In a proportional stratified sampling plan the number of items drawn from each stratum is proportional to the size of the strata. 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 500 is drawn the desired proportional sample may be obtained in the following manner.

Form 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

Form stratum four 5, 000(0.30) = 1,500 items

From stratum five 5,000 (0.25) = 1,250 items
                                             
Total = 5, 00 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 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 representation to one with small variation.

In disproportional stratified sampling an 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 the variation of the measurements differs greatly form stratum to stratum.

You’re given the following data of the number of lectures riders and professors in a university.

Length of service	Lecturers	Readers	Professors	Total
Less than 5 yrs	2000	250	50	2300
5 – 10 yrs	3000	220	80	3300
10 – 15 yrs.	1500	170	30	1700
More than 15 yrs.	880	80	40	1000
Total	7380	720	200	8302

Work out how many lecturers, readers and professors would be selected from each category if we follow stratified proportionate sampling method and take 10% of the universe equivalent to the sample size 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: - the sample size is 10% of the universe hence 830 persons would be selected in the sample. Since 12 strata reformed 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 5 yrs.	200	25	5	230
5-10 yrs.	300	22	8	330
10-15 yrs	150	17	3	170
More than 15 yrs.	88	8	4	100
Total	738	72	20	830

In the second case also the size of sample is 830 but the lecturer’s readers and professors are to be in the ratio of 5: 3: 2 of the sample we take 830 x 5 / 10 = 415 lecturers 830x3x / 10 = 249 readers and 830 x 2 / 10 = 166 professors. Since the weight age to length of service is 4: 3: 2: 1 the number selected from each category shall be as given in the table below:

Length of service	Lecturers	Readers	Professors	Total
Less than 5 yrs.	415x4x/10 = 166	249x4 / 10 = 99.6 or 100	166 x4 / 10 = 66.4 or 66	332
5 – 10 years	415 x 3 / 10 = 124.5 or 124	249 x 3 / 10 = 74.7 or 74	166 x 3 / 10 = 49.80 or 50	248
10-15 years	415 x 2 /10 = 83	249 x 2 / 10 = 49.8 or 50	166 x 2 / 10 = 33.2 or 33	166
Above 15 years	415 x 1 / 10 = 41.5 or 42	249 x 1 / 10 = 24.9 or 25	166 x 1 / 10 = 16.6 or 17	84
Total	415	249	166	830

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