Working with Normal Distributions

1. Working with Normal Distributions

2. 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

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

4. 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

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

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

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

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

9. 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

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

11. z=-1.625 Step 1: Draw a diagram Example: Find the P(Z < -1.625) Step 2: Use GC µ=0

12. z=-1.625 Graphics Calculator Stats Mode Dist = F5 Norm = F1 Ncd = normal distribution probability • Upper limit = -1.625 • Lower limit = -∞ = -99999 • µ =0 • σ = 1 P(Z < -1.625) = 0.052

13. Examples of Calculating Probabilities for Standard Normal Distributions: Find the probabilities that: • P(Z > 1.683) b) P(Z < 2.445) c) P(-1.774 < Z < 2.039) Answers: a) 0.046 b) 0.993 c)0.941