Basics of Sampling Theory P = { x 1, x 2, ……, x N } where P = population x 1, x 2, ……, x N are real numbers Assuming x is a random variable; Mean/Average of x, Σ x i x = N i=1 N

Basics of Sampling Theory Standard Deviation, Variance, Σ (x i – x) 2 σ x = N i=1 N √ Σ (x i – x) 2 σ x 2 = N i=1 N

Basics of Sampling Theory Theorem About Mean picking random numbers x, mean = x picking random numbers y, mean = y x = y Picking another number z, mean z = x = y z = c 1 x + c 2 y; c 1, c 2 are constants z = x + y

Basics of Sampling Theory Independence two events are independent if the occurrence of one of the events gives no information about whether or not the other event will occur; that is, the events have no influence on each other for example a, b and c are independent if: - a and b are independent; a and c are independent; and b and c are independent

Basics of Sampling Theory Theorem About Variances/Sampling Theorem z = (x + y)/2; σ z 2 = ? σ z 2 < σ x 2 Taking,z = (x + y)/2 σ z 2 = (σ x 2 + σ y 2 )/4 Taking k sample, z = (x + x’ + x’’ + …. + x’’ …k )/k σ z 2 = (kσ x 2 )/k 2 σ z 2 = σ x 2 )/k * Error depends on number of samples; bigger sample – less error; smaller sample – more error * This formula is true for sampling with replacement * This theory works only on independent variables; while mean theorem works on dependent variable

Basics of Sampling Theory Normal Distribution curve x x1x1 x2x2 xNxN Assuming numbers are sorted K = number of samples z = sample mean as k increases, z comes closer to x σ