Basics of Sampling Theory

P = { x1, x2, ……, xN}

where P = population

x1, x2, ……, xN are real numbers

Assuming x is a random variable;

Mean/Average of x, Σ xi

x =N

i=1

N

Basics of Sampling TheoryStandard Deviation,

Variance,

Σ (xi – x)2

σx = N

i=1

N

√ Σ (xi – x)2

σx2 =

N

i=1

N

Basics of Sampling Theory

Theorem About Mean

picking random numbers x, mean = xpicking random numbers y, mean = y

x = y

Picking another number z,

mean z = x = y

z = c1x + c2y ; c1, c2 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 TheoryTheorem About Variances/Sampling Theorem

z = (x + y)/2; σz2 = ? σz

2 < σx2

Taking, z = (x + y)/2

σz2 = (σx

2 + σy2)/4

Taking k sample, z = (x + x’ + x’’ + …. + x’’…k)/k

σz2 = (kσx

2 )/k2

σz2 = σx

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

xx1

x2 xN

Assuming numbers are sorted

• K = number of samples

• z = sample mean

• as k increases, z comes closer to x

σ