1 / 6

Normal Distribution

This guide provides a comprehensive overview of standard deviation and the normal distribution, including how to calculate mean and standard deviation from a data set. It explains the properties of the normal distribution and how to find probabilities based on standardized values (z-scores). Examples illustrate how to determine the probability of certain outcomes, like the weight of chickens, using the standard normal distribution and inverse normal distribution techniques. Understand the implications of mean and standard deviation on data analysis.

karina-robertson
Télécharger la présentation

Normal Distribution

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related