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§ 16.1 - 16.2 Approximately Normal Distributions; Normal Curves Approximately Normal Distributions of Data Suppose the following is a bar graph for the height distribution of 205 randomly chosen men. Approximately Normal Distributions of Data Notice that the graph is roughly ‘Bell-Shpaed’ Approximately Normal Distributions of Data Now look at the case with a sample size of 968 men: 70 60 50 40 30 20 10 87 84 81 78 75 72 69 66 63 60 57 54 51 48 0 Approximately Normal Distributions of Data Here the ‘Bell’ behaviour is more apparent: 70 60 50 40 30 20 10 87 84 81 78 75 72 69 66 63 60 57 54 51 48 0 Approximately Normal Distributions of Data Data that is distributed like the last two examples is said to be in an approximately normal distribution. If the ‘bell-shape’ in question were perfect then the data would be said to be a normal distribution. The bell-shaped curves are called normal curves. Normal Distributions Normal curves are all bell-shaped. However they can look different from one another: Normal Distributions: Properties Symmetry: Every normal curve is symmetric about a vertical axis. This axis is the line x = where is the mean/average of the data. Mean = Median Normal Distributions: Properties Symmetry: Every normal curve is Left-Half Right-Half symmetric about a vertical axis. 50% of data data is the This axis is the line 50% x= of where mean/average of the data. Mean = Median = mean = median Normal Distributions: Properties Standard Deviation: The data’s standard deviation, , is the distance between the curve’s points of inflection and the mean. (Inflection points are where a curve changes from ‘opening-up’ to ‘openingdown’ and vice-versa.) Normal Distributions: Properties Standard Deviation: The data’s standard deviation, , is thePoints distance between the curve’s points ofofinflection Inflection and the mean. (Inflection points are where a curve changes from ‘opening-up’ to ‘openingdown’ and vice-versa.) - + Normal Distributions: Properties Quartiles: The first and third quartiles for a normally distributed data set can be estimated by Q3 ≈ + (0.675) Q1 ≈ - (0.675) Normal Distributions: Properties Quartiles: The first and third quartiles for a normally distributed data set can be estimated by 50% Q3 ≈ + (0.675) 25% Q1 ≈ - (0.675) Q1 Q3 25% Example: Find the mean, median, standard deviation and the first and third quartiles. Point of Inflection 43 50 Example: Find the mean, median, standard deviation and the first and third quartiles. Points of Inflection 36 39 Example: Find the mean, median and standard deviation. 25% 64.6125 73.875 § 16.4 The 68-95-99.7 Rule The 68-95-99.7 Rule (For normal distributions) 1) (Roughly) 68% of all data is within one standard deviation of the mean, . (I.e. - 68% of the data lies between - and + ) The 68-95-99.7 Rule (For normal distributions) 1) (Roughly) 68% of all data is within one standard deviation of the mean, . (I.e. - 68% of the data lies between 68% - and + ) of Data 16% of Data 16% of Data - + The 68-95-99.7 Rule (For normal distributions) 1) 68% of all data is within one standard deviation of the mean, . 2) 95% of data is within two standard deviations of the mean. (I.e. - between - and + ) The 68-95-99.7 Rule (For normal distributions) 1) (Roughly) 68% of all data is within one standard deviation of the mean, . 95% 2) 95% of data is within two standard of deviations of the mean. Data (I.e. - between 2.5% of Data 2.5% of Data - 2 + 2 The 68-95-99.7 Rule (For normal distributions) 1) 68% of all data is within one standard deviation of the mean, . 2) 95% of data is within two standard deviations of the mean. 3) 99.7% of data is within three standard deviations of the mean. The 68-95-99.7 Rule (For normal distributions) 1) 68% of all data is within one standard deviation of the mean, . 2) 95% of data is within two deviations of the mean. 3) 0.15% of 99.7% Data 99.7% standard of Data of data is within three standard 0.15% deviations of the mean. of Data - 3 + 3 The 68-95-99.7 Rule (For normal distributions) 4) The range of the data R is estimated by R ≈ 6 Example: Find the mean, median, standard deviation and the first and third quartiles. 68% 36 52 Example: Find the standard deviation and the first and third quartiles. 84% 6.22 10.35 Example: Find the mean and standard deviation. 2.5% 25 125