Elementary Statstics

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.