Calculations with Normal Distributions (AQA A Level Maths): Revision Note

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

Calculating Normal Probabilities

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 so you do not need to worry about whether the inequality is strict (< or >) or weak (≤ or ≥)

• The equation of a normal distribution curve is complicated so the area must be calculated numerically

• You will be expected to use distribution functions on your calculator 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

• P(X = x ) = 0

• Your calculator 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 calculator 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 '' 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 square root 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

• P(X > a) 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 10⁹⁹)

• Similarly P(X < b) 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 -10⁹⁹)

• This works because the probability that X is more than 3 standard deviations bigger than the mean is less than 0.0015

  • This is the same for being 3 standard deviations less than the mean

  • This reduces to less than 0.000032 when using 4 standard deviations

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 calculator will have a function called "Inverse Normal Distribution"

  • Some calculators call this InvN

• Given that P(X < a) = p  you will need to enter:

  • The 'area' - this is the value p

    • Some calculators might ask for the 'tail' - this 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

• Always check your answer makes sense

  • If P(X < a)  is less than 0.5 then a should be smaller than the mean

  • If P(X < a) is more than 0.5 then a should be bigger than the mean

  • A sketch will help you see this

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

• Given P(X > a) = p

• Use P(X < a) = 1 - P(X > a)  to rewrite this as P(X < a) = 1 - p

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

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

  • Selecting 'right' for the tail

  • Entering the area as 'p'