A LevelMaths: StatisticsOCRRevision Notes4. Statistical Distributions4.3 Normal Distribution (A Level only)4.3.3 Standard Normal Distribution

Standard Normal Distribution (OCR A Level Maths: Statistics)

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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
    • begin mathsize 16px style Z tilde straight N left parenthesis space 0 comma space 1 squared right parenthesis end style

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:
    • begin mathsize 16px style Z equals fraction numerator X minus mu over denominator sigma end fraction end style
    • Where begin mathsize 16px style X tilde N left parenthesis mu comma sigma squared right parenthesis end style and begin mathsize 16px style Z tilde straight N left parenthesis space 0 comma space 1 squared right parenthesis end style
  • Probabilities are related by:
    • begin mathsize 16px style straight P left parenthesis X less than a right parenthesis equals straight P open parentheses Z less than fraction numerator a minus mu over denominator sigma end fraction close parentheses end style 
    • This will be useful when the mean or variance is unknown
  • If a value of x is less than the mean then the z-value will be negative
  • Some mathematicians use the function begin mathsize 16px style straight capital phi left parenthesis straight z right parenthesis end style to represent begin mathsize 16px style straight P left parenthesis Z less than z right parenthesis end style

The table of percentage points of the normal distribution

  • In your formula booklet you have the table of percentage points which provides information about specific values of the standard normal distribution that correspond to commonly used probabilities
    • begin mathsize 16px style straight P left parenthesis Z less or equal than z right parenthesis equals p end style
    • You are given the value of z to 3 decimal places when p  is:
      • 0.75, 0.9, 0.95, 0.975, 0.99, 0.995, 0.9975, 0.999, 0.9995
  • These values of z can be found using the "Inverse Normal Distribution" function on your calculator
    • If you are happy using your calculator then you can simply ignore this table
  • They are simply listed in your formula booklet as they are commonly used when:
    • Finding an unknown mean and/or variance for a normal distribution
    • Performing a hypothesis test on the mean of a normal distribution

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Finding Sigma and Mu

How do I find the mean (begin mathsize 26px style straight mu end style) or the standard deviation (begin mathsize 22px style sigma end style) if one of them is unknown?

  • If the mean or standard deviation of the begin mathsize 16px style X tilde N left parenthesis mu comma sigma squared right parenthesis end style is unknown then you will need to use the standard normal distribution
  • You will need to use the formula
    • begin mathsize 16px style z equals fraction numerator x minus mu over denominator sigma end fraction end style or its rearranged form begin mathsize 16px style x equals mu plus sigma z end style
  • You will be given a probability for a specific value of begin mathsize 16px style x left parenthesis P left parenthesis X less than x right parenthesis equals p space or space P left parenthesis X greater than x right parenthesis equals p right parenthesis end style 
  • 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 begin mathsize 16px style P left parenthesis Z less than z right parenthesis equals p end style or begin mathsize 16px style P left parenthesis Z greater than z right parenthesis equals p end style
    • Make sure the direction of the inequality for Z  is consistent with X
    • Try to use lots of decimal places for the z-value 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 or 
    • You will be given x 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 (begin mathsize 22px style straight mu end style) and the standard deviation (begin mathsize 22px style sigma end style) 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 size 16px x size 16px equals size 16px mu size 16px plus size 16px sigma size 16px 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 begin mathsize 16px style x end style

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 standard deviation 4 minutes.

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

4-3-3-standard-normal-distribution-we-solution

Examiner Tip

  • These questions are normally given in context so make sure you identify the key words in the question. Check whether your z-values are positive or negative and be careful with signs when rearranging.

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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.