Core answer: quartiles split ranked data at 25/50/75%, IQR=Q3−Q1, and values outside [Q1−1.5×IQR, Q3+1.5×IQR] are outliers; linear regression ŷ=a+bx fits by least squares with r² as explained variance; independent event probabilities multiply, mutually exclusive add; scientific notation writes a×10ⁿ with 1≤|a|<10.
Quartiles and box plots
After sorting: Q1 at 25%, median at 50%, Q3 at 75% (linear interpolation, matching Excel PERCENTILE.INC). Box plot five numbers: min, Q1, median, Q3, max, with outliers drawn as points. Fences: Q1−1.5×IQR and Q3+1.5×IQR; beyond 3×IQR is extreme.
Least-squares regression
Fit ŷ=a+bx:
- Slope b = Σ(x−x̄)(y−ȳ) ÷ Σ(x−x̄)²
- Intercept a = ȳ − b·x̄
- r² = share of y’s variance explained by x
| r² | Fit |
|---|---|
| <0.3 | weak |
| 0.3-0.7 | moderate |
| >0.7 | strong |
Three rules of probability
- Range: [0,1] — 0 impossible, 1 certain
- Independent → multiply: two heads in a row = 0.5×0.5 = 0.25
- Mutually exclusive → add: die shows 1 or 2 = 1/6+1/6 = 1/3
At least once goes through the complement: 10 draws at 5% each → 1−0.95¹⁰ = 40.1%, not 50%.
Sci notation and rounding
Scientific notation a×10ⁿ (1≤|a|<10): 299792458 m/s = 3.00×10⁸ (3 sig figs).
| Rule | 2.5 → | 3.5 → | Use |
|---|---|---|---|
| Half-up | 3 | 4 | everyday |
| Banker’s (half-even) | 2 | 4 | finance/stats de-biasing |
| Floor | 2 | 3 | conservative estimates |
Half-even pulls .5 to the nearest even digit, so long sums avoid systematic upward drift.
Example: salary outlier detection
Nine salaries (k): 6, 7, 8, 8, 9, 10, 11, 12, 45. Q1=7.5, Q3=11.5, IQR=4 → upper fence 17.5 → 45 flagged (the boss). Median 9 tells the truth; mean 12.9 is dragged up — the classic “average wage” illusion.
Example: area-to-price regression
Five homes: 60/80/90/110/130㎡, prices 180/230/260/310/350. Least squares gives ŷ=35+2.43x with r²≈0.99. Prediction at 100㎡ → ~278. High r² is not causation — area proxies location, and extrapolating to a 200㎡ villa breaks the model.
Common pitfalls
- “The mean is typical”: right-skewed income data demands the median — the quartile tool prints all five numbers plus the box.
- “High r² means causation”: ice cream vs drowning correlates strongly; the driver is summer. Correlation ≠ causation.
- “Multiplication always applies”: drawing without replacement breaks independence — the second draw is conditional.
- “0.5 must round up”: banker’s rounding sends 2.5→2 and 3.5→4, erasing bias in financial ledgers.