Finding the Parameters of a Normal Distribution (Cambridge (CIE) A Level Maths): Revision Note

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  is unknown then you will need to use the standard normal distribution

• You will need to use the formula

  • or its rearranged form 

• You will be given a probability for a specific value of  

• 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 

  • Use the table of the Normal Distribution to find the value of z such that  or 

  • Make sure the direction of the inequality for   is consistent with

  • The table gives the -value to three 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 3 decimal places throughout your calculations so that your final answer can be rounded to 3 significant figures

• 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 (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

• 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  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