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Normal Distribution

Links Standard Deviation The Normal Distribution Finding a Probability Standard Normal Distribution Inverse Normal Distribution. Normal Distribution. Mean = (12 + 8 + 7 + 14 + 4) ÷ 5 = 9. 25. 4. -5. 4. 7. -2. 25. 14. 5. 1. 8. Calculator function. -1. 9. 12. 3.

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Normal Distribution

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

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

  3. A lower mean A higher mean A smaller Std Dev. A larger Std Dev. 1st slide 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. 1 Std Dev either side of mean = 68% 2 Std Dev either side of mean = 95% 3 Std Dev either side of mean = 99% Distributions with different spreads have different STANDARD DEVIATIONS

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

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

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

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