2. Normal Distribution Links Standard Deviation The Normal Distribution Finding a Probability Standard Normal Distribution Inverse Normal Distribution

3. 4 7 14 12 8 25 1 9 4 Standard Deviation 1 st slide Given a Data Set 12, 8, 7, 14, 4 The standard deviation is a measure of the mean spread of the data from the mean. Mean = (12 + 8 + 7 + 14 + 4) ÷ 5 = 9 Calculate the mean -2 3 5 -5 How far is each data value from the mean? Square to remove the negatives Average = Sum divided by how many values Square root to ‘undo’ the squared (25 + 4 + 25 + 1 + 9) ÷ 5 = 12.8 Square root 12.8 = 3.58 Std Dev = 3.58 Calculator function

4. The Normal Distribution 1 st slide Key Concepts Total Area = 1 Distributions with different spreads have different STANDARD DEVIATIONS Area under the graph is the relative frequency = the probability The MEAN is in the middle. The distribution is symmetrical. A lower mean A higher mean A smaller Std Dev. A larger Std Dev. 1 Std Dev either side of mean = 68% 2 Std Dev either side of mean = 95% 3 Std Dev either side of mean = 99%

5. Finding a Probability 1 st slide Draw a distribution graph The mean weight of a chicken is 3 kg (with a standard deviation of 0.4 kg) Look up 2.5 Std Dev in tables (z = 2.5) How many Std Dev from the mean? Find the probability a chicken is less than 4kg 3kg 4kg 3kg 4kg 1 distance from mean standard deviation = = 2.5 1 0.4 3kg 4kg 0.50.4938 Probability = 0.5 + 0.4938 (table value) = 0.9938 So 99.38% of chickens in the population weigh less than 4kg

6. Standard Normal Distribution 1 st slide Draw a distribution graph The mean weight of a chicken is 2.6 kg (with a standard deviation of 0.3 kg) Look up z = 1.333 Std Dev in tables Change the distribution to a Standard Normal Find the probability a chicken is less than 3kg 2.6kg 3kg 0 z P(x < 3kg) Aim: Correct Working = = 1.333 0.4 0.3 distance from mean standard deviation z = = P(z < 1.333) = 0.5 + 0.4087 = 0.9087 = 1.333 The Question: Table value 0.5 Z = ‘the number of standard deviations from the mean’

7. Inverse Normal Distribution 1 st slide Draw a distribution graph The mean weight of a chicken is 2.6 kg (with a standard deviation of 0.3 kg) Look up the probability in the middle of the tables to find the closest ‘z’ value. 90% of chickens weigh less than what weight? (Find ‘x’) Corresponding ‘z’ value is: 1.281 Look up 0.400 z = 1.281 The closest probability is 0.3999 0 0.4 0.5 Z = ‘the number of standard deviations from the mean’ 2.6kg ‘x’ kg Area = 0.9 2.6kg 2.98 kg z = 1.281 The distance from the mean = ‘Z’ × Std Dev D = 1.281 × 0.3 D x = 2.6kg + 0.3843 = 2.9843kg