Download Basics of Sampling Theory

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,
N
Σ xi
x=
i=1
N
Basics of Sampling Theory
Standard Deviation,
√
σx =
Variance,
σx2
N
Σ (xi – x)2
i=1
N
N
Σ (xi – x)2
=
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 = 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 Theory
Theorem About Variances/Sampling Theorem
z = (x + y)/2; σz2 = ?
σz2 < σx2
Taking, z = (x + y)/2
σz2 = (σx2 + σy2)/4
Taking k sample, z = (x + x’ + x’’ + …. + x’’…k)/k
σz2 = (kσx2 )/k2
σz2 = σx2 )/k
* This theory works only on independent
variables; while mean theorem works on
dependent variable
* Error depends on number of
samples; bigger sample – less
error; smaller sample – more error
* This formula is true for sampling
with replacement
Basics of Sampling Theory
Normal Distribution curve
σ
x1 x2
• K = number of samples
• z = sample mean
xN
x
• as k increases, z comes closer to x
Assuming numbers
are sorted