## 3.1.10.2 Proportional and Disproportional Stratified Sample

 3.1.10.2 Proportional and Disproportional Stratified Sample

In a proportional stratified sampling plan, the number of items drawn from each stratum is proportional to the size of the stratum. For example, if the population is divided into five groups, their respective sizes being 10,15,20,30 and 25 per cent of the population and a sample of 5,000 is drawn, the desired proportional sample may be obtained in the following manner.

 From stratum one 5,000 (0.10) = 500 items From stratum two 5,000 (0.15) = 750 items From stratum three 5,000 (0.20) = 1,000 items From stratum four 5,000 (0.30) = 1,500 items From stratum five 5,000 (0.25) = 1,250 items Total = 5,000 items

Proportional stratification yields a sample that represents the universe with respect to the proportion in each stratum in the population. This procedure is satisfactory if there is no great difference in dispersion from stratum to stratum. But it is certainly not the most efficient procedure, especially when there is considerable variation in different strata. This indicates that in order to obtain maximum efficiency in stratification, we should assign greater representation to a stratum with a large dispersion and smaller representations to one with small variation.

In disproportional stratified sampling equal number of cases is taken from each stratum regardless of how the stratum is represented in the universe. Thus, in the above example, an equal number of items (1,000) from each stratum may be drawn. In practice disproportional sampling is common when sampling from a highly variable universe, wherein the variation of the measurements differs greatly from stratum to stratum.

The following is that information about data the number of lecturers, readers and professors in a University

 Length of service No of Asst.prof. No. of Assoc. prof. No. of Professors Total Less than 5 yrs. 2,000 250 50 2,300 5-10 yrs 3,000 220 80 3,300 10-15 yrs 1,500 170 30 1,700 More than 15 yrs 880 80 40 1,000 Total 7,380 720 200 8300

Work out how many lecturers, readers and Professors would be selected from each category if (i) we follow stratified proportionate sampling method and take 10% of the universe equivalent to the sample size. (ii) if the size of the sample is 10% of the universe but the lecturers, readers and professors are to be in the ratio of 5:3:2 and weight age of the length of service is to be in the ratio of 4:3:2:1.

Solution. (i) The sample size is 10% of the universe hence 830 persons would be selected in the sample. Since 12 strata are formed and we want to follow proportionate stratified sampling method, we will take 10% from each stratum. The number of persons selected shall be as follows:

 Length of service Lecturers Readers Professors Total Less than yrs. 200 25 5 230 5-10 yrs 300 22 8 330 10-15 yrs 150 17 3 170 More than yrs 88 8 4 100 Total 738 72 20 830

(ii) In the second case also the size of sample is 830 but the lecturers, readers and professors are to be in the ratio of 5:3:2 of the sample, i.e., we take 415 lectures, 249 readers and 166 professer. Then the length of service weightage in the ratio of 4:3:2:1 and to be taken as follows.

 Length of service Lecturers Readers Professors Total Less than 5 years 332 166 100 66 5-10 249 125 74* 50 10-15 166 83 50 33 Above 15 83 41 25 17 Total 415 249 166 830

Merits. 1. More representative. Since the population is first divided into various strata and then a sample is drawn from each stratum there is a little possibility of any essential group of the population being completely excluded. A more representative sample is thus secured. C.J. Grohman has rightly pointed out that this type of sampling balances the uncertainty of random sampling against the bias of deliberate selection.

2. Greater accuracy. Stratified sampling ensures greater accuracy. The accuracy is maximum if each stratum is formed so that it consists of uniform or homogenous items.

3. Greater geographical concentration. As compared with random sample, stratified samples can be more concentrated geographically, i.e. , the units from  different strata may be selected in such a way that all of them are localized in one geographical area. This would greatly reduce the time and expenses of interviewing.

Limitations. 1. Utmost care must be exercised in dividing the population into various strata. Each stratum must contain, as far as possible, homogenous items as otherwise the results may not be reliable. If proper stratification of the population is not done, the sample may have the effect of bias.

2. The items from each stratum should be selected at random. But this may be difficult to achieve in the absence of skilled sampling supervisors and a random selection within each stratum may not be ensured.

3. Because of the likelihood that a stratified sample will be more widely distributed geographically than a simple random sample cost per observation may be quite high.