Chapter 7Systematic Sampling

Selection of every kth case froma 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?If so, what is this equal chance?

Is this a simple random sample?

Yes

1/k

NO!!

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N = 100

Systematic Sampling - 41 26 51 762 27 52 773 28 53 784 29 54 795 30 55 806 31 56 817 32 57 828 33 58 839 34 59 8410 35 60 8511 36 61 8612 37 62 8713 38 63 8814 39 64 8915 40 65 9016 41 66 9117 42 67 9218 43 68 9319 44 69 9420 45 70 9521 46 71 9622 47 72 9723 48 73 9824 49 74 9925 50 75 100

N = 100

Want n = 20

Systematic Sampling - 51 26 51 762 27 52 773 28 53 784 29 54 795 30 55 806 31 56 817 32 57 828 33 58 839 34 59 8410 35 60 8511 36 61 8612 37 62 8713 38 63 8814 39 64 8915 40 65 9016 41 66 9117 42 67 9218 43 68 9319 44 69 9420 45 70 9521 46 71 9622 47 72 9723 48 73 9824 49 74 9925 50 75 100

N = 100

Want n = 20

k = N/n = 5

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N = 100

Want n = 20

k = N/n = 5

Select a random number between 1 and 5:

Choose 4

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

1 26 51 762 27 52 773 28 53 784 29 54 795 30 55 806 31 56 817 32 57 828 33 58 839 34 59 8410 35 60 8511 36 61 8612 37 62 8713 38 63 8814 39 64 8915 40 65 9016 41 66 9117 42 67 9218 43 68 9319 44 69 9420 45 70 9521 46 71 9622 47 72 9723 48 73 9824 49 74 9925 50 75 100

Systematic Sampling - 8

N = 100

Want n = 20

k = N/n = 5

Select a random number between 1 and 5:

Choose 4Start with #4 and select every 5th item

1 26 51 762 27 52 773 28 53 784 29 54 795 30 55 806 31 56 817 32 57 828 33 58 839 34 59 8410 35 60 8511 36 61 8612 37 62 8713 38 63 8814 39 64 8915 40 65 9016 41 66 9117 42 67 9218 43 68 9319 44 69 9420 45 70 9521 46 71 9622 47 72 9723 48 73 9824 49 74 9925 50 75 100

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

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

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.

Value of k?

Systematic Sampling - 11

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.

Value of k when N unknown?

Systematic Sampling - 12Estimation

Parameters of interest that typically desire

to estimate:

population mean

population total

population proportionp

How many people at this rally?

Systematic Sampling – 13Estimation of population mean

1

2

where the subscript signifies that systematic sampling was used.

Estimated

assuming a randomly ordered

variance o

populat o

f

i

Estimator of the population m

ˆ( 1

ean

ˆ

)

n

ii

sy

sy

sy

sy

y

yy

n

n sV y

N n

:

:

n.

Systematic Sampling – 14Estimation of population mean

2

From previous slide:

Identical to the estimated variance of

obtained by using simple random sampling.

This does NOT imply that ( ) ( )

(see following slid s

( ) 1

e )

ˆsy

sy

y

V y V y

n sV y

N n

Systematic Sampling – 152

( ) 1n

V yn N

2

( ) 1 ( 1)

1where 1 is a measure of the

1correlation between pairs of elements

within the same systematic sample.

syV y nn

n

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.

1. If ρ is close to 0, and N is fairly large, systematic sampling is roughly equivalent to simple random sampling.

Systematic Sampling – 16Summary: comparison of systematic and simple random sampling

2

( ) 1n

V yn N

2

( ) 1 ( 1)syV y nn

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.

1. Random order (If ρ is close to 0) Systematic and simple random sampling are approximately equal in precision.

Systematic Sampling – 172

( ) 1n

V yn N

2

( ) 1 ( 1)syV y nn

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 – 182

( ) 1n

V yn N

2

( ) 1 ( 1)syV y nn

2. Cyclic pattern in the y’s Systematic random sampling is worse than simple random sampling.

Systematic Sampling – 192

( ) 1n

V yn N

2

( ) 1 ( 1)syV y nn

3. Increasing or Decreasing order in the y’s Systematic random sampling is better than simple random sampling.

Systematic Sampling – 20Estimation of population total

1

22 2

where the subscript signifies that systematic sampling was used.

ˆEstimated variance o

assuming a randomly

f

o

:

rde

ˆ

ˆ

Estimator of the

ˆˆ(

population

) ( ) 1

total n

ii

sy

st

sy

yNy N

n

n sV N V y N

N n

:

red population.

Systematic Sampling – 21Estimation of population proportion p

1

where the subscript signifies that systematic sampling was used.

ˆEstimated variance of :

ˆwhere

Estimator of the popula

ˆ

ˆ ˆˆ ˆ( ) 1

tion proportion

ˆ=1 ,

1

n

ii

sy sy

sy sys

s

y

y

sy sy

sy

p

q

p

p

yp y

n

p qnV p

N n

:

assuming a randomly ordered population.

1, if i element has characteristic of interest Let

0, if i element does not have characteristic of interest

th

i thy

Systematic Sampling – 22Required Sample Size for Bound B

2

2

2

Sample size for

w

( 1

h e 4

)

er

Nn

N

BD

D

2

( 1

Sample size for

where 1

)

and 4

Npqn

N D pq

p

Bq p D