Lecture Notes: A-Level Statistics - Mr. Bison's Stats Series Part 3

Introduction

• Welcome to the third part of the A-Level maths series focused on statistics by Mr. Bison.
• Previous parts: Part 1 (Pure Maths), Part 2 (Mechanics).
• Importance: Students achieve varying marks—those who memorize formulas often perform better.

Data Collection

Key Terms:

• Census: Measures every member of a population.
  • Advantage: Accurate results.
  • Disadvantage: Expensive, time-consuming, may destroy testing units (e.g., avocados).
• Sampling Units: Individuals or items being sampled.
• Sampling Frame: List of sampling units.

Types of Sampling:

1. *Random Sampling:
   • *Simple Random Sampling:
     • Equal chance for all units.
     • Methods: Random number generator, lottery sampling.
     • Advantage: Bias-free.
     • Disadvantage: Requires a sampling frame.
   • Systematic Sampling:
     • Select every kth unit (formula: k = population size/sample size).
     • Must start with a random number between 1 and k.
     • Advantage: Quick.
     • Disadvantage: Requires a sampling frame.
   • Stratified Sampling:
     • Sample reflects groups (strata) of a population.
     • Method: Proportional sampling within strata.
     • Advantage: Reflects population accurately.
     • Disadvantage: Strata must be clearly defined.

Non-Random Sampling:

1. Quota Sampling:
   • Stratified sampling but not random.
   • Advantage: No need for a sampling frame.
   • Disadvantage: Potential bias.
2. Opportunity Sampling:
   • Selecting individuals available at the time (e.g., outside supermarkets).
   • Advantage: Easy, cheap.
   • Disadvantage: Unlikely to be representative.

Types of Data:

• Qualitative: Non-numerical (e.g., color).
• Quantitative: Numerical data.
  • Discrete: Fixed values.
  • Continuous: Any value within a range.

Large Data Set

Key Points:

• UK Weather Stations: Camborne, Heathrow, Hearn, Leeming, Leuchars.
  • Coastal stations: Windy, rainy.
  • Southern stations: Warmer, more sunshine.
• Recorded data: May to October (1987, 2015).
• International Stations: Perth (Australia), Beijing (China), Jacksonville (Florida, USA).
  • Notable events: Hurricanes in Jacksonville, great storm in the UK (Oct 1987).
• Data Types:
  • TR: Less than 0.05mm rain. Treated as zero.
  • NA: Reading not available, problematic for sampling.
  • Cloud cover: Measured in oktas (discrete, 0-8).
  • Max gust: Strongest wind speed in knots. 1 knot = 1.15 mph.

Measures of Central Tendency and Spread

Calculation Formulas:

• Mean (x̄): Σx/N or Σfx/Σf for grouped data.
• Quartiles & Percentiles:
  • Lower quartile (Q1): n/4
  • Median (Q2): n/2
  • Upper quartile (Q3): 3n/4
• Linear Interpolation: Required for grouped data.
• Variance: Mean of squares minus square of the means.
  • For groups: Σfx²/Σf - (Σfx/Σf)²
  • Standard Deviation: Square root of variance.
• Coding:
  • y = ax + b
  • Mean (ỹ) = a(x̄) + b
  • Standard Deviation (σy) = a(σx)

Data Representation

Diagrams:

• Cumulative Frequency and Box Plots: Used to visualize data distribution.
• Histograms: Continuous data representation.
  • Frequency density = Frequency / Class width.
• Comparison: One measure of location & one measure of spread.

Regression and Correlation

Key Concepts:

• Product Moment Correlation Coefficient (PMCC): Measures strength and direction (-1 to 1).
• Regression Line (y = a + bx): Line of best fit.
  • a: y-intercept (value of y when x=0).
  • b: Slope (change in y per unit change in x).
• Interpolation vs. Extrapolation:
  • Interpolation is reliable; estimating within range.
  • Extrapolation is less reliable; estimating outside range.

Probability

Venn Diagrams:

• Union (A ∪ B): Shade all parts.
• Intersection (A ∩ B): Shade overlap. Tree Diagrams:
• Multiply probabilities along branches.

Key Formulas:

• Mutual Exclusivity: P(A ∩ B) = 0
  • P(A ∪ B) = P(A) + P(B)
• Independence:
  • P(A ∩ B) = P(A) * P(B)
  • P(A|B) = P(A)
• Conditional Probability:
  • P(B|A) = P(A ∩ B) / P(A)
• Addition Law: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Discrete and Continuous Distributions

Discrete Uniform Distribution:

• Equal probability for all outcomes.

Binomial Distribution:

• X ~ B(n, p) (number of trials = n, probability = p).
• Conditions:
  • Fixed number of trials.
  • Fixed probability of success.
  • Trials are independent.
  • Two outcomes (success/failure).
• Cumulative Probability: Use tables or calculators.
  • P(X ≤ k) / P(X ≥ k)
  • P(a ≤ X ≤ b)

Normal Distribution:

• Y ~ N(μ, σ^2) (mean = μ, variance = σ^2).
• Standard Normal Distribution (Z):
  • Z ~ N(0, 1)
  • Z = (Y - μ) / σ
• Approximation of Binomial: If n is large and p ≈ 0.5.
  • Mean (μ) = np
  • Std. Dev (σ) = √(np(1 - p))
• Continuity Corrections: Discrete to continuous adjustments.
  • E.g., P(X > 5) becomes P(Y > 4.5)

Hypothesis Testing

Definitions:

• Null Hypothesis (H₀): Assumed true.
• Alternative Hypothesis (H₁): True if H₀ is incorrect.
• Significance Level (α): Threshold for rejecting H₀.

Testing Types:

• One-tailed: H₁ states > or <
• Two-tailed: H₁ states ≠
  • Split significance level.

Correlation Testing:

• Null: No correlation (ρ = 0).
• Alternative: Positive/Negative/Some correlation.
• Compare r with critical value.

Binomial Testing:

• H₀: p = k
• H₁: p > k or p < k (one-tailed); p ≠ k (two-tailed)
• Calculate probability, compare with α.

Normal Testing of Sample Mean:

• Sample mean (x̄) ~ Normal(μ, σ²/n)
• H₀: μ = k
• H₁: μ > k or μ < k
• Probability comparison with α.

Conclusion

• Comprehensive memorization aids in high performance.
• Explore other resources on the Bison Maths channel for more insights.