101 Ch. 6

# 101 Ch. 6 - Chapter 6 The Standard Deviation as a Ruler and...

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Chapter 6 The Standard Deviation as a Ruler and the Normal Model

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Normal Distributions Bell Curve Physical Characteristics Examples Heights Weights Lengths of Bird Wings Most important distribution in statistics
Normal Distributions Curve is always above the x-axis The tails never reach the x-axis They continue on to infinity and –infinity Area under entire curve = 1 or 100% Mean = Median This means the curve is symmetric

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Normal Distributions Two parameters (not calculated, i.e. do not come from the data) Mean μ (pronounced “meeoo”) Locates center of curve Splits curve in half Shifts curve along x-axis
Normal Distributions Standard deviation σ (pronounced “sigma”) Controls spread of curve Smaller σ makes graph tall and skinny Larger σ makes graph flat and wide Ruler of distribution Write as N( μ,σ)

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Perfect World Wouldn’t everyone like it if there was a perfect world? In Statistics there is!!!! We call it the Standard Normal Distribution.

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Standard Normal (Z) World Perfectly symmetric Centered at zero Half numbers below the mean and half above. Total area under the curve is 1. Can fill in as percentages across the curve.
Standard Normal Distribution Puts all normal distributions on same scale z has center (mean) at 0 z has spread (standard deviation) of 1 σ μ - = y z

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Standard Normal Distribution z = # of standard deviations away from mean μ Negative z, number is below the mean Positive z, number is above the mean Written as N(0,1)
Portal from X-world to Z-world z has no units (just a number) Puts variables on same scale Center (mean) at 0 Spread (standard deviation) of 1 Does not change shape of distribution s y y z - =

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Standardizing Variables z = # of standard deviations away from mean Negative z – number is below mean Positive z – number is above mean
Standardizing Y ~ N(70,3). Standardize y = 68. y = 68 is 0.67 standard deviations below the mean 67 . 0 3 70 68 - = - = - = σ μ y z

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Standardizing Notice the difference between Y and y Y denotes an entire distribution; all possible values of the distribution, the shape, the center, the spread y denotes a single value ONLY Can be generalized to all capital letters Z and z X and x
Standardizing N(70,3) N(0,1)

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Standardizing Y ~ N(70,3). Standardize y = 74 y = 74 is 1.33 standard deviations above mean 33 . 1 3 70 74 = - = - = σ μ y z
Standardizing N(70,3) N(0,1)

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## This note was uploaded on 03/30/2008 for the course STAT 101 taught by Professor Graham during the Spring '08 term at Iowa State.

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101 Ch. 6 - Chapter 6 The Standard Deviation as a Ruler and...

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