Normal Distribution

Standard Deviation Calculate the mean Given a Data Set 12, 8, 7, 14, 4 = 9 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 25 1 9 4 Std Dev = 3.58 4 7 14 12 8 -2 -1 3 5 -5 Average = Sum divided by how many values Calculator function 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. 1 Std Dev either side of mean = 68% A lower mean A higher mean 2 Std Dev either side of mean = 95% 3 Std Dev either side of mean = 99% A smaller Std Dev. A larger Std Dev. Distributions with different spreads have different STANDARD DEVIATIONS

Find the probability a chicken is less than 4kg 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 distance from mean standard deviation = = 2.5 1 0.4 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

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 Table value 0.5 Change the distribution to a Standard Normal distance from mean standard deviation z = = = 1.333 0.4 0.3 z = 1.333 Aim: Correct Working The Question: P(x < 3kg) = 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

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