Working with Normal Distributions

Measurements which occur in nature frequently have a normal distribution eg weight of new born babies Height of I year old apple trees Circumference of pine saplings Hand span of Y12 students Time to skip 100m This continuous data fits a bell-shaped curve

The mean, µ, is always in the middle The area under the curve represents probability

For all normal distributions you need to know two parameters: µ = mean measures the centre σ = standard deviation measures the spread i.e. how far each value is from the mean

An example of normal distribution: X = amount of milk in a 2L bottle Data collected might give: µ = 2005 ml σ = 10 ml

What are the differences between the two distributions below? A has the: Larger mean Smaller standard deviation a b

Each situation will have a different normal curve because their mean and sd will vary However, we can standardise (or transform) every normal distribution into a standard normal distribution by using a formula.

The Standard Normal Distribution This is a special normal distribution which always has: µ = 0 σ = 1 µ=o

We can compare any normal distribution to the standard normal distribution by using the formula: X = normal random variable µ = any value σ = any value Z = standard normal random variable µ = 0 σ = 1

Calculating Probabilities for Standard Normal Distributions Since probability = 1, area under the curve = 1 By symmetry RHS = LHS =0.5 µ = 0 σ = 1

Example: Find the P(Z < ) z= µ =0 Step 1: Draw a diagram Step 2: Use GC

Graphics Calculator Upper limit = Lower limit = -∞ = µ =0 σ = 1 Stats Mode Dist = F5 Norm = F1 Ncd = normal distribution probability z= P(Z < ) = 0.052

Examples of Calculating Probabilities for Standard Normal Distributions: Find the probabilities that: a)P(Z > 1.683) b) P(Z < 2.445) c) P( < Z < 2.039) Answers: a) b) c)0.941