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Normal Distribution Links Standard Deviation The Normal Distribution Finding a Probability Standard Normal Distribution Inverse Normal Distribution Standard Deviation Calculate the mean Mean = (12 + 8 + 7 + 14 + 4) ÷ 5 x =9 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. (25 + 4 + 25 + 1 + 9) ÷ 5 = 12.8 Square root 12.8 = 3.58 25 -5 7 4 n Calculator 8 function 4 5 6 7 Std Dev = 3.58 4 -2 25 5 1 -1 8 9 3 12 How far is each data value from the mean? xx x x 2 x x 2 n 14 9 10 11 12 13 14 x 1st slide Square to remove the negatives Average = Sum divided by how many values x x 2 n Square root to ‘undo’ the squared The Normal Distribution Key Concepts 1st slide Area under the graph is the relative frequency = the probability Total Area = 1 The MEAN is in the middle. The distribution is symmetrical. x A lower mean 1 Std Dev either side of mean = 68% A higher mean x x A smaller Std Dev. 2 Std Dev either side of mean = 95% 3 Std Dev either side of mean = 99% A larger Std Dev. Distributions with different spreads have different STANDARD DEVIATIONS Finding a Probability 1st slide The mean weight of a chicken is 3 kg (with a standard deviation of 0.4 kg) Find the probability a chicken is less than 4kg x 4kg 3kg Draw a distribution graph 1 How many Std Dev from the mean? x 1 distance from mean = = 2.5 0.4 standard deviation Look up 2.5 Std Dev in tables (z = 2.5) Probability = 0.5 + 0.4938 (table value) = 0.9938 4kg 3kg 0.5 0.4938 x 3kg So 99.38% of chickens in the population weigh less than 4kg 4kg Standard Normal Distribution 1st slide The mean weight of a chicken is 2.6 kg (with a standard deviation of 0.3 kg) Find the probability a chicken is less than 3kg x 3kg 2.6kg Draw a distribution graph Table value 0.5 Change the distribution to a Standard Normal z= distance from mean 0.4 = = 1.333 standard deviation 0.3 x x z 0 z = 1.333 Aim: Correct Working The Question: P(x < 3kg) = P(z < 1.333) Look up z = 1.333 Std Dev in tables Z = ‘the number of standard deviations from the mean’ = 0.5 + 0.4087 = 0.9087 Inverse Normal Distribution The mean weight of a chicken is 2.6 kg (with a standard deviation of 0.3 kg) 1st slide Area = 0.9 90% of chickens weigh less than what weight? (Find ‘x’) x ‘x’ kg 2.6kg Draw a distribution graph Look up the probability in the middle of the tables to find the closest ‘z’ value. 0.5 0.4 Z = ‘the number of standard deviations from the mean’ 0 The closest probability is 0.3999 Look up 0.400 Corresponding ‘z’ value is: 1.281 z = 1.281 The distance from the mean = ‘Z’ × Std Dev D = 1.281 × 0.3 z = 1.281 D x 2.98 kg x = 2.6kg + 0.3843 = 2.9843kg 2.6kg