Revision

⚡ Statistics 1: Revision Refresher (V5)

The “Late Night” guide for rapid formula recall and conceptual checks.

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🏗️ Descriptive Foundations (Weeks 1-4)

• Scales: Nominal (ID) $\to$ Ordinal (Order) $\to$ Interval ($+,-$) $\to$ Ratio ($\times ,÷$).
• The “Corrected Sum” Logic:
  • $\sum x_{new}=\sum x_{old}+x_{corr}-x_{wrong}$
  • $\sum x_{new}^2=\sum x_{old}^2+x_{corr}^2-x_{wrong}^2$
• Variance Identity: $s^2=\frac{\sum x^2}{n-1}-\frac{(\sum x{)}^2}{n(n-1)}$.
• Outlier Check: $[Q_1-1.5IQR,Q_3+1.5IQR]$.

🔗 Correlations & Bivariate Data

Change	Effect on Mean	Effect on Variance	Effect on $r$
Add $c$	$\bar{x}+c$	Unchanged	Unchanged
Mult. $k$	$k\bar{x}$	$k^2{s}^2$	Unchanged (if $k>0$)
Mult. $-k$	$-k\bar{x}$	$k^2{s}^2$	Sign Flips

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🎲 Probability & Bayes (Weeks 5-8)

• Independence: $P(A\cap B)=P(A)P(B)$.
• Mutual Exclusive: $P(A\cap B)=0$.
• Bayes Shortcut: $P(B|A)=\frac{Likelihood\times Prior}{TotalProbability}$.

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🧪 Random Variables (Weeks 9-12)

• Expectation: $E[X]=\sum xf(x)$.
• Variance: $Var(X)=E[X^2]-(E[X]{)}^2$.
• Linearity: $E[\sum a_i{X}_i]=\sum a_{i}E[X_i]$.

📦 Distribution Cheat Sheet

Distribution	Parameter	Mean	Variance	Note
Binomial	$n,p$	$np$	$np(1-p)$	$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k{q}^{n-k}$
Poisson	$\lambda$	$\lambda$	$\lambda$	$P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$
Uniform	$a,b$	$\frac{a+b}{2}$	$\frac{(b-a{)}^2}{12}$	$f(x)=\frac{1}{b-a}$
Exponential	$\lambda$	$1/\lambda$	$1/{\lambda }^2$	$P(X>t)=e^{-\lambda t}$

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⚠️ Final Trap Alerts

1. Read the Denominator: Standard deviation is $\sqrt{Var}$.
2. Bessel Correction: For sample, use $n-1$.
3. Area Principle: In graphs, area = count.
4. Memorylessness: For Exponential, $P(X>s+t|X>s)=P(X>t)$.