Overview of Systematic Sampling

 

SYSTEMATIC SAMPLING

By: UJJAINI DALAL

CHRIST UNIVERSITY, BANGALORE

Systematic sampling is a commonly employed technique if the complete and up-to-date list of the sampling units in available. This consists in selecting only the first unit at random, the rest being automatically selected according to some predetermined pattern involving regular spacing of units. Systematic sampling is a statistical method that researchers use to zero down on the desired population they want to research.  Systematic sampling is an extended implementation of probability sampling in which each member of the group is selected at regular periods to form a sample.

How Systematic Sampling Works

When we  are sampling, ensure you represent the population fairly. Systematic sampling is a symmetrical process where the researcher chooses the samples after a specifically defined interval. Sampling like this leaves the researcher no room for bias regarding choosing the sample.

Let us suppose that N sampling units are serially numbered from 1 to N some order and a sample of size n is to be drawn such that N=nk => k=(N/n) where k, usually called the sampling interval , is an integer.
Systematic sampling consists in drawing a random number, says i<= k and selecting the unit corresponding to this number and every kth unit subsequently. Thus the sample of size n will consist of units I, i+k, i+2k, …., i+(n-1)k. The random number ‘I’ is called the random start and its value determines, as a matter of the fact, the whole sample.

Types of Systematic Sampling : Here are the types of systematic sampling:

 (a) Systematic random sampling (b) Linear systematic sampling (c) Circular systematic sampling.

Important Formulas and few important Results

Estimation of population mean : When N = nk:

• The sample mean provides an unbiased estimate of the population mean.

 

• The variance of the estimate is

• Comparison with SRSWOR:

Estimation of population mean : When N not equal to nk:

   ·   The sample mean provides an unbiased estimate of the population mean.

• The variance is given as

·       

• In systematic sampling with sampling interval k from a population with size N not equal to nk, an unbiased estimator of the population mean Y is given by

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

  When to use systematic sampling

     Some practical situations where systematic sampling has been found very useful are given as follows:
·   The selection of every kth strip in the forest survey for estimation of timber.
·  The selection of every kth village in rural surveys.
·  The selection of cornfields every kth mile/km apart for observation on incidence of borer.
·  The selection of every kth time interval for the estimation of the total catch of fish in fishers.

     Because of its operational convenience, the job of collecting the systematic sample can be entrusted to the field worker.

 Here are 4 other situations of when to use Systematic Sampling:   
1. Budget restrictions: In comparison to other sampling methods like simple random sampling, this sampling technique is more suitable for conditions where there are budget restrictions and also the extremely uncomplicated accomplishment of the study.
2.  Uncomplicated implementation: As systematic sampling depends on the defined sampling intervals to decide the sample, it becomes simple for the researchers and statisticians to manage samples with more respondents. This is because the time invested in creating samples is minimal, and the cost spent is also restricted due to the periodic nature of systematic sampling.
3.  Absence of data pattern: There are specific data that don’t have an arrangement in place. This data can be analyzed in an unbiased manner, using systematic sampling.      
4.  Low risk of data manipulation in research: It is highly productive while researching a broad subject, especially when there’s a negligible risk of data manipulation.

 

   Advantage of Systematic Sampling

     Here are the advantages of systematic sampling:

·   Systematic sampling is operationally more convenient than simple random sampling or stratified random sampling. Time and work involved is also relatively much less.

·   Systematic Sampling yields a sample which is evenly spread over the entire population.

·   It’s extremely simple and convenient for the researchers to create, conduct, analyze samples.

·   As there’s no need to number each member of a sample, it is better for representing a population in a faster and simpler manner.

 

     Limitations of Systematic Sampling

    Systematic sampling has a number of disadvantages as mentioned below:

  ·The main disadvantage of systematic sampling is that systematic samples are not in general random samples since the requirement in  merit two is rarely fulfilled.

  · If N is not a multiple of n, then
(i) the actual sample size is different from that required.
(ii) sample mean is not an  unbiased estimate of the population mean.

  · It is not possible to obtain unbiased estimate of the variance of systematic sampling on the basis of a single sample because a systematic sample is regarded as a sample of one unit(cluster).

  · Systematic sampling may yield highly biased estimates if there are periodic features associated with the sampling interval i.e, if the frame has a periodic feature and k is equal to or a multiple of the period.

 

EXPLAINATION OF SYSTEMATIC SAMPLING USING R

We have considered a dataset which represent the report of the number of forest fires in India divided by states. The series comprises the period of 3 years (2008 to 2011). The data were obtained from the official website of the India government. Suppose we want select a random sample of Year 2010-2011 after every 5^th draw.

PROCEDURE            

COMMENT: The variance of the estimator is given by 163227.9. The 95% confidence interval is given as (-673.4306,1126.8592).

 While reaching to conclusion about a large volume of data, we prefer to take samples from the whole population and then we analyze them and reach to a conclusion. We want to use our judgment as less as possible as the judgment sometimes can lead towards biasness. As the Simple Random Sampling involves more judgment and Stratified Random Sampling needs complex process of classification of the data into different classes, we use Systematic Random Sampling. We can also say that this method is the hybrid of two other methods (viz. Simple Random Sampling and Stratified Random Sampling). Systematic sampling is used in many aspects in real life and also we can inculcate   the  dataset in R and compute the estimate the variable using systematic sampling.