The Normal Distribution (DP IB Maths: AI HL)

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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
    • straight P left parenthesis X equals k right parenthesis equals 0 for any value k
    • Instead we define the probability density function straight f left parenthesis x right parenthesis  for a specific value
      • This is a function that describes the relative likelihood that the random variable would be close to that value
    • We talk about the probability of X 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 equals a and x equals b is equal to straight P left parenthesis a less or equal than X less or equal than b right parenthesis
    • The total area under the graph equals 1
  • As straight P left parenthesis X equals k right parenthesis equals 0for any value k, it does not matter if we use strict or weak inequalities
    • straight P left parenthesis X less or equal than k right parenthesis equals straight P left parenthesis X less than k right parenthesisfor any value k when X is a continuous random variable

What is a normal distribution? 

  • A normal distribution is a continuous probability distribution
  • The continuous random variable X 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 X tilde straight N left parenthesis mu comma space sigma squared right parenthesis
    • 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 increases then the graph is widened horizontally and made shorter vertically to maintain the same area
    • A small variance leads to a tall curve with a narrow centre
    • A large variance leads to a short curve with a wide centre

4-3-1-the-normal-distribution-diagram-1

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 equals mu
    • Mean = Median = Mode = μ
  • 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)

4-3-1-the-normal-distribution-diagram-2

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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

Examiner Tip

  • An exam question might involve different types of distributions so make it clear which distribution is being used for each variable

Worked example

The random variable S represents the speeds (mph) of a certain species of cheetahs when they run. The variable is modelled using straight N left parenthesis 40 comma space 100 right parenthesis.

a)
Write down the mean and standard deviation of the running speeds of cheetahs.

4-6-1-ib-ai-aa-sl-modelling-normal-a-we-solution

b)
State two assumptions that have been made in order to use this model.

4-6-1-ib-ai-aa-sl-modelling-normal-b-we-solution

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.