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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 Std Dev = 3.58 4 -2 25 5 1 8 Calculator -1 function 4 5 6 7 8 9 3 9 10 11 12 13 14 x xx x x 2 x x 2 n 14 12 How far is each data value from the mean? 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 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 A smaller Std Dev. x 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 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 4kg x 3kg Draw a distribution graph 1 How many Std Dev from the mean? distance from mean standard deviation x 1 = = 2.5 0.4 Look up 2.5 Std Dev in tables (z = 2.5) 4kg 3kg 0.5 Probability = 0.5 + 0.4938 (table value) = 0.9938 So 99.38% of chickens in the population weigh less than 4kg 0.4938 x 3kg 4kg Standard Normal Distribution 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 3kg 2.6kg x Draw a distribution graph Table value 0.5 Change the distribution to a Standard Normal z= distance from mean standard deviation z 0.4 = = 1.333 0.3 x Look up z = 1.333 Std Dev in tables Z = ‘the number of standard deviations from the mean’ 0 z = 1.333 Aim: Correct Working The Question: P(x < 3kg) = P(z < 1.333) = 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) Area = 0.9 90% of chickens weigh less than what weight? (Find ‘x’) ‘x’ kg 2.6kg x 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 D = 1.281 × 0.3 The distance from the mean = ‘Z’ × Std Dev x = 2.6kg + 0.3843 = 2.9843kg z = 1.281 D 2.98 kg 2.6kg x

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