Statistics Comprehensive Review — Essential Formulas and Practical Applications

Statistics: The Full Workflow

• Data Collection — Sample Design
• Descriptive Statistics — Mean, Std Dev
• Probability Theory — Distributions, CLT
• Inferential Statistics — Estimation, Testing

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Essential Formula Reference

Central Tendency and Dispersion

$$\begin{aligned} \text{Arithmetic mean}: \bar{x} &= \sum x_{i} / n \\ \text{Sample variance}: s^2 &= \sum (x_{i}-\bar{x})^2 / (n-1) \\ \text{Standard deviation}: s &= \sqrt{s^2} \\ \text{Coefficient of variation}: CV &= s/\bar{x} \times 100\% \end{aligned}$$

Probability Fundamentals

$$\begin{aligned} \text{Addition rule}: P(A∪B) &= P(A) + P(B) − P(A∩B) \\ \text{Multiplication}: P(A∩B) &= P(A) \times P(B \mid A) \\ \text{Conditional}: P(A \mid B) &= P(A∩B) / P(B) \\ \text{Independence}: P(A∩B) &= P(A) \times P(B) \end{aligned}$$

Key Probability Distributions

$$\begin{aligned} \text{Binomial}: E(X) &= np, Var(X) = np(1-p) \\ \text{Poisson}: E(X) &= Var(X) = \lambda \\ \text{Normal}: X~N(\mu ,\sigma ^2) \to Z &= (X-\mu )/\sigma ~ N(0,1) \\ \text{t-dist}: t &= \frac{\bar{x}-\mu _0}{\frac{s}{\sqrt{n}}}, df = n-1 \end{aligned}$$

Estimation and Testing

$$\begin{aligned} \text{Standard error}: SE &= \frac{\sigma}{\sqrt{n}} \\ &\text{95\% CI}: \bar{x} \pm 1.96 \times \frac{\sigma}{\sqrt{n}} \\ &\text{99\% CI}: \bar{x} \pm 2.576 \times \frac{\sigma}{\sqrt{n}} \\ \text{Z-test statistic}: Z &= \frac{\bar{x}-\mu _0}{\frac{\sigma}{\sqrt{n}}} \\ \text{F-statistic}: F &= MSB/MSW \\ \text{Chi-square}: \chi ^2 &= \sum (O-E)^2/E \end{aligned}$$

Regression Analysis

$$\begin{aligned} \text{Slope}: \beta_{1} &= r \times (\frac{\text{sy}}{\text{sx}}) \\ \text{Intercept}: \beta_{0} &= ȳ − \beta_{1}x̄ \\ \text{Coefficient of det}.: R^2 &= \frac{SSR}{SST} = 1 − \frac{SSE}{SST} \end{aligned}$$

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Key Values to Memorize

Quantity	Value
Z₀.₀₂₅ (95% two-tailed)	1.96
Z₀.₀₀₅ (99% two-tailed)	2.576
Z₀.₀₅ (90% two-tailed)	1.645
68–95–99.7% rule	±1σ, ±2σ, ±3σ
Binomial → Poisson approx.	n ≥ 30, p ≤ 0.05, λ = np

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Common Statistical Fallacies and Traps

1. Confusing Correlation with Causation

Ice cream sales ↑ → Drowning deaths ↑ (Common cause: summer, not causation)

2. Misinterpreting the p-value

p = 0.03 does NOT mean H₀ has only a 3% probability of being true
→ It means “assuming H₀ is true, this data has only a 3% chance of occurring”

3. Statistical Significance ≠ Practical Importance

With very large n, tiny differences become statistically significant
→ Always report effect size (Cohen’s d, η²) alongside the p-value

4. Survivorship Bias

Analyzing only businesses that survived ignores failures → distorted conclusions
→ Classic example: WWII analysis of returning aircraft showed bullet holes only where planes survived hits — armor was needed elsewhere

5. Extreme Values in Small Samples

Outliers in small samples have outsized effects on the mean → use the median

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Real-World Applications of Statistics

Quality Control

• Six Sigma: Defect rate < 3.4 ppm (within μ ± 6σ)
• Process capability index: Cp = (USL − LSL) / 6σ

A/B Testing

Statistically compares two versions (A and B) — used in website conversion rate optimization, pharmaceutical clinical trials, and advertising effectiveness.

Use t-test or Z-test to compare two proportions or means
Ensure sufficient sample size → adequate statistical power

Economic Indicator Interpretation

• GDP growth rate: % change from same period prior year (seasonally adjusted)
• Consumer Price Index (CPI): Price level relative to a base year (published monthly by BLS)
• Unemployment rate: Unemployed / Labor force (published monthly by BLS, Current Population Survey)

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Key Concept Cards

Central Limit Theorem (Review) ★★★★★ : Regardless of population distribution, when n ≥ 30, the sample mean approximates a normal distribution. The theoretical foundation of all statistical inference. Memory tip: CLT = the most important theorem in statistics

Limits of Statistical Significance ★★★★★ : p < 0.05 does not guarantee practical significance. Always interpret alongside effect size. Memory tip: significant ≠ important; large n can make small differences statistically significant

Survivorship Bias ★★★★☆ : Drawing conclusions only from observable (surviving) cases, ignoring those that were eliminated. Memory tip: WWII aircraft — only planes that came back were analyzed, leading to misplaced armor reinforcement

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Practice Questions

Q. An online course has a mean satisfaction rating of 7.2 (σ = 1.5, n = 400). What is the 95% confidence interval for the population mean?

SE = 1.5/√400 = 0.075. 95% CI = 7.2 ± 1.96 × 0.075 = [7.053, 7.347].

Q. Two marketing strategies have click-through rates of A = 5.2% (n₁ = 1,000) and B = 5.8% (n₂ = 1,000). Is the difference statistically significant?

Using a two-proportion Z-test: difference = 0.006, pooled SE = √[0.052×0.948/1000 + 0.058×0.942/1000] ≈ 0.0070. Z ≈ 0.857. p > 0.05 → not statistically significant. The absolute difference is too small relative to sampling variability given these sample sizes.