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The Normal Family
Let W = Wi = Weight (lbs) of the i-th randomly selected college-aged male, i = 1, 2, …
Assume W has
Central Location
Dispersion
Shape
population mean
population standard deviation
Normal
𝜇 = 160,
σ = 30
~N
Normal Distribution / Normal Random Variable
W~ N(𝛍 = 160, 𝝈 = 30)
Location:

Population Mean =Expected Value of W = μ = μW = E(W) = 160
Dispersion:


Population Standard Deviation = Standard Deviation of W = σ = σW = 30
Range is infinite
Shape:


Symmetric about μ = 160, i.e., P{W > 160 + a} = P{W < 160 − a}, for any number a
Inflection points at 130 and 190
Normal Distn / Random Variable, X~ N(μ, σ)
Location:

Population Mean = Expected Value of X = E (X) = μ =μX
Dispersion:

Population Standard Deviation = Standard Deviation of X = σ = σX

Range is infinite
Shape:

Symmetric about μ

P{X > μ + a} = P{X < μ < −a} for any number a

Inflection points at (μ − σ ) and (μ + σ)
Standard Normal Distn / Random Variable, Z ~ N(0,1)
Location:

Population Mean = Expected Value of Z = μ = μZ = E(Z) = 0
Dispersion:

Population Standard Deviation= Standard Deviation of Z = σ = σZ = 1

Range is infinite
Shape:

Symmetric about μ = 0, P{ Z > a} = P{Z < −a} for any number a

Inflection points at −1 and +1
"Normal" Refers to Shape
A Normal r.v. can have any location and any dispersion.



Different locations, μX ≠ μY
Different dispersions, σX ≠ σY
Same shape, functional form, or family.

Different locations, μX ≠ μY

Same dispersion, σX = σY

Same shape, functional form, or family

Same locations, μX = μY

Different dispersions, σX ≠ σY

Same shape, functional form, or family
Standard Normal Cumulative Distribution Function (CDF)
The standard normal cumulative distribution function (CDF) is the s-shaped curve in the figure
above. For any real-number value z, the standard normal CDF is the probability that a standard
normal random variable is less than or equal to z.
Advanced students may be helped by the mathematical definitions.
Golde I. Holtzman, Department of Statistics, College of Arts and Sciences, Virginia Tech (VPI)
Last updated: September 8, 2009 © Golde I. Holtzman, all rights reserved.
URL: http://courseware.vt.edu/users/holtzman/STAT5605/normal.html