This is a technique to break down the variation of a random variable into useful components (called stratum) in order to decrease experimental variation and increase accuracy of results. It has been found that a more accurate estimate of population mean m can often be obtained by taking measurements from naturally occurring subpopulations and combining the results using weighted averages. For example, suppose an accurate estimate of the mean weight of sixth grade students is desired for a large school system. Suppose (for cost reasons) we can only take a random sample of m = 100 students, Instead of taking a simple random sample of 100 students from the entire population of all sixth grade students, we use stratifi­ed sampling as follows. The school system under study consists three large schools. School A has N₁ = 310 sixth grade students, School B has N₂ = 420 sixth grade students, and School C has N₃ = 516 sixth grade students. This is a total population of 1246 sixth grade students in our study and we have strata consisting of the 3 schools. A preliminary study in each school with relatively small sample size has given estimates for the sample standard deviation s of sixth grade student weights in each school. These are shown in the following table.

How many students should we randomly choose from each school for a best estimate m for the population mean weight? A lot of mathematics goes into the answer. Fortunately, Bill Williams of Bell Laboratories wrote a book called A Sampler on Sampling (John Wiley and Sons, publisher), which provides an answer. Let n₁ be the number of students randomly chosen from School A, n₂ be the number chosen from School B, and n₃ be the number chosen from School C. This means our total sample size will be m = n₁ + n₂ + n₃. What is the formula for n_i? A popular and widely used technique is the following.

The n_i are usually not whole numbers, so we need to round to the nearest whole number. This formula allocates more students to schools that have a larger population of sixth graders and/or have larger sample standard deviations. Remember, this is a popular and widely used technique for stratifi­ed sampling. It is not an absolute rule. There are other methods of strati­fied sampling also in use. In general practice, according to Bill Williams, the use of naturally occurring strata seems to reduce overall variability in measurements by about 20% compared to simple random samples taken from the entire (un-stratifi­ed) population.

Now suppose you have taken a random sample size n_i from each appropriate school and you got a sample mean weight x̅_i from each school. How do you get the best estimate for population mean weight m of the all 1246 students? The answer is, we use a weighted average.

This is an example with three strata. Applications with any number of strata can be solved in a similar way with obvious extensions of formulas.

(a) Compute the size of the random samples n₁, n₂, n₃ to be taken from each school. Round each sample size to the nearest whole number and make sure they add up to m = 100.

(b) Suppose you took the appropriate random sample from each school and you got the following average student weights:

x̅₁ = 82 lb, x̅₂ = 115 lb, x̅₃ = 90 lb. Compute your best estimate for the population mean weight m.

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