Edexcel IAL Math

Statistics 1

Learning Notes | 60 min | AI powered

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1. Mathematical Models in Probability and Statistics

1.1 Statistical Modeling

• Purpose:
  

• To represent real-world situations mathematically
  
• To make predictions based on data
  
• To test hypotheses about populations
  

• Process:
  

• Identify the problem or question
  
• Collect relevant data
  
• Choose an appropriate statistical model
  
• Analyze the data using the model
  
• Interpret results and draw conclusions
  
• Refine the model if necessary
  

• Examples:
  

• Probability distributions for random phenomena
  
• Regression models for relationships between variables
  
• Hypothesis tests for making decisions based on data
  

2. Representation and Summary of Data

2.1 Graphical Representations

• Histograms:
  

• Used for continuous data
  
• Rectangles represent frequency within class intervals
  
• Area of each rectangle proportional to frequency
  
• Vertical axis can show frequency, relative frequency, or frequency density
  

• Stem and Leaf Diagrams:
  

• Displays both the distribution shape and the actual data values
  
• Stem: Leading digit(s)
  
• Leaf: Final digit
  
• Back-to-back diagrams compare two distributions
  

• Box Plots:
  

• Box shows interquartile range (IQR) with median marked
  
• Whiskers extend to minimum and maximum values, or to 1.5×IQR
  
• Shows median, quartiles, range, and potential outliers
  
• Multiple box plots allow for easy comparison of distributions
  

2.2 Measures of Location

• Mean:
  

• Sum of values divided by number of values
  
• x̄ = Σx/n for ungrouped data
  
• x̄ = Σfx/Σf for grouped data
  
• Influenced by extreme values (not resistant)
  

• Median:
  

• Middle value when data is arranged in order
  
• For n values: (n+1)/2 position if n is odd; average of n/2 and (n/2)+1 positions if n is even
  
• More resistant to outliers than the mean
  

• Mode:
  

• Most frequently occurring value(s)
  
• Can have multiple modes (bimodal, multimodal)
  
• For grouped data, modal class has highest frequency
  

• Data Coding:
  

• Transformation of data to simplify calculations
  
• y = (x - a)/b where a and b are constants
  
• Effect on mean: ȳ = (x̄ - a)/b
  
• No effect on shape of distribution
  

2.3 Measures of Dispersion

• Range:
  

• Difference between maximum and minimum values
  
• Simple but affected by outliers
  

• Interquartile Range (IQR):
  

• Difference between upper and lower quartiles: Q₃ - Q₁
  
• Covers the middle 50% of data
  
• More resistant to outliers than range
  

• Variance:
  

• Average of squared deviations from the mean
  
• σ² = Σ(x - x̄)²/n for a population
  
• s² = Σ(x - x̄)²/(n-1) for a sample
  
• Units are square of original data units
  

• Standard Deviation:
  

• Square root of variance
  
• σ or s
  
• Same units as original data
  
• Measures average distance from the mean
  

• Interpercentile Ranges:
  

• Difference between specific percentiles
  
• Examples: 10-90 percentile range, 5-95 percentile range
  
• Used to exclude extreme values
  

2.4 Skewness and Outliers

• Skewness:
  

• Measure of asymmetry in a distribution
  
• Positive (right) skew: Mean > Median, long tail to the right
  
• Negative (left) skew: Mean < Median, long tail to the left
  
• Symmetric: Mean ≈ Median
  

• Outliers:
  

• Values that differ significantly from other observations
  
• Can be identified using box plots
  
• Common rule: Value is an outlier if it's more than 1.5×IQR below Q₁ or above Q₃
  
• Should be investigated to determine if they represent errors or important phenomena
  

3. Probability

3.1 Elementary Probability

• Basic Definition:
  

• Probability of event A = P(A) = Number of favorable outcomes / Total number of possible outcomes
  
• 0 ≤ P(A) ≤ 1
  
• P(certain event) = 1
  
• P(impossible event) = 0
  

• Rules:
  

• P(not A) = 1 - P(A)
  
• P(A or B) = P(A) + P(B) - P(A and B)
  
• For mutually exclusive events: P(A or B) = P(A) + P(B)
  

3.2 Sample Space and Events

• Sample Space:
  

• Set of all possible outcomes of an experiment
  
• Denoted by S or Ω
  

• Event:
  

• A subset of the sample space
  
• Collection of outcomes
  

• Types of Events:
  

• Exclusive (mutually exclusive): Cannot occur simultaneously (A ∩ B = ∅)
  
• Complementary: A' or Ā contains all outcomes not in A, and A ∪ A' = S
  
• Conditional: Probability of an event given that another event has occurred
  

• Conditional Probability:
  

• P(A|B) = P(A ∩ B) / P(B)
  
• Probability of A given that B has occurred
  

3.3 Independence

• Independent Events:
  

• Occurrence of one event does not affect the probability of the other
  
• P(A|B) = P(A) and P(B|A) = P(B)
  
• P(A ∩ B) = P(A) × P(B)
  

• Testing Independence:
  

• Check if P(A ∩ B) = P(A) × P(B)
  
• Check if P(A|B) = P(A)
  

• Applications:
  

• Multiple trials of the same experiment
  
• Events from different experiments
  
• Selecting items with replacement
  

3.4 Probability Laws

• Sum Law:
  

• P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
  
• For n events: P(A₁ ∪ A₂ ∪ ... ∪ Aₙ) = ΣP(Aᵢ) - ΣP(Aᵢ ∩ Aⱼ) + ΣP(Aᵢ ∩ Aⱼ ∩ Aₖ) - ... + (-1)ⁿ⁺¹P(A₁ ∩ A₂ ∩ ... ∩ Aₙ)
  

• Product Law:
  

• P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)
  
• For n events: P(A₁ ∩ A₂ ∩ ... ∩ Aₙ) = P(A₁) × P(A₂|A₁) × P(A₃|A₁ ∩ A₂) × ... × P(Aₙ|A₁ ∩ A₂ ∩ ... ∩ Aₙ₋₁)
  

• Tools:
  

• Tree diagrams for sequential events
  
• Venn diagrams for set relationships
  
• Tables for two-way classifications
  

• Sampling:
  

• With replacement: Independent selections
  
• Without replacement: Dependent selections
  
• Probability changes after each selection if without replacement
  

4. Correlation and Regression

4.1 Scatter Diagrams and Linear Regression

• Scatter Diagram:
  

• Graphical representation of bivariate data
  
• Each point represents a pair of values (x, y)
  
• Reveals patterns, trends, and possible relationships
  

• Linear Relationship:
  

• When points approximately follow a straight line
  
• Positive: y tends to increase as x increases
  
• Negative: y tends to decrease as x increases
  
• Strength indicated by how closely points cluster around the line
  

• Regression Line:
  

• Line of best fit through data points
  
• Minimizes the sum of squared vertical distances from points to line
  
• Equation: y = a + bx
  
• Least squares method gives b = Σ[(x - x̄)(y - ȳ)] / Σ[(x - x̄)²] and a = ȳ - bx̄
  

4.2 Explanatory and Response Variables

• Definitions:
  

• Explanatory (independent) variable: Typically plotted on x-axis
  
• Response (dependent) variable: Typically plotted on y-axis
  
• Relationship: Changes in explanatory variable may cause or be associated with changes in response variable
  

• Applications:
  

• Prediction: Using the regression line to estimate y for a given x value
  
• Interpolation: Predictions within the range of observed x values
  
• Extrapolation: Predictions outside the range of observed x values (potentially unreliable)
  

• Variable Transformation:
  

• Linear change of variable: y = ax + b
  
• Logarithmic transformation: ln(y) = a + b ln(x) for power relationships
  
• Used to linearize non-linear relationships
  

4.3 Correlation Coefficient

• Product Moment Correlation Coefficient:
  

• Measures strength and direction of linear relationship
  
• Formula: r = Σ[(x - x̄)(y - ȳ)] / √[Σ(x - x̄)² × Σ(y - ȳ)²]
  
• Range: -1 ≤ r ≤ 1
  

• Interpretation:
  

• r ≈ 1: Strong positive correlation
  
• r ≈ -1: Strong negative correlation
  
• r ≈ 0: Little or no linear correlation
  
• |r| > 0.8: Strong correlation
  
• 0.5 < |r| < 0.8: Moderate correlation
  
• |r| < 0.5: Weak correlation
  

• Limitations:
  

• Only measures linear relationship
  
• Sensitive to outliers
  
• Correlation does not imply causation
  
• May hide non-linear relationships
  

5. Discrete Random Variables

5.1 Concept of a Discrete Random Variable

• Definition:
  

• A variable that can take discrete values with specific probabilities
  
• Each value has a non-zero probability
  
• Sum of all probabilities equals 1
  

• Examples:
  

• Number of heads in coin tosses
  
• Number of defective items in a sample
  
• Number of accidents per day
  

5.2 Probability and Distribution Functions

• Probability Function:
  

• p(x) = P(X = x)
  
• Gives probability for each possible value of X
  
• Properties:
  

• p(x) ≥ 0 for all x
  
• Σp(x) = 1 for all possible x values
  

• Cumulative Distribution Function (CDF):
  

• F(x) = P(X ≤ x) = Σp(t) for all t ≤ x
  
• Gives probability that X is less than or equal to x
  
• Properties:
  

• 0 ≤ F(x) ≤ 1
  
• F(x) is non-decreasing
  
• lim(x→-∞) F(x) = 0 and lim(x→∞) F(x) = 1
  

5.3 Mean and Variance

• Expected Value (Mean):
  

• E(X) = μ = Σ[x × p(x)] for all possible x values
  
• Represents the long-run average value
  

• Variance:
  

• Var(X) = σ² = E[(X - μ)²] = Σ[(x - μ)² × p(x)]
  
• Alternative formula: Var(X) = E(X²) - [E(X)]²
  
• Where E(X²) = Σ[x² × p(x)]
  

• Properties:
  

• For constant a and b:
  

• E(aX + b) = aE(X) + b
  
• Var(aX + b) = a²Var(X)
  

5.4 Discrete Uniform Distribution

• Definition:
  

• Equal probability for each possible value
  
• X ~ U(a, b) where a, b are integers and a < b
  

• Probability Function:
  

• p(x) = 1/(b - a + 1) for x = a, a+1, a+2, ..., b
  
• p(x) = 0 otherwise
  

• Mean and Variance:
  

• E(X) = (a + b)/2
  
• Var(X) = [(b - a + 1)² - 1]/12
  

• Examples:
  

• Fair die roll: X ~ U(1, 6)
  
• Random selection of a day in a week: X ~ U(1, 7)
  

6. The Normal Distribution

6.1 Properties of the Normal Distribution

• Definition:
  

• Continuous probability distribution
  
• Notation: X ~ N(μ, σ²)
  
• Parameters: Mean μ and variance σ²
  

• Characteristics:
  

• Bell-shaped and symmetric about the mean
  
• Mean = median = mode
  
• Asymptotic to the x-axis (never touches)
  
• Approximately 68% of data within μ ± σ
  
• Approximately 95% of data within μ ± 2σ
  
• Approximately 99.7% of data within μ ± 3σ
  

• Standard Normal Distribution:
  

• Z ~ N(0, 1)
  
• Transformation: Z = (X - μ)/σ
  
• Used with tables of the cumulative distribution function
  

6.2 Using Normal Distribution Tables

• Standardization:
  

• Convert X ~ N(μ, σ²) to Z ~ N(0, 1) using Z = (X - μ)/σ
  
• Find probabilities using standard normal tables
  

• Finding Probabilities:
  

• P(X ≤ a) = P(Z ≤ (a - μ)/σ)
  
• P(X > a) = 1 - P(X ≤ a)
  
• P(a < X ≤ b) = P(X ≤ b) - P(X ≤ a)
  

• Finding Values:
  

• For probability p, find z value where P(Z ≤ z) = p
  
• Convert back to x using x = μ + zσ
  

• Applications:
  

• Calculating probabilities for normal random variables
  
• Finding confidence intervals
  
• Hypothesis testing