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Normal Distribution
Adult male heights
Adult male heightsNational Health Statistics Report (2008) recorded the heights of4,482 males age 20 and older
Heights (inches)
Density
55 60 65 70 75 80 85
0.00
0.02
0.04
0.06
0.08
0.10
Adult male heightsNational Health Statistics Report (2008) recorded the heights of4,482 males age 20 and older
Heights (inches)
Density
55 60 65 70 75 80 85
0.00
0.02
0.04
0.06
0.08
0.10
Adult male heightsNational Health Statistics Report (2008) recorded the heights of4,482 males age 20 and older
Heights (inches)
Density
55 60 65 70 75 80 85
0.00
0.02
0.04
0.06
0.08
0.10
What is a normal distribution?I Unimodal, symmetric distribution (bell-shaped)I Denoted N(µ, σ): Normal with mean µ and standard
deviation σI Many large populations are very close to normally distributed
Adult male heights
Heights (inches)
Density
55 60 65 70 75 80 85
0.00
0.02
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0.06
0.08
0.10
N(µ = 68.7, σ = 3.7)
Changing µSuppose we start out with a N(0, 1) distribution and change themean to 5. What will happen?
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Density
Changing µSuppose we start out with a N(0, 1) distribution and change themean to 5. What will happen?
-10 -5 0 5 10
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Density
-10 -5 0 5 10
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Densitymean=0,sd=1mean=5,sd=1
Changing σSuppose we start out with a N(0, 1) distribution and change thestandard deviation to 4. What will happen?
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Density
Changing σSuppose we start out with a N(0, 1) distribution and change thestandard deviation to 4. What will happen?
-10 -5 0 5 10
0.0
0.1
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0.5
Density
-10 -5 0 5 10
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Densitymean=0,sd=1mean=0,sd=4
Empirical RuleFor normally distributed data,
I 68.2% of data fall within 1 standard deviation of the mean
I 95.4% of data fall within 2 standard deviations of the mean
I 99.7% of data fall within 3 standard deviations of the mean
Adult male heightsN(µ = 68.7, σ = 3.7)
Heights (inches)
Density
55 60 65 70 75 80 85
0.00
0.02
0.04
0.06
0.08
0.10
Adult male heightsN(µ = 68.7, σ = 3.7)
Heights (inches)
Density
55 60 65 70 75 80 85
0.00
0.02
0.04
0.06
0.08
0.10 99.68% of observations 79.857.6
4468 out of 4482
Who is taller?
I Veronica: 67 inches; Coach K: 71 inches
Who is taller relative to their gender?
Who is taller?
I Veronica: 67 inches; Coach K: 71 inches
Who is taller relative to their gender?
Who is taller? (cont.)
But...the two heights come from different distributions.
55 60 65 70 75 80 85
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Density
Women: N(63.8,2.9)
Veronica: 67 inches
55 60 65 70 75 80 85
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Density
Men: N(68.7,3.7)
Coach K: 71 inches
Who is taller? (cont.)
But...the two heights come from different distributions.
55 60 65 70 75 80 85
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0.15
Density
Women: N(63.8,2.9)
Veronica: 67 inches
55 60 65 70 75 80 85
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Density
Men: N(68.7,3.7)
Coach K: 71 inches
Standardizing
We can compare data from different distributions using Z scores
I Standardize a value: z = xobs−µσ
I Gives the number of standard deviations above (or below) themean an observation is.
I Relates to standard normal distribution: N(0, 1)
Standardizing
We can compare data from different distributions using Z scores
I Standardize a value: z = xobs−µσ
I Gives the number of standard deviations above (or below) themean an observation is.
I Relates to standard normal distribution: N(0, 1)
Standardizing
We can compare data from different distributions using Z scores
I Standardize a value: z = xobs−µσ
I Gives the number of standard deviations above (or below) themean an observation is.
I Relates to standard normal distribution: N(0, 1)
Standardizing
We can compare data from different distributions using Z scores
I Standardize a value: z = xobs−µσ
I Gives the number of standard deviations above (or below) themean an observation is.
I Relates to standard normal distribution: N(0, 1)
Standardizing (cont.)So, we can obtain our two z-scores:
I ZVeronica = 67−63.82.9 = 1.10
I ZCoachK = 71−68.73.7 = 0.89
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Height (inches)
Density
Coach K Veronicaz=.89 z=1.1
Standardizing (cont.)So, we can obtain our two z-scores:
I ZVeronica = 67−63.82.9 = 1.10
I ZCoachK = 71−68.73.7 = 0.89
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0.1
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Height (inches)
Density
Coach K Veronicaz=.89 z=1.1
Standardizing (cont.)So, we can obtain our two z-scores:
I ZVeronica = 67−63.82.9 = 1.10
I ZCoachK = 71−68.73.7 = 0.89
-3 -2 -1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
Height (inches)
Density
Coach K Veronicaz=.89 z=1.1
Standardizing (cont.)So, we can obtain our two z-scores:
I ZVeronica = 67−63.82.9 = 1.10
I ZCoachK = 71−68.73.7 = 0.89
-3 -2 -1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
Height (inches)
Density
Coach K Veronicaz=.89 z=1.1
