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Feb 27: Expectation, Variance, and Standard Deviation In-class - - PowerPoint PPT Presentation
Feb 27: Expectation, Variance, and Standard Deviation In-class - - PowerPoint PPT Presentation
Feb 27: Expectation, Variance, and Standard Deviation In-class Midterm Exam MOVED to 3/10 Goals for today What are mean, variance, and standard deviation? What is the difference between distribution mean/variance and sample mean/variance?
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Goals for today
What are mean, variance, and standard deviation? What is the difference between distribution mean/variance and sample mean/variance? When are mean and variance informative, and when are they misleading? What is the 68/95/99.7 rule?
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Mean is a balance point
torque = force × distance
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Mean is a balance point
torque = force × distance
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Mean is a balance point
torque = force × distance
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Mean is a balance point
torque = force × distance balance point is where we get equal torque on both sides
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Mean is a balance point
torque = force × distance balance point is where we get equal torque on both sides
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6.44
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Mean is a balance point
torque = force × distance balance point is where we get equal torque on both sides
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x - μ = 6 - 6.44 = -0.44
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Mean is a balance point
torque = force × distance balance point is where we get equal torque on both sides
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4 × -0.44
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Mean is a balance point
torque = force × distance balance point is where we get equal torque on both sides
Σ (x - μ) = 0 Σ x = Nμ (Σ x)/N = μ
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mean = average
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Mean is a balance point
torque = force × distance balance point is where we get equal torque on both sides
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6.44
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Mean is a balance point
torque = force × distance balance point is where we get equal torque on both sides
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Mean is a balance point
torque = force × distance balance point is where we get equal torque on both sides
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Mean is sensitive to outliers
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Median ignores values
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Median ignores values
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5 5 6 6 6 6 7 7 328
Median ignores values
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The sum of squared distances to the mean
x = [2, 3, 7]
2 3 7
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2 3 7
The sum of squared distances to the mean
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2 3 7
The sum of squared distances to the mean
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2x2 1x1 3x3
2 3 7
Σ (x - μ)2 N = (4 + 1 + 9)/3 = 4.66
Variance: mean squared distances to the mean
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2 3 7
Σ (x - μ)2 N = (4 + 1 + 9)/3 = 4.66
Variance: mean squared distances to the mean
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2.16x2.16
2 3 7
Σ (x - μ)2 N = (4 + 1 + 9)/3 = 4.66
Variance: mean squared distances to the mean
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2.16
2 3 7
Standard deviation: square root of mean squared distances to the mean
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2.64x2.64
2 3 7
Σ (x - μ)2 N-1 = (4 + 1 + 9)/2 = 7
Variance: alternative form
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2x2 1x1 3x3
2 3 7
Mean is the point that minimizes variance for a fixed data set
d/dμ Σ (x - μ)2 = 2 Σ (x - μ) Σ (x - μ) = 0
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Goals for today
What are mean, variance, and standard deviation? What is the difference between distribution mean/variance and sample mean/variance? When are mean and variance informative, and when are they misleading? What is the 68/95/99.7 rule?
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Mean is a balance point for a distribution
torque = force × distance balance point is where we get equal torque on both sides
P(2) P(3) P(4) P(10)
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Mean is a balance point for a distribution
torque = force × distance balance point is where we get equal torque on both sides
μ = Σ x P(x)
P(2) P(3) P(4) P(10)
mean = average = eypectation
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What are the expectations of these two dice?
P(6)=1/2 P(6)=1/6
μ = E[x] = Σ x P(x)
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What are the expectations of these two dice?
P(6)=1/2 P(6)=1/6
μ = E[x] = Σ x P(x) "eypectation of x"
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What are the expectations of these two dice?
P(6)=1/6
E[x] = Σ xP(x) = 1×.16 + 2×.16 + ... + 6×.16 = (1 + 2 + ... + 6) × .16 = 21 / 6 = 3.5
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What are the expectations of these two dice?
P(6)=1/6
μ = E[x] = Σ x P(x) = Σ x / N
- nly if P(x) is
uniform for all x
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What are the expectations of these two dice?
P(6)=1/2
E[x] = Σ xP(x) = 1×.1 + 2×.1 + ... + 6×.5 = .1 × (1 + 2 + ... + 5) + 3 = 1.5 + 3 = 4.5
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What are the variances of these two dice?
P(6)=1/2 P(6)=1/6
σ2 = E[Σ (x-μ)2] = Σ (x-μ)2 P(x)
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Which has greater variance?
P(6)=1/2 P(6)=1/6
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Variance of uniform distribution
P(6)=1/6
var[x] = Σ (x-μ)2 P(x) = (1-3.5)2×.16 + ... + (6-3.5)2×.16 = -2.52×.16 + -1.52×.16 + ... + 2.52×.16 = 2.916
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Variance of non-uniform distribution
P(6)=1/2
var[x] = Σ (x-μ)2 P(x) = (1-4.5)2×.1 + ... + (6-4.5)2×.5 = -3.52×.1 + ... + 1.52×.5 = 3.25
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Which has greater variance?
P(6)=1/2 P(6)=1/6
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Sample mean/var vs. Distribution mean/var
sample distribution mean x ̄ = Σ x/N μ = Σ x P(x) variance s2 = Σ(x-x ̄ )2/N σ2 = Σ (x-μ)2 P(x)
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Sample mean/var vs. Distribution mean/var
sample distribution mean x ̄ = Σ x/N μ = Σ x P(x) variance s2 = Σ(x-x ̄ )2/N σ2 = Σ (x-μ)2 P(x)
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Distribution vs. Sample with dice
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Mean and variance for distributions
mean variance binomial np np(1-p) geometric 1/p (1-p)/p2 Poisson λ λ
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Distribution vs. Sample with parametric distributions
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Goals for today
What are mean, variance, and standard deviation? What is the difference between distribution mean/variance and sample mean/variance? When are mean and variance informative, and when are they misleading? What is the 68/95/99.7 rule?