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CIE A-Level Maths Study Notes

4.5.1 Fundamentals of the Normal Distribution

The normal distribution is a fundamental concept in statistics and a key model for continuous random variables. Its importance stems from its natural occurrence in many real-world phenomena and its central role in the Central Limit Theorem. This theorem suggests that the distribution of sample means approximates a normal distribution, regardless of the population's original distribution, provided the sample size is large.

Understanding the Normal Distribution

  • Symmetry Around the Mean: The curve is symmetrical around the mean, where mean, median, and mode are equal.
  • Defined by Mean and Variance: The mean (μ) sets the center; variance (σ²) defines the spread.
  • Asymptotic: The tails extend indefinitely, indicating all values are possible but increasingly unlikely.
Standard normal distribution

Image courtesy of scribbr

Standard Normal Distribution

Centralized around zero, standard deviation of one.

Applications

  • Measurement Errors: In scientific experiments.
  • Biological Attributes: Like population heights.
  • Financial Models: Like stock market returns.

Using Normal Distribution Table

  • Standardization: Convert to Z-score using Z=XμσZ = \frac{X - μ}{σ}.
  • Lookup: Find Z-score in the table for the probability.
Z-score tableZ-score table

Image courtesy of byjus

Sketching Normal Curves

  • Draw Axis: Mark variable range.
  • Indicate Mean: Center curve at the mean.
  • Shape Curve: Symmetrical, bell-shaped.

Example: Widget Weights

  • Given: Mean = 100 grams, Standard Deviation = 15 grams.
  • Find: Probability of > 120 grams.
  • Solution: Z=12010015=1.33Z = \frac{120 - 100}{15} = 1.33.
  • Probability: 9.12≈ 9.12%.
Probability Graph

Example: Test Scores

  • Given: Mean = 70, Standard Deviation = 8.
  • Find: Probability of < 60.
  • Solution: Z=60708=1.25Z = \frac{60 - 70}{8} = -1.25.
  • Probability: 10.56≈ 10.56%.
Probability Graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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