Basics of Sampling Theory

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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
S xi x N
i1
2
Basics of Sampling Theory

• Standard Deviation,
• Variance,

N
v
S (xi x)2 sx N
i1
N
S (xi x)2 sx2 N
i1
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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

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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

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Basics of Sampling Theory

• Theorem About Variances/Sampling Theorem
• z (x y)/2 sz2 ? sz2 lt sx2
• Taking, z (x y)/2
• sz2 (sx2 sy2)/4
• Taking k sample, z (x x x .
  xk)/k
• sz2 (ksx2 )/k2
• sz2 sx2 )/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
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Basics of Sampling Theory

• Normal Distribution curve

s
x2
xN
x1
x
Assuming numbers are sorted

• K number of samples
• z sample mean
• as k increases, z comes closer to x