Calculations with Normal Distribution (DP IB Applications & Interpretation (AI)): Revision Note

Calculating normal probabilities

Throughout this section we will use the random variable . For X distributed normally, X can take any real number value. Therefore any values mentioned in this section will be assumed to be real numbers.

How do I find probabilities using a normal distribution?

• The area under a normal curve between the points  and  is equal to the probability 

  • Remember for a normal distribution you do not need to worry about whether the inequality is strict (< or >) or weak (≤ or ≥)

    • 

• You will be expected to use distribution functions on your GDC to find the probabilities when working with a normal distribution

How do I calculate P(X = x): the probability of a single value for a normal distribution? 

• The probability of a single value is always zero for a normal distribution

  • You can picture this as the area of a single line is zero

• 

• Your GDC is likely to have a "Normal Probability Density" function

  • This is sometimes shortened to NPD, Normal PD or Normal Pdf

  • IGNORE THIS FUNCTION for this course!

  • This calculates the probability density function at a point NOT the probability

How do I calculate P(a < X < b): the probability of a range of values for a normal distribution? 

• You need a GDC that can calculate cumulative normal probabilities

• You want to use the "Normal Cumulative Distribution" function

  • This is sometimes shortened to NCD, Normal CD or Normal Cdf

• You will need to enter:

  • The 'lower bound' - this is the value a

  • The 'upper bound' - this is the value b

  • The 'μ' value - this is the mean

  • The 'σ' value - this is the standard deviation

• Check the order carefully as some calculators ask for standard deviation before mean

  • Remember it is the standard deviation

    • so if you have the variance then take the square root of it

• Always sketch a quick diagram to visualise which area you are looking for

How do I calculate P(X > a) or P(X < b) for a normal distribution? 

• You will still use the "Normal Cumulative Distribution" function

• can be estimated using an upper bound that is sufficiently bigger than the mean

  • Using a value that is more than 4 standard deviations bigger than the mean is quite accurate

  • Or an easier option is just to input lots of 9's for the upper bound (99999999...) or to use another 'very big number' like 10⁹⁹

• can be estimated using a lower bound that is sufficiently smaller than the mean

  • Using a value that is more than 4 standard deviations smaller than the mean is quite accurate

  • Or an easier option is just to input lots of 9's for the lower bound with a negative sign (-99999999...), or to use another 'very big negative number' like -10⁹⁹

Are there any useful identities?

• 

• As  you can use:

  • 

  • 

  • 

• These are useful when:

  • The mean and/or standard deviation are unknown

  • You only have a diagram

  • You are working with the inverse distribution

Inverse normal distribution

Given the value of P(X < a) how do I find the value of a? 

• Your GDC will have a function called "Inverse Normal Distribution"

  • Some calculators call this InvN

• Given that  you will need to enter:

  • The 'area' - this is the value p

    • Some calculators might ask for the 'tail'

    • is the left tail as you know the area to the left of a

  • The 'μ' value - this is the mean

  • The 'σ' value - this is the standard deviation

Given the value of P(X > a)  how do I find the value of a? 

• If your calculator does have the tail option (left, right or centre) then you can use the "Inverse Normal Distribution" function straightaway by:

  • Entering the area as 'p'

  • Selecting 'right' for the tail

    • is the right tail as you know the area to the right of a

• If your calculator does not have the tail option (left, right or centre) then:

  • Given 

  • Use to rewrite this as

    • 

  • Then use the method for P(X < a) to find a