Chapter 7
Systematic Sampling

Selection of every kth case from
a list of possible subjects.
Systematic Sampling - 2
Definition:
A sample obtained by randomly selecting 1
element from among the first k elements in
the frame and every kth element
thereafter is called a 1-in-k systematic
sample with a random start. (Assumes
population is randomly ordered).
Does each element in the frame have an equal chance to be selected? Yes
If so, what is this equal chance?
1/k
Is this a simple random sample?
NO!!
Systematic Sampling - 3
N = 100
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Systematic Sampling - 4
N = 100
Want n = 20
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Systematic Sampling - 5
N = 100
Want n = 20
k = N/n = 5
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Systematic Sampling - 6
N = 100
Want n = 20
k = N/n = 5
Select a random number between 1 and 5:
Choose 4
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Systematic Sampling - 7
N = 100
Want n = 20
k = N/n = 5
Select a random number between 1 and 5:
Choose 4
Start with #4 and select every 5th item
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Systematic Sampling - 8
There are actually only 5 distinct
systematic random samples which
are:
1. {1,6,11,…,91,96}
2. {2,7,12,…,92,97}
3. {3,8,13,…,93,98}
4. {4,9,14,…,94,99}
5. {5,10,15,…,95,100}
We are simply choosing 1 of these
5 groups at random
N = 100
Want n = 20
k = N/n = 5
Select a random number between 1 and 5:
Choose 4
Start with #4 and select every 5th item
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Systematic Sampling - 9

Advantages
– Easier to perform in the field, especially if
a good frame is not available
– Frequently provides more information per
unit cost than simple random sampling, in
the sense of smaller variances.
Example. A systematic sample was drawn from a batch of
produced computer chips. The first 400 chips are fine but,
due to a fault in the machine later in the production process,
the last 300 chips are defective. Systematic sampling will
select uniformly over the non-defective and defective items
and would give a very accurate estimate of the fraction of
defective items.
Systematic Sampling - 10
Value of k?
 In general, for a systematic random
sample of n elements from a population
(or frame) of size N, choose k ≤ N/n.
Example: From a population of 90,000
students we desire a sample of 12,000
students. Since 90,000/12,000 = 7.5, we
can select a 1-in-7 systematic sample.
Systematic Sampling - 11
Value of k when N unknown?
Must guess the value of k to achieve
a sample size n.
 If k is too large, in some cases can go
back and select another 1-in-k sample
until the sample size n is attained.

Systematic Sampling - 12
Estimation
Parameters of interest that typically desire
to estimate:
 population mean
 population total
p population proportion
How many
people at
this rally?
Systematic Sampling – 13
Estimation of population mean 
Estimator of the population mean  :
n
ˆ  ysy 
y
i 1
i
n
where the subscript sy signifies that systematic sampling was used.
Estimated variance of ysy :
2
n
s


Vˆ ( ysy )   1  
N n

assuming a randomly ordered population.
Systematic Sampling – 14
Estimation of population mean 
From previous slide:
2
n
s


ˆ
V ( ysy )  1  
 N n
Identical to the estimated variance of y
obtained by using simple random sampling.
This does NOT imply that V ( y )  V ( ysy )
(see following slides)
Systematic Sampling – 15
 
n
V ( y) 
1  
n  N
2
1. If ρ is close to 0, and
N is fairly large,
systematic sampling is
roughly equivalent to
simple random sampling.
V ( ysy ) 
2
1  (n  1)  
n
1
where
   1 is a measure of the
n 1
correlation between pairs of elements
within the same systematic sample.
2. If ρ is close to 1, then the elements within the sample are quite
similar wrt the characteristic being measured, and systematic
sampling will yield a higher variance of the sample mean than will
simple random sampling.
3. If the elements in the systematic sample tend to be very
different, then ρ is negative and systematic sampling may be more
precise than simple random sampling.
Systematic Sampling – 16
Summary: comparison of systematic
and simple random sampling
2 
n
V ( y) 
1  
n  N
V ( ysy ) 
2
n
1  (n  1)  
1. Random order (If ρ is close to 0) Systematic and simple random
sampling are approximately equal in precision.
2. Cyclic pattern in the y’s Systematic random sampling is worse than
simple random sampling.
3. Increasing or Decreasing order in the y’s Systematic random
sampling is better than simple random sampling.
Systematic Sampling – 17
 
n
V ( y) 
1  
n  N
2
V ( ysy ) 
2
n
1  (n  1)  
1. Random order: if ρ
is close to 0, and N is
fairly large, systematic
sampling is roughly
equivalent to simple
random sampling.
Systematic Sampling – 18
 
n
V ( y) 
1  
n  N
2
V ( ysy ) 
2
n
1  (n  1)  
2. Cyclic pattern in the y’s
Systematic random sampling
is worse than simple random
sampling.
Systematic Sampling – 19
 
n
V ( y) 
1  
n  N
2
V ( ysy ) 
2
n
1  (n  1)  
3. Increasing or Decreasing
order in the y’s Systematic
random sampling is better
than simple random
sampling.
Systematic Sampling – 20
Estimation of population total 
Estimator of the population total  :
n
ˆ  Nysy  N
y
i 1
i
n
where the subscript sy signifies that systematic sampling was used.
Estimated variance of ˆ :
2
n
s


Vˆ (ˆ)  N 2Vˆ ( yst )  N 2 1  
N n

assuming a randomly ordered population.
Systematic Sampling – 21
Estimation of population proportion p
 1, if ith element has characteristic of interest
Let yi  
th
0,
if
i
element does not have characteristic of interest

Estimator of the population proportion p :
n
pˆ sy  ysy 
y
i 1
i
n
where the subscript sy signifies that systematic sampling was used.
Estimated variance of pˆ sy :
ˆ sy qˆ sy
p
n


Vˆ ( pˆ sy )  1  
N  n 1

ˆ sy , assuming a randomly ordered population.
where qˆ sy =1  p
Systematic Sampling – 22
Required Sample Size for Bound B
Sample size for 
Sample size for p
N
n
2
( N  1) D  
Npq
n
( N  1) D  pq
2
2
B
where D 
4
2
B
where q  1  p and D 
4