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

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Presentation transcript:

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

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 = ( ) ÷ 5 = 9 Calculate the mean 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 ( ) ÷ 5 = 12.8 Square root 12.8 = 3.58 Std Dev = 3.58 Calculator function

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%

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 = = kg 4kg Probability = (table value) = So 99.38% of chickens in the population weigh less than 4kg

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 = 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 = = distance from mean standard deviation z = = P(z < 1.333) = = = The Question: Table value 0.5 Z = ‘the number of standard deviations from the mean’

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: Look up z = The closest probability is Z = ‘the number of standard deviations from the mean’ 2.6kg ‘x’ kg Area = kg 2.98 kg z = The distance from the mean = ‘Z’ × Std Dev D = × 0.3 D x = 2.6kg = kg