The Normal Distribution (CIE A Level Maths: Probability & Statistics 1)

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
  • 

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