Standard Normal Distribution (CIE A Level Maths: Probability & Statistics 1)

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
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Why is the standard normal distribution important?

• Calculating probabilities for the normal distribution can be difficult and lengthy due to its complicated probability density function
• The probabilities for the standard normal distribution have been calculated and laid out in the table of the normal distribution which can be found in your formula booklet
  • Nowadays, many calculators can calculate probabilities for any normal distribution, if yours does it is a good idea to learn how to use it to check your answers but you must still use the tables of the normal distribution and show all your working clearly
• It is possible to map any normal distribution onto the standard normal distribution curve
• Mapping different normal distributions to the standard normal distribution allows distributions with different means and standard deviations to be compared with each other

How is any normal distribution mapped to the standard normal distribution?

• Any normal distribution curve can be transformed to the standard normal distribution curve by a horizontal translation and a horizontal stretch
  • Therefore, for  and , we have the relationship:

• Probabilities are related by:
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  • This is a very useful relationship for calculating probabilities for any normal distribution
  • As it is a normal distribution  so you do not need to worry about whether the inequality is strict (< or >) or weak (≤ or ≥)
• A value of z = 1 corresponds with the x-value that is 1 standard deviation above the mean and a value of z = -1 corresponds with the x-value that is 1 standard deviation below the mean
• If a value of x is less than the mean then the z -value will be negative
• The function is used to represent 

How is the table of the normal distribution function used?

• In your formula booklet you have the table of the normal distribution which provides probabilities for the standard normal distribution
  • The probabilities are provided for 
  • To find other probabilities you should use the symmetry property of the normal distribution curve
• The table gives probabilities for values of z between 0 and 3
  • For negative values of z, the symmetry property of the normal distribution is used
  • For values greater than z = 3 the probabilities are small enough to be considered negligible
• The tables give the probabilities to 4 decimal places
• To read probabilities from the normal distribution table for a z value of up to 2 decimal places:
  • The very first column lists all z values to 1 decimal place from z = 0.0 to z = 2.9
  • The top row gives the second decimal place for each of these z values
  • So the value of  would be found at the point where the ‘1.2’ row meets the ‘3’ column
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• To read probabilities from the normal distribution table for a z value of 3 decimal places:
  • There is an extra section to the right of the tables that gives the amount to add on to the probabilities for the third decimal place
  • The values given in the columns represent one ten-thousandth
    • If the value is 7 we add 0.0007 to the probability
    • If the value is 23 we add 0.0023 to the probability
  • To find the value of  we would need to find the amount to add on to 0.8907
  • Find the point where the 1.2 row meets the ADD 4 column, this gives us the number 7
  • Add the value 0.0007 to the probability for 
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How is the table used to find probabilities that are not listed?

• The property that the area under the graph is 1 allows probabilities to be found for P( Z > z)
  • Use the formula 
• The symmetrical property of the normal distribution gives the following results:
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• This allows probabilities to be found for negative values of z or for 
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  • Therefore:
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• The four cases in terms of  are:
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• Drawing a sketch of the normal distribution will help find equivalent probabilities

How are z values found from the table of the normal distribution function?

• To find the value of z for which look for the value of p from within the table and find the corresponding value of z
  • If the probability is given to 4 decimal places most of the time the value will exist somewhere in the tables
  • Occasionally you may have to use the ADD columns to find the exact value
  • If the values in the ADD columns don’t exactly match up use the closest value or find the midpoint of the z values that are either side of the probability
• If your probability is 0.5 or greater look through the tables to find the corresponding z value
  • For   use the z value found in the table
  • For take the negative of the z value found in the table
• If the probability is less than 0.5 you will need to subtract it from one before using the tables to find the corresponding z value
  • For   take the negative of the z value found in the table
  • For  use the z value found in the table
• Always draw a sketch so that you can see these clearly
• The formula booklet also contains a table of the critical values of z
  • This gives z values to 3 decimal places for common probabilities
  • The probabilities in this table are 0.75, 0.9, 0.95, 0.975, 0.99, 0.995, 0.9975, 0.999 and 0.9995