systematic sampling

Systematic sampling is a method of sampling $n$ items from apopulation of $N$ items with the following sequence of procedures:

1.
label each item in the population as $x_{i}$ , where $1\\leq i\\leq N$ ,
2.
randomly select one item from the first $k$ items in thepopulation, where $k=\\lceil N/n\\rceil$ , the smallest integergreater than or equal to $N/n$ ,
3.
pick the $k$ th item thereafter, $x_{i},x_{i+k},x_{i+2k},\\ldots$ ,until all $n$ items are picked.

If $N\\equiv 0\\mod n$ , then all $n$ items can be picked beforereaching the end of the population. Otherwise, define $x_{j+N}:=x_{j}$ . Using this, we can continue to pick our sampleunits until all $n$ items are picked.

The above method suggests that we do not have to make our first pickfrom among the first $k$ items. By the above definition, we canstart anywhere in the population and still end up $n$ units ofsample.

Remarks.

•
If $n$ divides $N$ , then systematic sampling can be viewed asgrouping the population into $k=N/n$ strata, and picking one samplefrom each stratum. The difference between systematic sampling andstratified sampling is that in systematic sampling, only the firstsample is picked randomly, all other samples are picked based on theposition of the first pick.
•
Again, if $n\\mid N$ , one can view systematic sampling as aone-stage cluster sampling, where a primary sampling unit is definedas the set of units $x_{i},x_{i+k},\\ldots,x_{i+(n-1)k}$ in thepopulation, where $k=N/n$ and $1\\leq i\\leq k$ . In this way, thereare $k$ primary sampling units. A simple random sample of one unitcan then be drawn from these $k$ units. All of the items within theselected primary sample will then be the complete sample drawn bysystematic sampling.
•
If some linear trend exists in the population, it is a goodidea to order the population so the linearity is preserved by theorder, so that when a systematic sampling is performed, thelinearity gets carried over to the samples.
•
When ordering the population, care should be taken so that theredoes not exhibit any periodic patterns (seasonality, etc…) thatwould greatly bias the sample.

单词	SystematicSampling
释义	systematic sampling Systematic sampling is a method of sampling $n$ items from apopulation of $N$ items with the following sequence of procedures: 1. label each item in the population as $x_{i}$ , where $1\\leq i\\leq N$ , 2. randomly select one item from the first $k$ items in thepopulation, where $k=\\lceil N/n\\rceil$ , the smallest integergreater than or equal to $N/n$ , 3. pick the $k$ th item thereafter, $x_{i},x_{i+k},x_{i+2k},\\ldots$ ,until all $n$ items are picked. If $N\\equiv 0\\mod n$ , then all $n$ items can be picked beforereaching the end of the population. Otherwise, define $x_{j+N}:=x_{j}$ . Using this, we can continue to pick our sampleunits until all $n$ items are picked. The above method suggests that we do not have to make our first pickfrom among the first $k$ items. By the above definition, we canstart anywhere in the population and still end up $n$ units ofsample. Remarks. • If $n$ divides $N$ , then systematic sampling can be viewed asgrouping the population into $k=N/n$ strata, and picking one samplefrom each stratum. The difference between systematic sampling andstratified sampling is that in systematic sampling, only the firstsample is picked randomly, all other samples are picked based on theposition of the first pick. • Again, if $n\\mid N$ , one can view systematic sampling as aone-stage cluster sampling, where a primary sampling unit is definedas the set of units $x_{i},x_{i+k},\\ldots,x_{i+(n-1)k}$ in thepopulation, where $k=N/n$ and $1\\leq i\\leq k$ . In this way, thereare $k$ primary sampling units. A simple random sample of one unitcan then be drawn from these $k$ units. All of the items within theselected primary sample will then be the complete sample drawn bysystematic sampling. • If some linear trend exists in the population, it is a goodidea to order the population so the linearity is preserved by theorder, so that when a systematic sampling is performed, thelinearity gets carried over to the samples. • When ordering the population, care should be taken so that theredoes not exhibit any periodic patterns (seasonality, etc…) thatwould greatly bias the sample.
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