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Normal Distribution

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Edexcel International A Level

Mathematics

Statistics 1

Normal Distribution

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Table of Contents

1. The Normal Distribution

Continuous Random Variables

Definition: A continuous random variable (CRV) is a random variable that can take any value within a range of infinite values.
Key Characteristics
  • Can take any value within a range (not just isolated values)
  • Usually measure something (height, weight, time, temperature)
  • Probability of any single value = 0: P(X = k) = 0
  • We talk about probability of being within a range

Continuous Probability Distributions

Key Concept: For a continuous distribution:
  • We use a probability density function f(x)
  • Area under the curve between x = a and x = b equals P(a ≤ X ≤ b)
  • Total area under the curve = 1
  • P(X ≤ k) = P(X < k) for any value k (strict vs weak inequalities don't matter)

What is a Normal Distribution?

Definition: A normal distribution is a continuous probability distribution that is:
  • Symmetrical (mirror image on both sides)
  • Bell-shaped (highest at the center, tails extend infinitely)
Notation: If X follows a normal distribution: X ~ N(μ, σ²)
  • μ = mean (center of distribution)
  • σ² = variance
  • σ = standard deviation (square root of variance)

Important Properties of Normal Distribution

Property Description
Mean = Median = Mode All three measures of center equal μ
Symmetrical about x = μ Mirror image on both sides of the mean
Points of inflection At x = μ ± σ (one standard deviation from mean)
68-95-99.7 Rule 68% within μ ± σ, 95% within μ ± 2σ, 99.7% within μ ± 3σ

Effect of Mean and Variance

Changing Parameters:
  • If μ changes → curve translates horizontally
  • If σ changes → curve stretches/compresses horizontally
  • Small variance → tall, narrow curve
  • Large variance → short, wide curve

Modeling with Normal Distribution

When can you use normal distribution?
  • Population must be large enough
  • Data must be symmetrical
  • Data must have one mode
When NOT to use normal distribution:
  • Multiple modes or no mode (e.g., random number generator)
  • Not symmetrical (e.g., human lifespan - skewed left)
Examples
✓ CAN Model ✗ CANNOT Model
Heights of adults Lifetimes (skewed)
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Also in the full note
  • 2. Standard Normal Distribution
  • 3. Using Normal Distribution Tables
  • 4. Calculating Normal Probabilities
  • 5. Inverse Normal Distribution
  • 6. Finding Distribution Parameters (μ and σ)
  • 7. Key Facts Summary
  • 8. Pre-Exam Preparation
  • What is the Standard Normal Distribution?
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More Statistics 1 topics

Statistical Measures
Basic Statistical Measures
Discrete Random Variables
Discrete Probability Distributions
Calculating Probabilities & Events
Basic Probability
Conditional Probability
Frequency Tables
Working with Data
Correlation & Regression