A Population is made up of groups that have wide variations within the groups and less variations from group to group. Which is the appropriate type of sampling method?
The question appears to have some confusion. The groups within a populations are not always fixed. A population may be divided in different groups in several alternate ways. It is for the researcher to decide if there is a need to identify these groups separately, and if so, what criteria is to be used for grouping.
However, assuming that there is a need to divide the population in groups, it should be done in such a way that the variation within the group is minimized while that between the group may be high. The question assumes the reverse regarding variation within and between the group.
Having clarified this issue, I will get down to answering the question of appropriate method of sampling for different groups within a population. There are two basic method of sampling - random and systematic. What method is appropriate in a given situation is not really dependent on the nature of groups within a population. Only requirement is that the method should be common for all the groups within the population.
Another aspect of sample selection whether the total sample is to be drawn from the complete population without differentiating between group, or to draw sample from each group separately. The First method is appropriate when differences between the groups is not high, or when it is not necessary to analyse data group wise.
It is best to draw sample form each group separately when, it is important to to do detailed analysis of each group separately, or when there is high variation between groups. When this is the selected method, then the additional question of sample size for each group need to be answered. There are three possible methods for this. First alternative is to take equal number of samples from each group. This method is appropriate when the size of each group is large and variation within groups is comparable, and sample size is decided based on desired level of accuracy and confidence.
Second method is to decide sample size separately for each group based on variation within each group - higher the variation, larger the group. This method is appropriate when the variation within groups differ from group to group. Third method is to have have sample size proportional to the group size. This method is appropriate when size of each group is not very large and variations within groups is not is comparable for different groups.