Statistical Convergence

For a random stationary variable x,

Expected value of x,

Sample mean of x,

Statistical convergence error,

1) Normal data

Student-t Statistic

or the confidence interval

Confidence interval size (limit):

For a 95% confidence level,

t_n;α/2 = 2 for n > 30

2) Non-normal with unknown distribution

Tchebycheff inequality

or

where the limit value

• c = 2 for about 75% probability (i.e., 75% confidence level)

• c = 4.5 for about 95% probability (i.e., 95% confidence level)

Note: the assumption, σ_x ≈ s_x, is not justified when s_x is increasing with N.

3) Statistical convergence error

For a 95% confidence level:

c = 2.0 if x is normal, by using the Student-t statistics,

c = 4.5 if x is not normal with unknown distribution, by using the Tchebycheff inequality.