Calculating Probabilities using Normal Distribution (AQA Level 3 Mathematical Studies (Core Maths))

Calculating Probabilities using Normal Distribution

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

• When working with a normal distribution, you will be expected to find the probabilities by either using:

  • the distribution function on your calculator

  • or the statistical tables that are provided to you in the exam

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 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 '𝜎' value - this is the standard deviation

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

  • Remember it is asking for the standard deviation

  • If you have the variance make sure that you 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 on your calculator

• 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 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

How can I use the statistical tables to calculate probabilities?

• Sometimes you may be required to use statistical tables provided to you in the exam to calculate probabilities

  • Use Table 1: Normal distribution function

• To calculate probabilities using this table you must first find the appropriate z-value using the formula

• To find , trace along the relevant row and column to find the given probability

  • E.g. , find the value where the row 0.3 and the column 0.02 meet

• To find , use the identity

  • I.e. First find the probability that the random variable is less than the given value from the table, then subtract it from

• To find , use the identity

  • I.e. Find the probabilities that the random variable is less than both given values, then subtract the lower one from the higher one

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 you will need to enter:

  • The 'area' - this is the value

    • Some calculators might ask for the 'tail' - this is the left tail as you know the area to the left of

  • The 'μ' value - this is the mean

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

• Always check your answer makes sense

  • If  is less than 0.5 then should be smaller than the mean

  • If is more than 0.5 then should be bigger than the mean

  • A sketch will help you see this

• If you are required to use the statistical tables to find a value of for , use Table 2: Percentage points of the normal distribution

  • E.g. To find , find the row 0.6 and the column 0.03, the value that corresponds to both is 0.3319

• Remember that values found from the table are z-values from the standardised normal distribution

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

• Given , use to rewrite this as

• Then use the method for to find

  • You can use your calculator or the statistical tables

• 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 ''