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

Throughout this section we will use the random variable X tilde straight N left parenthesis mu comma sigma squared right parenthesis. For a normal distribution, X can take any real number. Therefore any values mentioned in this section will be assumed to be real numbers.

Did this video help you?

Calculating Normal Probabilities

How do I find probabilities using a normal distribution?

  • The area under a normal curve between the points begin mathsize 16px style x equals a end style and begin mathsize 16px style x equals b end style is equal to the probability begin mathsize 16px style P left parenthesis a less than X less than b right parenthesis end style

    • 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 'begin mathsize 16px style sigma end style' 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?

  • straight P left parenthesis X less than mu right parenthesis space equals space straight P left parenthesis X greater than mu right parenthesis equals 0.5

  • As P left parenthesis X equals a right parenthesis equals 0 you can use:

    • straight P left parenthesis X less than a right parenthesis plus straight P left parenthesis X greater than a right parenthesis space equals 1

    • straight P left parenthesis X greater than a right parenthesis equals 1 minus straight P left parenthesis X less than a right parenthesis space

    • straight P left parenthesis a less than X less than b right parenthesis space equals P left parenthesis X less than b right parenthesis minus P left parenthesis X less than a right parenthesis

  • These are useful when:

    • The mean and/or standard deviation are unknown

    • You only have a diagram

    • You are working with the inverse distribution

Worked Example

The random variable Y tilde straight N left parenthesis 20 comma 5 squared right parenthesis. Calculate:

(a) straight P left parenthesis Y space equals space 20 right parenthesis,

(b) straight P left parenthesis 18 less or equal than Y less than 27 right parenthesis,

(c) straight P left parenthesis Y greater than 29 right parenthesis.

4-3-2-calculating-normal-probabilities-we-solution-part-1
4-3-2-calculating-normal-probabilities-we-solution-part-2
4-3-2-calculating-normal-probabilities-we-solution-part-3

Did this video help you?

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'

Worked Example

The random variable  W space tilde straight N left parenthesis 50 comma space 36 right parenthesis.

Find the value of w such that  straight P left parenthesis W space greater than space w right parenthesis equals 0.175 .

4-3-2-inverse-normal-distribution-we-solution

Examiner Tips and Tricks

Always ask yourself two questions when using your calculator:

  • Have you entered the mean and the standard deviation in the correct order?

  • Have you entered the standard deviation correctly and not the variance?

👀 You've read 1 of your 5 free revision notes this week
An illustration of students holding their exam resultsUnlock more revision notes. It's free!

By signing up you agree to our Terms and Privacy Policy.

Already have an account? Log in

Did this page help you?

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

Download notes on Calculations with Normal Distributions