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Probability & Statistics 1 - Coggle Diagram
Probability & Statistics 1
Exploring Data
Concepts
Mean (µ or x̄): sum of all values ÷ number of values
Median: middle value when data is ordered
Mode: most frequently occurring value
Range: largest – smallest
Interquartile range (IQR): Q₃ – Q₁
Variance (σ² or s²)
Standard deviation (σ or s): √variance
Outliers: values > Q₃ + 1.5×IQR or < Q₁ – 1.5×IQR
Formula
Ungrouped mean: x̄ = Σx / n
Grouped mean: x̄ = Σxf / Σf
Ungrouped variance: σ² = Σ(x – x̄)² / n = Σx²/n – x̄²
Grouped variance: σ² = Σ(x – x̄)²f / Σf = Σx²f/Σf – x̄²
Coding: E(aX+b) = aE(X)+b, Var(aX+b) = a²Var(X)
Representing and Interpreting Data
Concepts
Stem‑and‑leaf diagrams
Box‑and‑whisker plots (box plots)
Shows: min, Q₁, median, Q₃, max
Histograms
Area of bar ∝ frequency
Frequency density = frequency / class width (vertical axis)
Cumulative frequency curves
Used to find median, quartiles, percentiles graphically
Formula
Frequency density = frequency / class width
Common Mistakes
Histograms: labelling vertical axis as "frequency" instead of "frequency density"
Cumulative frequency: plotting at the upper class boundary, not the midpoint
Probability
Formula
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
P(A|B) = P(A ∩ B) / P(B) [provided P(B) > 0]
Independent: P(A ∩ B) = P(A)P(B)
Mutually exclusive: P(A ∩ B) = 0
Common Mistakes
Confusing mutually exclusive (cannot happen together) with independent (one doesn't affect the other)
reversing the order (P(A|B) vs P(B|A))
Forgetting to subtract intersection when using P(A∪B) formula
Discrete Random Variables
Concepts
Random variable: a variable whose value depends on the outcome of an experiment
Discrete vs continuous random variables
Probability distribution table
Condition for a valid distribution: ΣP(X=x) = 1
Formula
E(X) = Σ x p(x)
E(X²) = Σ x² p(x)
Var(X) = E(X²) – [E(X)]²
E(aX + b) = aE(X) + b
Var(aX + b) = a² Var(X)
Common Mistakes
Var(X) formula: using [E(X)]² instead of E(X)²
Expectation of a function: E(g(X)) = Σ g(x)·P(X=x), not g(E(X))
Permutations and Combinations
Concepts
Factorial notation: n! = n × (n–1) × … × 1
Permutations: order matters
Combinations: order does not matter
Problems with restrictions (e.g., "together", "apart", "at least")
Arrangements with repeated items (divide by factorials of repeats)
Formula
⁰⁰Pᵣ = n! / (n – r)! (permutations of r from n)
ⁿCᵣ = n! / [r!(n – r)!] (combinations of r from n)
Arrangements with repeats: n! / (a! b! c! …)
Discrete Probability Distributions
Concepts
Binomial distribution B(n, p)
Conditions: fixed n, independent trials, two outcomes, constant p
Geometric distribution Geo(p)
Conditions: independent trials, constant p, waiting for first success
Recognising which distribution to apply in context
Formula
Binomial: P(X = r) = ⁿCᵣ · pʳ · (1–p)ⁿ⁻ʳ
Binomial: E(X) = np, Var(X) = np(1–p)
Geometric: P(X = r) = (1–p)ʳ⁻¹ · p (r = 1, 2, 3, …)
Geometric: E(X) = 1/p (Var(X) = (1–p)/p²
Common Mistakes
Choosing binomial when the number of trials is not fixed (should be geometric, or vice versa)
The Normal Distribution
Concepts
Continuous distribution with mean μ and variance σ²
Standard normal distribution: Z ~ N(0, 1)
Using normal tables (Φ(z)) to find probabilities
Formula
Standardisation: Z = (X – μ) / σ
Φ(z) = P(Z ≤ z) (given in MF19 tables)
P(X < a) = Φ((a – μ) / σ)
Symmetry: Φ(–z) = 1 – Φ(z)
Common Mistakes
Normal approximation to binomial: incorrectly applying the ±0.5 continuity correction (use X ≤ k → Z ≤ (k+0.5–μ)/σ; X ≥ k → Z ≥ (k–0.5–μ)/σ)
Table reading: using wrong tail (P(Z > z) = 1 – Φ(z))