# Normal curves properties of normal curves example

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Normal Curves Properties of Normal Curves : Example : Draw a Normal curve with a mean of 5, then draw another Normal curve with a mean of 10 and a standard deviation equal to that of the first curve. 25
Normal Curves Properties of Normal Curves : 4) The standard deviation controls the spread of the Normal curve. The larger is, the more spread out the σ curve is. 26
Normal Curves Properties of Normal Curves : 5) We can measure on a Normal curve by taking the distance from the mean to the “change of curvature points.” (Where the curve starts to flatten out) σ 27
28 Normal Curves Imagine that you are skiing down a mountain that has the shape of a Normal curve. At first, you descend at an ever-steeper angle as you go out from the peak:
Normal Curves Properties of Normal Curves : Example : Draw a Normal curve where =5 and =2. μ σ 29
Normal Distribution Definition : A Normal distribution is a distribution that is described by a Normal curve, and is described in full by its mean and variance ( and ). μ σ Remember that and alone do not specify the shape of μ σ most distributions, and that the shape of density curves in general does not reveal . These are special σ properties of Normal distributions . 30
Normal Distribution Why are Normal distributions important ? Three reasons: 1. Normal distributions are good descriptions for real data . Test scores 2. Normal distributions can approximate results of events of chance (such as those in Vegas). 3. Many effective tools we will learn in this course are based off of Normal distributions. 31
Normal distributions have a simple rule to follow the spread of the data without making complex area calculations: The 68-95-99.7 Rule (Also known as The Empirical Rule ) 32 68-95-99.7 Rule for Any Normal Curve 68% of the observations fall within one standard deviation of the mean 95% of the observations fall within two standard deviations of the mean 99.7% of the observations fall within three standard deviations of the mean
33 68-95-99.7 Rule for Any Normal Curve 68% + σ - σ µ +2 σ -2 σ 95% µ +3 σ -3 σ 99.7% µ
34 68-95-99.7 Rule for Any Normal Curve
35 68-95-99.7 Rule 68% [ μ ± σ ] 95% [ μ ± 2 σ ] 99.7% [ μ ± 3 σ ] 68-95-99.7 Rule for Any Normal Curve
36 Health and Nutrition Examination Study of 1976-1980 Heights of adult men, aged 18-24 mean: 70.0 inches standard deviation: 2.8 inches heights follow a normal distribution, so we have that heights of men are N(70, 2.8).
37 Health and Nutrition Examination Study of 1976-1980 68-95-99.7 Rule for men’s heights 68% are between 67.2 and 72.8 inches [ μ ± σ = 70.0 ± 2.8 ] 95% are between 64.4 and 75.6 inches [ μ ± 2 σ = 70.0 ± 2(2.8) = 70.0 ± 5.6 ] 99.7% are between 61.6 and 78.4 inches [ μ ± 3 σ = 70.0 ± 3(2.8) = 70.0 ± 8.4 ]