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IBMaths: AA HLDPRevision Notes4. Statistics & Probability4.6 Normal Distribution4.6.3 Standardisation of Normal Variables

Standardisation of Normal Variables (DP IB Maths: AA HL)

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Standard Normal Distribution

What is the standard normal distribution?

  • The standard normal distribution is a normal distribution where the mean is 0 and the standard deviation is 1
    • It is denoted by Z
    • Z tilde text N end text left parenthesis 0 comma space 1 squared right parenthesis

Why is the standard normal distribution important?

  • Any normal distribution curve can be transformed to the standard normal distribution curve by a horizontal translation and a horizontal stretch
  • Therefore we have the relationship:
    • Z equals fraction numerator X minus mu over denominator sigma end fraction 
    • Where X tilde text N end text left parenthesis mu comma space sigma squared right parenthesis and Z tilde text N end text left parenthesis 0 comma space 1 squared right parenthesis
  • Probabilities are related by:
    • straight P left parenthesis a less than X less than b right parenthesis equals straight P stretchy left parenthesis fraction numerator a minus mu over denominator sigma end fraction less than Z less than fraction numerator b minus mu over denominator sigma end fraction stretchy right parenthesis 
    • This will be useful when the mean or variance is unknown
  • Some mathematicians use the function straight capital phi left parenthesis z right parenthesis to represent straight P left parenthesis Z less than z right parenthesis

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z-values

What are z-values (standardised values)?

  • For a normal distribution X tilde straight N left parenthesis mu comma space sigma squared right parenthesisthe z-value (standardised value) of an x-value tells you how many standard deviations it is away from the mean
    • If z = 1 then that means the x-value is 1 standard deviation bigger than the mean
    • If z = -1 then that means the x-value is 1 standard deviation smaller than the mean
  • If the x-value is more than the mean then its corresponding z-value will be positive
  • If the x-value is less than the mean then its corresponding z-value will be negative
  • The z-value can be calculated using the formula:
    • z equals fraction numerator x minus mu over denominator sigma end fraction 
    • This is given in the formula booklet
  • z-values can be used to compare values from different distributions

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Finding Sigma and Mu

How do I find the mean (μ) or the standard deviation (σ) if one of them is unknown? 

  • If the mean or standard deviation of X tilde straight N left parenthesis mu comma space sigma squared right parenthesis is unknown then you will need to use the standard normal distribution
  • You will need to use the formula
    • z equals fraction numerator x minus mu over denominator sigma end fraction or its rearranged form x equals mu plus sigma z
  • You will be given a probability for a specific value of
    • straight P left parenthesis X less than x right parenthesis equals p or straight P left parenthesis X greater than x right parenthesis equals p
  • To find the unknown parameter:
  • STEP 1: Sketch the normal curve
    • Label the known value and the mean
  • STEP 2: Find the z-value for the given value of x  
    • Use the Inverse Normal Distribution to find the value of z such that straight P left parenthesis Z less than z right parenthesis equals p or straight P left parenthesis Z greater than z right parenthesis equals p
    • Make sure the direction of the inequality for Z is consistent with the inequality for X
    • Try to use lots of decimal places for the z-value or store your answer to avoid rounding errors
      • You should use at least one extra decimal place within your working than your intended degree of accuracy for your answer
  • STEP 3: Substitute the known values into z equals fraction numerator x minus mu over denominator sigma end fraction or x equals mu plus sigma z
    • You will be given and one of the parameters (μ or σ) in the question
    • You will have calculated z in STEP 2
  • STEP 4: Solve the equation

How do I find the mean (μ) and the standard deviation (σ) if both of them are unknown? 

  • If both of them are unknown then you will be given two probabilities for two specific values of x
  • The process is the same as above
    • You will now be able to calculate two z -values
    • You can form two equations (rearranging to the form x equals mu plus sigma z is helpful)
    • You now have to solve the two equations simultaneously (you can use your calculator to do this)
    • Be careful not to mix up which z-value goes with which value of x

Worked example

It is known that the times, in minutes, taken by students at a school to eat their lunch can be modelled using a normal distribution with mean μ minutes and standard deviation σ minutes.

Given that 10% of students at the school take less than 12 minutes to eat their lunch and 5% of the students take more than 40 minutes to eat their lunch, find the mean and standard deviation of the time taken by the students at the school.

4-6-3-ib-aa-sl-finding-mu-sigma-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.