Edexcel International A Level Maths: Statistics 1

Revision Notes

3.2.4 Finding Sigma and Mu

Test Yourself

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 the X tilde N left parenthesis mu comma 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 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 
  • 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 table of the Normal Distribution to find the value of z such that P left parenthesis Z less than z right parenthesis equals p or P left parenthesis Z greater than z right parenthesis equals p
    • Make sure the direction of the inequality for Z  is consistent with X
    • The table gives the z-value to four decimal places to avoid rounding errors
      • Use the sketch to help you decide whether your z value is positive or negative
      • You should use the 4 decimal places throughout your calculations so that your final answer can be rounded to 3 significant figures
  • STEP 3: Substitute the known values into z equals fraction numerator x minus mu over denominator sigma end fraction or bold italic x bold equals bold italic mu bold plus bold italic sigma bold italic z
    • 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 (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 
    • 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 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.



Exam 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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Author: Dan

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.