The Normal Distribution (Edexcel A Level Maths): Revision Note

Properties of Normal Distribution

The binomial distribution is an example of a discrete probability distribution. The normal distribution is an example of a continuous probability distribution.

What is a continuous random variable?

• A continuous random variable (often abbreviated to CRV) is a random variable that can take any value within a range of infinite values

  • Continuous random variables usually measure something

  • For example, height, weight, time, etc

What is a continuous probability distribution?

• A continuous probability distribution is a probability distribution in which the random variable X is continuous

• The probability of X being a particular value is always zero

  • P(X = k) = 0 for any value k

  • Instead we define the probability density function f(x) for a specific value

  • We talk about the probability of being within a certain range

• A continuous probability distribution can be represented by a continuous graph (the values for X along the horizontal axis and probability density on the vertical axis)

• The area under the graph between the points x = a and x = b is equal to P(a ≤ X ≤ b)

  • The total area under the graph equals 1

• As P(X = k) = 0 for any value , it does not matter if we use strict or weak inequalities

  • P(X ≤ k) = P(X < k) for any value k

What is a normal distribution?

• A normal distribution is a continuous probability distribution

• The continuous random variable can follow a normal distribution if:

  • The distribution is symmetrical

  • The distribution is bell-shaped

• If X follows a normal distribution then it is denoted 

  • μ is the mean

  • σ² is the variance

  • σ  is the standard deviation

• If the mean changes then the graph is translated horizontally

• If the variance changes then the graph is stretched horizontally

  • A small variance leads to a tall curve with a narrow centre

  • A large variance leads to a short curve with a wide centre

What are the important properties of a normal distribution?

• The mean is μ

• The variance is σ²

  • If you need the standard deviation remember to square root this

• The normal distribution is symmetrical about x = μ

  • Mean = Median = Mode = μ

• The normal distribution curve has two points of inflection

  • x = μ ±  σ (one standard deviation away from the mean)

• There are the results:

  • Approximately two-thirds (68%) of the data lies within one standard deviation of the mean (μ ±  σ)

  • Approximately 95% of the data lies within two standard deviations of the mean (μ ± 2σ)

  • Nearly all of the data (99.7%) lies within three standard deviations of the mean (μ ± 3σ)

• For any value x a z-score (or z-value) can be calculated which measures how many standard deviations x is away from the mean

  • 

Modelling with Normal Distribution

What can be modelled using a normal distribution?

• A lot of real-life continuous variables can be modelled by a normal distribution provided that the population is large enough and that the variable is symmetrical with one mode

• For a normal distribution X can take any real value, however values far from the mean (more than 4 standard deviations away from the mean) have a probability density of practically zero

  • This fact allows us to model variables that are not defined for all real values such as height and weight

What can not be modelled using a normal distribution?

• Variables which have more than one mode or no mode

  • For example, the number given by a random number generator

• Variables which are not symmetrical

  • For example, how long a human lives for