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Shapes of Distributions The Normal Distribution The three histograms shown below represent the marks scored by students sitting 3 different tests. A Left-Skewed B C Symmetrical Right-Skewed A symmetrical distribution occurs often and is known as the NORMAL distribution. It is characterized by the bell-shaped curve that can be drawn through the top of each bar. Which of the distribution below are approximately normal, right skewed, left skewed, or neither? A B C Right Skewed D E F Neither Normal G H Neither Neither Normal Left Skewed I Right Skewed Normal The Normal Distribution In a normal distribution most of the data is grouped about the mean and the rest of the data tails off symmetrically on either side. Approximately 68% of the data lies within one standard deviation of the mean and 95% of the data within two standard deviations of the mean. The Normal Distribution In a normal distribution most of the data is grouped about the mean and the rest of the data tails off symmetrically on either side. Approximately 68% of the data lies within one standard deviation of the mean and 95% of the data within two standard deviations of the mean. 68% x -1σ x x x +1σ The Normal Distribution In a normal distribution most of the data is grouped about the mean and the rest of the data tails off symmetrically on either side. Approximately 68% of the data lies within one standard deviation of the mean and 95% of the data within two standard deviations of the mean. 68% x -1σ x x +1σ 95% x -2σ x x +2σ The Normal Distribution As an example consider a test in which the results are normally distributed. If the mean is 20 and the standard deviation is 3 then roughly: are between 17 and 23. 68% of the marks 11 14 17 20 23 26 are between 14 and 26. 95% of the marks 11 29 14 17 20 23 26 29 The Normal Distribution Example 1 The results of a test are normally distributed with a mean score of 56 and a standard deviation of 4. (a) What percentage of students should score between 52 and 60 marks? 68% Ans: 68% (b) What percentage of students should score between 48 and 64 marks? Ans: 95% (c) What percentage of students should score over 64 . Ans: 2.5% x -1σ x x +1σ Questions 95% x-2σ x x +2σ The Normal Distribution Example 2 3000 bars of chocolate have a mean weight of 120 g and a s.d. of 3 g. Assuming that they are normally distributed: (a) How many bars would you expect to weigh between 117 g and 123 g? Ans: 2040 (b) How many bars would you expect to weigh between 114 g and 126 g? Ans: 2850 (c) How many bars would you expect to weigh less than 114 g? Ans: 75 68% x -1σ x x +1σ 95% x-2σ x x +2σ The Normal Distribution Example 3 In a sample of 2000 adult males, the mean height was found to be 175 cm and the s.d. was 5 cm. Assuming a normal distribution: (a) How many adults would you expect to be between 170 cm and 180 cm? 68% x -1σ x x +1σ Ans: 1360 (b) Find the heights between which the central 95% of the distribution should lie. 95% Ans: 165 – 185 cm (c) How many adults would you expect to be taller than 185 cm? Ans: 50 x-2σ x x +2σ The Normal Distribution Example 4 The time taken for Jenny to walk to school is normally distributed with a mean time of 18 minutes and a s.d. of 4 minutes. Calculate: (a) The number of days last academic year (192 days) on which you would expect Jenny to take between 10 and 26 minutes to arrive at school. Ans: 182 (b) The number of days on which you would expect her to take less than 14 minutes. 68% x -1σ x x +1σ 95% Ans: 31 x-2σ x x +2σ The Normal Distribution Example 5 The mean lifetime of a particular type of battery is 8½ hours with a s.d. of 30 minutes. In a batch of 5 000 batteries and assuming the lifetimes are normally distributed : (a) How many of the batteries would you expect to last for between 8 and 9 hours? Ans: 3400 (b) How many of the batteries would you expect to last for between 7½ and 9½ hours? 68% x -1σ x x +1σ 95% Ans: 4750 x-2σ x x +2σ Let’s add some more information… Example Question 1 Worksheet 1 The results of a test are normally distributed with a mean score of 56 and a standard deviation of 4. Find: (a) What percentage of students score between 52 and 60 marks? (b) What percentage of students score between 48 and 64 marks? (c) What percentage of students score over 64 . Example Question 2 3000 bars of chocolate have a weight of 120 g and a s.d. of 3 g. Assuming that they are normally distributed: (a) How many bars would you expect to weigh between 117 g and 123 g? (b) How many bars would you expect to weigh between 114 g and 126 g? (c) How many bars would you expect to weigh less than 114 g? Question 1 Worksheet 2 In a sample of 2000 adult males, the mean height was found to be 175 cm and the s.d. was 5 cm. Assuming a normal distribution: (a) How many adults would you expect to be between 170 cm and 180 cm? (b) Find the heights between which the central 95% of the distribution should lie. (c) How many adults would you expect to be taller than 185 cm? Question 2 The time taken for Jenny to walk to school is normally distributed with a mean time of 18 minutes and a s.d. of 4 minutes. Calculate: (a) The number of days last academic year (192 days) on which you would expect Jenny to take between 10 and 26 minutes to arrive at school. (b) The number of days on which you would expect her to take less than 14 minutes. Question 3 The mean lifetime of a particular type of battery is 8½ hours with a s.d. of 30 minutes. In a batch of 5 000 batteries and assuming the lifetimes are normally distributed : (a) How many of the batteries would you expect to last for between 8 and 9 hours? (b) How many of the batteries would you expect to last for between 7½ and 9½ hours?