On this page

1. Unit 1 — Probability and descriptive statistics
2. Unit 2 — Random variables and expectation
3. Unit 3 — Distributions
4. Unit 4 — Correlation and regression
5. Unit 5 — Inference and hypothesis testing
6. Critical values
7. Excel functions

Everything from the five units on one page. Print it, and use it for the last week before the exam. Each section links back to the unit that explains it.

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Unit 1 — Probability and descriptive statistics

Probability rules

	Formula
Classical probability	P(A) = favourable outcomes / total outcomes
Complement	P(A′) = 1 − P(A)
Addition	P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Addition, mutually exclusive	P(A ∪ B) = P(A) + P(B)
Multiplication	P(A ∩ B) = P(A) · P(B|A)
Multiplication, independent	P(A ∩ B) = P(A) · P(B)
Conditional	P(A|B) = P(A ∩ B) / P(B)
Independence test	P(A ∩ B) = P(A) · P(B)

Bayes' theorem (examined but not in the syllabus — finding D1)

Central tendency

	Formula
Mean	x̄ = Σxᵢ / n
Median	middle value when sorted; average the two middle if n is even
Mode	most frequent value
Empirical relation	Mode ≈ 3·Median − 2·Mean

Dispersion

	Formula
Range	max − min
IQR	Q₃ − Q₁
Outlier fences	Q₁ − 1.5·IQR and Q₃ + 1.5·IQR
Population variance	σ² = Σ(xᵢ − μ)² / N
Sample variance	s² = Σ(xᵢ − x̄)² / (n − 1)
Standard deviation	σ = √σ² or s = √s²
Coefficient of variation	CV = (s / x̄) × 100%
Pearson skewness	3(mean − median) / s

The n vs n−1 rule: population → divide by n; sample → divide by n − 1. This is the most common arithmetic error in the course.

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Unit 2 — Random variables and expectation

	Discrete	Continuous
Distribution	PMF p(x) = P(X = x)	PDF f(x)
Total probability	Σ p(x) = 1	∫ f(x) dx = 1
CDF	F(x) = Σ_{t ≤ x} p(t)	F(x) = ∫_{−∞}^{x} f(t) dt
Expectation	E(X) = Σ x·p(x)	E(X) = ∫ x·f(x) dx

Variance

Properties

Expectation	Variance
E(c) = c	Var(c) = 0
E(aX) = a·E(X)	Var(aX) = a²·Var(X)
E(aX + b) = a·E(X) + b	Var(aX + b) = a²·Var(X)
E(X + Y) = E(X) + E(Y) (always)	Var(X + Y) = Var(X) + Var(Y) (independent only)
E(XY) = E(X)·E(Y) (independent only)

Moments

Moment	Meaning
μ′₁ = E(X)	Mean
μ₂ = E[(X−μ)²]	Variance
μ₃	Skewness
μ₄	Kurtosis

• β₁ = μ₃²/μ₂³ (skewness) · β₂ = μ₄/μ₂² (kurtosis)
• β₂ = 3 mesokurtic · > 3 leptokurtic (peaked) · < 3 platykurtic (flat)
• MGF: Mₓ(t) = E(e^{tX}), and μ′ᵣ = d^r M/dt^r at t = 0

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Unit 3 — Distributions

Discrete

Distribution	PMF	Mean	Variance
Binomial(n, p)	ⁿCₖ pᵏ(1−p)ⁿ⁻ᵏ	np	np(1−p)
Poisson(λ)	e^{−λ} λᵏ / k!	λ	λ
Geometric(p)	(1−p)^{k−1} p	1/p	(1−p)/p²
Negative binomial(r, p)	^{k−1}C_{r−1} pʳ(1−p)^{k−r}	r/p	r(1−p)/p²

Continuous

Distribution	PDF	Mean	Variance
Uniform(a, b)	1/(b−a)	(a+b)/2	(b−a)²/12
Normal(μ, σ)	(1/(σ√(2π)))e^{−(x−μ)²/(2σ²)}	μ	σ²
Exponential(λ)	λe^{−λx}	1/λ	1/λ²
Gamma(α, λ)	(λ^α/Γ(α))x^{α−1}e^{−λx}	α/λ	α/λ²

Normal distribution essentials

Interval	Probability
μ ± 1σ	68.27%
μ ± 2σ	95.45%
μ ± 3σ	99.73%

Exponential CDF: F(x) = 1 − e^{−λx}, so P(X > x) = e^{−λx} Memorylessness: P(X > s+t | X > s) = P(X > t) — exponential and geometric only

Joint distributions

• Marginal: p(x) = Σ_y p(x, y)
• Conditional: p(y|x) = p(x, y) / p(x)
• Independent when p(x, y) = p(x)·p(y) for every pair

Central Limit Theorem

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Unit 4 — Correlation and regression

	Formula
Covariance	Cov(X,Y) = Σ(xᵢ−x̄)(yᵢ−ȳ)/(n−1)
Pearson r	r = Σ(x−x̄)(y−ȳ) / √[Σ(x−x̄)²·Σ(y−ȳ)²]
equivalently	r = Cov(X,Y)/(sₓ·s_y)
Spearman ρ	ρ = 1 − 6Σd² / [n(n²−1)]
Regression slope	b₁ = Σ(x−x̄)(y−ȳ) / Σ(x−x̄)²
Regression intercept	b₀ = ȳ − b₁x̄
Fitted line	ŷ = b₀ + b₁x
Residual	eᵢ = yᵢ − ŷᵢ
R²	SS_reg/SS_total = 1 − SS_res/SS_total
Adjusted R²	1 − [(1−R²)(n−1)/(n−k−1)]
SE of slope	SE(b₁) = √(MS_res / Σ(x−x̄)²)
t for slope	t = b₁ / SE(b₁), df = n − 2

Checks that catch arithmetic errors:

• Σ(x − x̄) = 0 and Σ(y − ȳ) = 0 always
• Residuals sum to zero
• For simple regression: R² = r² and t² = F

Regression ANOVA

Source	SS	df	MS	F
Regression	SS_reg	k	SS_reg/k	MS_reg/MS_res
Residual	SS_res	n−k−1	SS_res/(n−k−1)
Total	SS_tot	n−1

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Unit 5 — Inference and hypothesis testing

Confidence intervals

Test statistics

Test	Statistic	df
z-test	z = (x̄ − μ₀)/(σ/√n)	—
One-sample t	t = (x̄ − μ₀)/(s/√n)	n − 1
Two-sample t (pooled)	t = (x̄₁−x̄₂)/√(s²ₚ(1/n₁+1/n₂))	n₁+n₂−2
pooled variance	s²ₚ = [(n₁−1)s₁² + (n₂−1)s₂²]/(n₁+n₂−2)
Paired t	t = d̄/(s_d/√n)	n − 1
Chi-square	χ² = Σ(O−E)²/E	(r−1)(c−1)
expected frequency	E = (row total × column total)/grand total
F-test	F = s₁²/s₂², larger on top	(n₁−1, n₂−1)

Errors and power

	H₀ true	H₀ false
Reject H₀	Type I error (α)	Correct — power = 1 − β
Fail to reject	Correct (1 − α)	Type II error (β)

Decision rule: p < α → reject H₀. Otherwise fail to reject — never "accept".

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Critical values

Standard normal (z)

Confidence	α	Two-tailed z(α/2)	One-tailed z(α)
90%	0.10	1.645	1.282
95%	0.05	1.960	1.645
99%	0.01	2.576	2.326

Student's t — two-tailed at α = 0.05

df	t	df	t
5	2.571	18	2.101
8	2.306	20	2.086
10	2.228	24	2.064
12	2.179	30	2.042
15	2.131	∞	1.960

(One-tailed at α = 0.05: df 10 → 1.812, df 24 → 1.711.)

Chi-square — right tail at α = 0.05

df	χ²	df	χ²
1	3.841	5	11.070
2	5.991	6	12.592
3	7.815	8	15.507
4	9.488	10	18.307

F — right tail at α = 0.05

df₁ \ df₂	5	9	10
1	6.61	5.12	4.96
2	5.79	4.26	4.10
3	5.41	3.86	3.71

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Excel functions

Task	Function
Mean, median, mode	AVERAGE, MEDIAN, MODE.SNGL
Sample sd / variance	STDEV.S, VAR.S
Population sd / variance	STDEV.P, VAR.P
Quartile	QUARTILE.INC(range, 1 or 3)
Correlation	CORREL, PEARSON
Covariance	COVARIANCE.S, COVARIANCE.P
Rank (for Spearman)	RANK.AVG
Regression	SLOPE, INTERCEPT, RSQ, FORECAST.LINEAR
Binomial	BINOM.DIST(k, n, p, cumulative)
Poisson	POISSON.DIST(k, λ, cumulative)
Normal	NORM.DIST(x, μ, σ, TRUE), NORM.INV(p, μ, σ)
Standard normal	NORM.S.DIST(z, TRUE), NORM.S.INV(p)
Exponential	EXPON.DIST(x, λ, TRUE)
Confidence margin	CONFIDENCE.T(α, s, n), CONFIDENCE.NORM(α, σ, n)
t-test	T.TEST(r1, r2, tails, type)
Chi-square	CHISQ.TEST(observed, expected) → p-value
F-test	F.TEST(r1, r2) → p-value

The last two return p-values, not test statistics. Reporting a CHISQ.TEST result as χ² is a standard mistake.