**We have understood that in simple random sampling, the variance of the sample estimate of the population is: **

(a) Inversely proportional to the sample size,

(b) Directly proportional to the variability of the sampling units in the population.

We also know that the precision is defined as reciprocal of its sampling variance. Therefore as sample size increases precision increases. Apart from increasing the sample size or sampling fraction n/ N, the only way of increasing the precision of sample mean is to devise a sampling technique which will effectively reduce variance, the population heterogeneity. One such technique is Stratified Sampling.

**Stratification Means Division into Layers****:**

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**Past data or some other information related to the character under study may be used to divide the population into various groups given below: **

(i) Units within each group are as homogeneous as possible.

(ii) The group means are as widely different as possible. Thus, if we have a population consisting of N sampling units, it is divided into k relatively homogeneous mutually disjoint (non-overlapping) sub-groups, termed as strata, of sizes N_{1}, N_{2} ^{_____ }N such that N = “N if or i = 1 to k.

**Principle Advantages of Stratified Random Sampling****:**

**1. ****More Representatives: **

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In a non-stratified random sample some strata may be over represented, others may be under-represented while some may be excluded altogether. Stratified sampling ensures any desired representation in the sample of the various strata in the population. It over-rules the possibility of any essential group of the population being completely excluded in the sample. Stratified sampling thus provides a more representative cross section of the population and is frequently regarded as the most efficient system of sampling.

**2. ****Greater Accuracy: **

Stratified sampling provides estimates with increased precision. Moreover, stratified sampling enables us to obtain the results of known precision for each stratum.

**3. ****Administrative Convenience: **

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As compared with simple random sample, the stratified random samples are more concentrated geographically. Accordingly, the time and money involved in collecting the data and interviewing the individuals may be considerably reduced and the supervision of the field work could be allocated with greater ease and convenience.

**Sampling Problems****:**

Sometimes you will notice that sampling problems may differ markedly in different parts of population. For example- Literates and Illiterate people live in ordinary homes and people living in institutions, hostels, hospitals etc.

In such cases we will deal with the problem through stratified sampling by regarding the different parts of the population as stratum and tackling the problems of the survey within each stratum independently.

**Note: **

**You can allocate the sample sizes for different strata which can be done in two ways: **

1. Proportional allocation.

2. Optimum allocation.

In proportional allocation, allocation of n, the sample size of each stratum is called proportional if the sample fraction is constant for each stratum i.e.,

N_{1}= n_{2} =—————-= n_{k}

n_{1 }N_{2 }N_{k}

**Optimum allocation is another guiding principle in the determination of the n _{1} is to choose so as to: **

1. Minimize the variance (i.e., maximize the variance) of the estimate for-(i) fixed sample size, (ii) fixed cost

2. Minimize the total cost for fixed desired precision.

**Systematic Random Sampling**:

Let us consider the population size is N. Number all the sampling units from 1 to N in some order and a sample of size n is drawn in such a way that N = nk i.e. k = N/n, Where k, usually called the sampling interval and is an integer.

In systematic random sampling if a number drawn randomly, and suppose that the number drawn is i d. k and selecting the unit corresponding to this number and every k^{th }unit subsequently. Thus the systematic sample of size n will consist of the units.

Here i, I + k, I + 2k, ——————————-, I + (n – 1) k are units. The random number i is called the random start and its value determines the whole sample.

**Merits and Demerits of Systematic Random Sampling****:**

**Merits and demerits of systematic random sampling are:**

**Merits****:**

(i) Systematic sampling is operationally more convenient than simple random sampling or stratified random sampling. It saves your time and work involved.

(ii) This sampling is more efficient to simple random sample, provided the frame (the list from which you have drawn the sample units) is arranged wholly at random.

**Demerits****:**

(i) The main disadvantage of systematic sampling is that it they are not in general random samples since the requirement in merit two is rarely fulfilled.

(ii) If N is not a multiple of n, then;

a. The actual sample size is different from that required,

b. Sample mean is not an unbiased estimate of the population mean.

(iii) It does not provide the sampling error.

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