The Normal Distribution

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Understand the use of a normal distribution to model a continuous random variable, and use normal distribution tables. You may need to sketch normal curves to show distributions or probabilities.

  2. Solve problems concerning a variable X, where X ~ N(μ, σ²), including:

    • Finding the value of P(X > x₁), or a related probability, given the values of x₁, μ, and σ
    • Finding a relationship between x₁, μ, and σ given the value of P(X > x₁) or a related probability
    • For calculations involving standardisation, you must show full details of working (e.g., Z = (X - μ) / σ)

  3. Recall conditions under which the normal distribution can be used as an approximation to the binomial distribution, and use this approximation, with a continuity correction, in solving problems. The condition is that n is sufficiently large to ensure that both np > 5 and nq > 5.

1. What is a Normal Distribution?

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A normal distribution is a special pattern that appears in many real-life situations involving continuous data. Continuous data means measurements that can take any value within a range (like heights, weights, test scores, or temperatures).

When we collect large amounts of continuous data, they often form a bell-shaped curve when we draw them on a graph. This curve is called a normal curve or Gaussian curve.

Examples of data that often follow a normal distribution:

  • Heights of people in a population
  • Exam scores in a large class
  • Weights of apples from a tree
  • Errors in measurements
  • IQ scores

Key features of the normal distribution curve:

  • It is symmetrical (the left side is a mirror image of the right side)
  • It has a bell shape (highest in the middle, tails off on both sides)
  • The highest point of the curve is at the mean (average) value
  • The curve never actually touches the horizontal axis (it extends infinitely in both directions, though the tails get very close to zero)
  • The total area under the curve equals 1 (representing 100% probability)

2. The Normal Distribution: Notation and Parameters

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When we say a variable X follows a normal distribution, we write:

X ~ N(μ, σ²)

This means "X follows a normal distribution with mean μ and variance σ²"

Let's break down what each symbol means:

  • X = the variable we're measuring (like height, weight, or test score)
  • ~ = "follows" or "is distributed as"
  • N = Normal distribution
  • μ (the Greek letter "mu") = the mean (average value). This is the center of the distribution.
  • σ² (the Greek letter "sigma" squared) = the variance (a measure of spread)
  • σ (sigma) = the standard deviation (the square root of variance, also measures spread)

Important: The notation uses σ² (variance), but when we do calculations, we usually work with σ (standard deviation).

Understanding the parameters:

  1. The mean (μ) tells us where the center of the distribution is located. If μ increases, the whole curve shifts to the right. If μ decreases, it shifts to the left.

  2. The standard deviation (σ) tells us how spread out the data is:

    • Small σ → data clustered tightly around the mean → tall, narrow curve
    • Large σ → data spread widely → short, wide curve

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