measurement Scale

Not all averages are meaningful. Match your summary to the data scale.

Scale	Typical statistics	Meaningful central tendency	Meaningful spread	Notes
Nominal	Counts, proportions, mode	Mode	None (just frequencies)	Never take means or SDs of labels.
Ordinal	Median, percentiles, IQR, rank correlation	Median	IQR	Preserve order; spacing not uniform.
Interval	Mean, SD, z-score	Mean	SD, variance	Ratios (×2) meaningless (no true zero).
Ratio	Mean, geometric mean, SD, CV	Mean or geometric mean	SD, CV	Ratios and proportions valid.

Minimal Pandas Examples

import pandas as pd

# Example dataset
df = pd.DataFrame({
    "color": ["red","blue","blue","red","green"],
    "satisfaction": [1,2,3,4,5],
    "temp_C": [20,22,19,23,21],
    "income": [30_000, 50_000, 45_000, 70_000, 60_000]
})

# Nominal
df["color"].value_counts(normalize=True)

# Ordinal
df["satisfaction"].median()

# Interval
df["temp_C"].mean(), df["temp_C"].std()

# Ratio
df["income"].mean(), df["income"].std() / df["income"].mean()  # CV

• Q1: For “favorite color,” what valid descriptive statistic exists?
  • → Mode or frequency table only.
• Q2: For “pain level (1–5),” should you compute a mean?
  • → No, use median or percentiles (ordinal scale).
• Q3: For “age in years,” which measure of spread is valid?
  • → Standard deviation or coefficient of variation (ratio scale).