6.2 Normal Distribution. Normal Distributions

Transcription

1 6.2 Normal Distribution Normal Distributions 1

2 Homework Read Sec 6-1, and 6-2. Make sure you have a good feel for the normal curve. Do discussion question p302 2

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4 Objective Identify Complete normal model using empirical rule. 4

5 Normal Distribution The normal distribution is a probability distribution. The normal probability distribution requires Continuous Variables Many naturally occurring variables are approximately normally distributed. The normal distribution requires an infinite number of samples. As the sampling increases in size, a frequency distribution approaches the normal distribution, also called a Gaussian distribution, or what uninformed persons call the bell curve. 5

6 Symmetric The normal distribution is unimodal and symmetric about the mean. Remember? Negatively skewed distributions are clustered toward the high end of the sample, and the median is nearly always above (greater than) the mean. Positively skewed distributions are clustered toward the low end of the sample, and the median is nearly always below (less than) the mean. Negatively (Left) Skewed Symmetric Positively (Right) Skewed Mean < Median Mean = Median Mean > Median 6

7 Equation The normal distribution curve is a function of two values µ and σ with two constant irrational numbers e and π. e (x µ) f (x ) = e 2σ 2 π µ = population mean σ 2π σ = population standard deviation The important information here is that the curve (the shape) is determined by the mean and standard deviation of the population data. 7

8 Normal Distribution Each normally distributed variable has its own normal distribution curve, determined by the values of that variable s mean and standard deviation. Equal means but different standard deviations σ < σ 8

9 Normal Distribution Different means but equal standard deviations σ = σ μ < μ 9

10 Normal Distribution Different means and different standard deviations σ < σ μ < μ 10

11 Applet 11

12 Area The area under the normal curve is determined using calculus and you are taking this class to avoid calculus so we will not go into the derivation. What you need to know is the total area under the curve is 1. More importantly for us, the areas between differing standard deviations can be determined. An approximation is known as the empirical rule. We met the empirical rule a few sections ago. 12

13 Empirical Rule About 99.7% of observations (data values) will fall within 3σ of the mean. -1σ to 1σ About 68% of observations (data values) will fall within 1σ of the mean. -2σ to 2σ About 95% of observations (data values) will fall within 2σ of the mean. -3σ to 3σ 13

14 Empirical Rule 2.14% (about 2.35%) of observations will fall between 2σ and 3σ, or between -2σ and -3σ. µ to 1σ % (about 34%) of observations will fall between the mean and +1σ or between µ and -1σ. 1σ to 2σ 13.59% (about 13.5%) of observations will fall between +1σ and +2σ or between -1σ to -2σ. 2σ to 3σ 14

15 Normal Distribution 99.7% 95% % 0.1 3σ 2σ 1σ μ 1σ 2σ 3σ 15

16 Normal Distribution 49.85% % 47.5% % 34% 34% 3σ 2σ 1σ μ 1σ 2σ 3σ 16

17 Normal Distribution 50% of the data falls above the mean, 50% of the data falls below the mean. Symmetric about the mean (mean = median = mode) Continuous (no gaps) - the data are continuous 50% Asymptotic to the x-axis (independent axis) Total area under the curve = 1 3σ 2σ 1σ μ 1σ 2σ 3σ Empirical Rule (68%, 95%, 99.7%) 17