Edexcel A Level Maths: Statistics:复习笔记4.3.3 Standard Normal Distribution

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

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:
    • Where and
  • Probabilities are related by:
    • 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   to represent

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
    • You are given the value of to 4 decimal places when p  is:
      • 0.5, 0.4, 0.3, 0.2, 0.15, 0.1, 0.05, 0.025, 0.01, 0.005, 0.001, 0.005
  • 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

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 or
  • 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 or
    • 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 (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  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

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.

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.