You need to know/approximate the intra-cluster correlation (say, rho). Then the design effect is calculated as deff = 1 + (average cluster size - 1)*rho. Typically, the average cluster size is 20 to 30. For example, the demographic and health survey considers 30, the multiple cluster indicator survey considers 20 on average. Using the deff value (typically 1.5 to 2.5), you can calculate the total sample size. Then the number of clusters = total sample size/average cluster size.

There are 3 major methods for determining number of clusters. Please refer to the link: https://www.datanovia.com/en/lessons/determining-the-optimal-number-of-clusters-3-must-know-methods/

The most common method is elbow method, heuristic but effective and useful.

Jannah -

Are you asking quantitative questions where you might be interested in mean responses? If so, using simple random sampling at all three stages is discussed in Cochran(1977), Sampling Techniques, 3rd ed, Wiley, section 10.8 Three-Stage Sampling, pages 285-288. (But there should be other such textbooks you might find, or online notes such as those from the Pennsylvania State University.) However, in that section, all clusters have the same number of second-stage units, each of which has the same number of third-stage units. So I think your situation is more complex than you might find written out anywhere. (Statistics Canada, I think, has or had some software that does or did such complex variance calculations.) Note, anyway, that the variance at each stage is considered, and a finite population correction is made at each stage.

From that you see that the larger the variance between your N = 50 clusters, the more of them you would need. But because of the finite population correction factor there, as n approaches N, the need becomes less urgent. You would need to know something of the variance at each stage, and the finite population correction factors.

I also wonder about your second-stage. Does it not matter which of several class-levels are drawn?

Variance is very involved here. Even getting a mean involves careful weighting when the size of the clusters, size of the schools, and size of the classrooms (number of students in each classroom) varies. Each child will not count the same because they don't each have the same chance of selection. A child in a small classroom in a school with few classrooms in a small cluster of schools would have a much bigger chance of selection (and should therefore be given a smaller weight) than a child in a large classroom in a large school in a large cluster. That former child would represent fewer children than the latter one.

Of course if you wanted the schools to count the same, and they are not the same size, then the students would not count the same. A modification of that thinking is the reason that more than once a president of the US has been elected when that candidate lost the popular vote. The smaller States were disproportionately represented. Each person's vote did not count the same.

:-(

But still, you don't want to forget your small schools.

If you are looking at qualitative information, so that weighting quantitative results will not help, it would be best here, I think, if your clusters were all the same size, schools were all the same size, and classrooms were all the same size. But since you cannot control school and classroom sizes, I would then suggest modifying your clusters so that each cluster contains roughly the same number of students.

Just some thoughts. Again, regarding your specific question, the more the clusters vary from each other, the more you need to include in your sample.

Best wishes - Jim

PS - You noted that you want to have 970 students in your overall sample. That number is usually determined after you have a way of estimating the overall variance you will have, and here that is made more complex and involves tradeoffs because of the stages. On page 243 in Cochran(1977) it even notes that variance in a cluster is a function of cluster size (aside: which is a fact that explains heteroscedasticity in regression in a very different context - Ken Brewer). So your sample size of 970 is mysterious.