# 3.3.3 Normal Distribution - Calculations

## Normal Distribution - Calculations

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

## 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 P(a < X < b )
• 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 standardise all normal distributions to  and use the table of the normal distribution to find the probabilities
• It is likely that your calculator has a function that can find normal probabilities, if so it is a good idea to learn to use it so that you can check your probabilities
• However you must show your calculations to get the z values and use the tables to get all the marks

#### How do I calculate the probability for a normal distribution?

• A random variable   can be coded to model the standard normal distribution  using the formula

• You can calculate a probability  using the relationship
• Always sketch a quick diagram to visualise which area you are looking for
• Once you have determined the z value use the table of the normal distribution to find the probability
• Refer to your sketch to decide if you need to subtract the probability from one

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
• You can look at which side of the mean x is on and the direction of the inequality to decide if your answer should be greater or less than 0.5
• As you can use:

#### Worked example

The random variable . Calculate:

(a)
,

(b)
(a)
,

(b)

## Inverse Normal Distribution

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

• Given a probability you will have to look through the table of the normal distribution to locate the z-value that corresponds with that probability
• Look at whether your probability is greater or less than 0.5 and the direction of the inequality to determine whether your z-value will be positive or negative
• If  is more than 0.5 or is less than 0.5 then a should be bigger than the mean
• z will be positive
• If is less than 0.5 or  is more than 0.5 then a  should be smaller than the mean
• z will be negative
• You do not need to remember these, a sketch will help you see it
• Always sketch a diagram

• If your probability is less than 0.5 you will need to subtract it from one to find the corresponding z value
• Remember that the position of the z-value will not change, only the direction of the inequality
• Once you have the correct value substitute it into the formula   and solve to find the value of a
• Always check that your answer makes sense by considering where a is in relation to the mean

#### Given the value of P(µ- a < X < µ + a) I find the value of a  ?

• A sketch making use of the symmetry of the graph is essential
• If you are given   then  will be
• This is easier to see from a sketch than to remember
• You can then look through the tables for the corresponding z-value and substitute into the formula

#### Worked example

The random variable

Find the value of  such that

#### Exam Tip

• The most common mistake students make when finding values from given probabilities is forgetting to check whether the z-value should be negative or not.  Avoid this by checking early on using a sketch whether z is positive or negative and writing a note to yourself before starting the other calculations.

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### Author:Amber

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.