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The Normal Distribution C A If the three histograms shown below represent the marks scored by students sitting 3 different tests, comment briefly on the difficulty level of each test. Left-SkewedRight-Skewed Symmetrical Easy Difficult In test A most students scored high marks. In test B the marks are evenly distributed. B Normal In histogram B, most students found the test neither easy nor difficult. This type of 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. In test C most students scored Low marks.

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Frequency Marks 102030405060 5 10 15 20 70 80 Mean Test Scores Mode Median The Normal Distribution A Left-skewed distribution is said to be positively skewed. As a rule of thumb the mode and median will lie somewhere to the left of the mean. Left-skewed distribution

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Frequency Marks 102030405060 5 10 15 20 70 80 Mean Mode Median The Normal Distribution A Right-skewed distribution is said to be negatively skewed. As a rule of thumb the mode and median will lie somewhere to the right of the mean. Test Scores Right-skewed distribution

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Frequency Marks 102030405060 5 10 15 20 70 In a normal distribution the mean, mode and median all coincide through the centre of the distribution. The Normal Distribution Median Mean Mode Test Scores Normal distribution

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The Normal Distribution Which of the distribution below are approximately, normal, right or left skewed. A BC D EF G HI Right Skewed Left SkewedNormal Neither

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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. x

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x + 1 s.d x - 1 s.d 68% x x 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.

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The Normal Distribution x + 1 s.d x - 1 s.d x + 2 s.d x - 2 s.d 68% 95% x x 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.

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The Normal Distribution As an example consider the results of a test in which the results are normally distributed. If the mean is 20 and the standard deviation is 3 then roughly: 20 23 17 26 14 29 11 68% of the marks are between 17 and 23. 20 23 17 26 14 29 11 95% of the marks are between 14 and 26.

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Questions The Normal Distribution 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? (b) What percentage of students should score between 48 and 64 marks? (c) What percentage of students should score over 64. Example Question 1. Ans: 68% (since 52 and 60 are within 1 s.d. of the mean). Ans: 95% (Since 48 and 64 are within 2 s.d.s of the mean). Ans: 2½% (Since 5% of the marks are outside 2 s.d.s of the mean). x x + 2 s.d. x - 2 s.d. x x + 1 s.d. x - 1 s.d. 68% 95%

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The Normal Distribution 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? Example Question 2. Ans: 2040 (Since 117 and 123 are within 1 s.d. of the mean, we take 68% of 3000). Ans: 2850 (Since 114 and 126 are within 2 s.d.s of the mean, we take 95% of 3000). Ans: 75 (5% of the bars are outside 2 s.d.s of the mean, so we take half of 3000 - 2850). x x + 2 s.d. x - 2 s.d. x x + 1 s.d. x - 1 s.d. 68% 95%

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The Normal Distribution 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 1. Ans: 1360 (Since 170 and 180 are within 1 s.d. of the mean, we take 68% of 2000). Ans: 165 – 185 cm (Since this is within 2 s.d.s of the mean). Ans: 50 (5% of the heights are outside 2 s.d.s of the mean so we take 2½ % of 2000) x x + 2 s.d. x - 2 s.d. x x + 1 s.d. x - 1 s.d. 68% 95%

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The Normal Distribution 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 2. Ans: 182 (Since 10 and 26 are within 2 s.d. of the mean, we take 95% of 192). Ans: 31 (Since 14 and 22 are within 1 s.d. of the mean, we take ½ of 32% of 192). x x + 2 s.d. x - 2 s.d. x x + 1 s.d. x - 1 s.d. 68% 95%

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The Normal Distribution 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? Question 3. Ans: 3400 (Since 8 and 9 are within 1 s.d. of the mean, we take 68% of 5 000). Ans: 4750 (Since 7½ and 9½ are within 2 s.d. of the mean, we take 95% of 5 000). x x + 2 s.d. x - 2 s.d. x x + 1 s.d. x - 1 s.d. 68% 95%

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Worksheets Example Question 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? Worksheet 1

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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? Worksheet 2 Question 1 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?

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Chapter 2: The Normal Distributions

Chapter 2: The Normal Distributions

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