Systematic sampling - 2, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Comparison of systematic sampling, stratified sampling and SRS with population with linear trend: We assume that the values of units in the population increase according to linear trend. So the  values of successive units in the population increase in accordance with a linear model so that

Now we determine the variances of under this linear  trend.

Under SRSWOR

Under systematic sampling Earlier y_ij denoted the value of study variable with the  j^th unit in the i^th systematic sample. Now y_ij represents the value of  [i + ( j− 1)k ]^th unit of the population, so

Under stratified sampling

If k is large, so that 1/k is negligible, then comparing 

Thus,

So stratified  sampling is best for linearly trended population.  Next  best is systematic sampling.

Estimation of variance:

As such there is only one cluster, so variance in principle, cannot be estimated. Some approximations have been suggested.

1. Treat systematic sample as if it were a random sample. In this case, an estimate of variance is 

This estimator under-estimates the true variance.

2. Use of successive differences of the values gives the estimate of variance as 

This estimator is a biased estimator of true variance.

3. Use the balanced difference of  y₁ ,y₂ , ..., y_n to get the estimate of variance as 

4. The interpenetrating subsamples can be utilized by dividing the sample into C groups each of 

Systematic sampling when  N ≠ nk.

When  N is not expressible as  nk then suppose  N can be expressed as N = nk + p; p< k.

Then consider the following sample mean as an estimator of population mean

In this case

So   is a biased estimator of   .

An unbiased estimator of is

where   is the total of values of the i^th column.

Now we consider another procedure which is opted when N ≠ nk.

When population size  N is not expressible as the product of  n and  k , then let

N = nq + r.

Then take the sampling interval as

Let denotes the largest integer contained in 

number of units expected in sample

If q = q*, then we get

Similarly if  = q* + 1, then  

Example: Let N = 17 and  n = 5. Then  q = 3 and r = 2 .  Since

Then sample sizes would be 

This can be verified from the following example:

We now prove the following theorem which shows how to obtain an unbiased estimator of the population mean when  N ≠ nk.

Theorem: In systematic sampling with sampling interval k from a population with size  N ≠ nk , an unbiased estimator of the population mean   is given by

where  i stands for the i th systematic sample, i = 1, 2, ..., k andn ' denotes the size of  i^th systematic sample.

Proof. Each systematic sample has probability1/k.  Hence

Now, each unit occurs in only one of the k possible systematic samples. Hence

which on substitution in  proves the theorem.

When N ≠ nk , the systematic samples are not of the same size and the sample mean is not an unbiased estimator  of the population mean.  To overcome these disadvantages of systematic  sampling when N ≠ nk , circular systematic sampling is proposed. Circular  systematic sampling consists of selecting a random number from 1 to N and then selecting the unit corresponding to this random number.
Thereafter  every k^th unit in a cyclical manner is selected till a sample of n units is obtained, k being the nearest integer to N/n

In other words, if  i is a number selected at random from 1 to N , then the circular systematic sample consists of units  with serial numbers

This sampling scheme ensures equal probability of inclusion in the sample for every unit.

Example: Let N = 14 and n = 5. Then, k = nearest integer to 14/5 = 3 Let the first number selected at random from 1 to 14 be 7.  Then, the circular systematic sample consists of units with serial numbers 7,10,13, 16-14=2, 19-14=5.

This procedure is illustrated diagrammatically in following  figure.

Theorem: In circular systematic sampling, the sample mean is an unbiased estimator of the population mean.

Proof: If  i is the number selected at random, then the circular systematic sample mean is 

where   denotes the total of  y values in the i^th circular systematic sample, i = 1, 2, ...,N . We note here that in circular systematic sampling,  there are N circular systematic samples, each having probability 1/N of its selection.  Hence,

Clearly, each unit of the population occurs in  n of the N possible circular systematic sample means.  Hence,

which on substitution in E () proves the theorem.

What to do when  N ≠ nk One of the following possible procedures may be adopted  when  N ≠ nk.

(i) Drop one unit at random if sample has (n + 1) units.
(ii) Eliminate some units so that N = nk .
(iii) Adopt circular systematic sampling scheme.
(iv) Round off the fractional interval k .