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? When are mean and variance informative, and when are they misleading? What is the 68/95/99.7 rule?

Mean is a balance point torque = force × distance

Mean is a balance point torque = force × distance

Mean is a balance point torque = force × distance

Mean is a balance point torque = force × distance balance point is where we get equal torque on both sides

Mean is a balance point torque = force × distance 6 balance point is where we 6 7 get equal torque on both 6 5 7 sides 6 5 7 10 6.44

Mean is a balance point torque = force × distance 6 balance point is where we 6 7 get equal torque on both 6 5 7 sides 6 5 7 10 x - μ = 6 - 6.44 = -0.44

Mean is a balance point torque = force × distance 6 balance point is where we 6 7 get equal torque on both 6 5 7 sides 6 5 7 10 4 × -0.44

Mean is a balance point torque = force × distance 6 balance point is where we 6 7 get equal torque on both 6 5 7 sides 6 5 7 10 Σ (x - μ) = 0 Σ x = Nμ mean = average ( Σ x)/N = μ

Mean is a balance point torque = force × distance 6 balance point is where we 6 7 get equal torque on both 6 5 7 sides 6 5 7 10 6.44

Mean is a balance point torque = force × distance 6 balance point is where we 6 7 get equal torque on both 6 5 7 sides 6 5 7 8

Mean is a balance point torque = force × distance 6 balance point is where we 6 7 get equal torque on both 6 5 7 sides 6 5 7 8

Mean is sensitive to outliers 6 6 7 6 5 7 6 5 7 17

Median ignores values 5 5 6 6 6 6 7 7 10

Median ignores values 5 5 6 6 6 6 7 7 10

Median ignores values 5 5 6 6 6 6 7 7 328

The sum of squared distances to the mean x = [2, 3, 7] 2 3 7

The sum of squared distances to the mean 2 3 7

The sum of squared distances to the mean 2 3 7

Variance: mean 3x3 squared distances to the mean 2x2 1x1 Σ (x - μ) 2 2 3 7 N = (4 + 1 + 9)/3 = 4.66

Variance: mean squared distances to the mean Σ (x - μ) 2 2 3 7 N = (4 + 1 + 9)/3 = 4.66

Variance: mean squared distances to 2.16x2.16 the mean Σ (x - μ) 2 2 3 7 N = (4 + 1 + 9)/3 = 4.66

Standard deviation: square root of mean squared distances to the mean 2.16 2 3 7

Variance: alternative form 2.64x2.64 Σ (x - μ) 2 2 3 7 N-1 = (4 + 1 + 9)/2 = 7

Mean is the point that 3x3 minimizes variance for a fixed data set 2x2 1x1 d/dμ Σ (x - μ) 2 2 3 7 = 2 Σ (x - μ) Σ (x - μ) = 0

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?

Mean is a balance point for a distribution torque = force × distance balance point is where we get equal torque on both sides P(10) P(2) P(3) P(4)

Mean is a balance point for a distribution torque = force × distance balance point is where we get equal torque on both sides P(10) P(2) P(3) P(4) μ = Σ x P(x) mean = average = eypectation

What are the expectations of these two dice? P(6)=1/6 μ = E[x] = Σ x P(x) P(6)=1/2

What are the expectations of these two dice? P(6)=1/6 "eypectation of x" μ = E[x] = Σ x P(x) P(6)=1/2

What are the expectations of these two dice? E[x] = Σ xP(x) P(6)=1/6 = 1×.16 + 2×.16 + ... + 6×.16 = (1 + 2 + ... + 6) × .16 = 21 / 6 = 3.5

What are the expectations of these two dice? P(6)=1/6 μ = E[x] = Σ x P(x) only if P(x) is = Σ x / N uniform for all x

What are the expectations of these two dice? E[x] = Σ xP(x) = 1×.1 + 2×.1 + ... + 6×.5 P(6)=1/2 = .1 × (1 + 2 + ... + 5) + 3 = 1.5 + 3 = 4.5

What are the variances of these two dice? P(6)=1/6 σ 2 = E[ Σ (x-μ) 2 ] = Σ (x-μ) 2 P(x) P(6)=1/2

Which has greater variance? P(6)=1/6 P(6)=1/2

Variance of uniform distribution var[x] = Σ (x-μ) 2 P(x) P(6)=1/6 = (1-3.5) 2 ×.16 + ... + (6-3.5) 2 ×.16 = -2.5 2 ×.16 + -1.5 2 ×.16 + ... + 2.5 2 ×.16 = 2.916

Variance of non-uniform distribution var[x] = Σ (x-μ) 2 P(x) = (1-4.5) 2 ×.1 + ... + (6-4.5) 2 ×.5 = -3.5 2 ×.1 + ... + 1.5 2 ×.5 P(6)=1/2 = 3.25

Which has greater variance? P(6)=1/6 P(6)=1/2

Sample mean/var vs. Distribution mean/var sample distribution x ̄ = Σ x/N μ = Σ x P(x) mean s 2 = Σ(x-x ̄ ) 2 /N σ 2 = Σ (x-μ) 2 P(x) variance

Sample mean/var vs. Distribution mean/var sample distribution x ̄ = Σ x/N μ = Σ x P(x) mean s 2 = Σ(x-x ̄ ) 2 /N σ 2 = Σ (x-μ) 2 P(x) variance

Distribution vs. Sample with dice

Mean and variance for distributions mean variance binomial np np(1-p) (1-p)/p 2 geometric 1/p Poisson λ λ

Distribution vs. Sample with parametric distributions

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?