Stratification: how it works & consequences


I am going to put some caveats up front on this post. I am going to use stats vocab without explanation and absolutely reach above the AP Stats curriculum to make the understanding of the curriculum we teach more apparent.

With that said, let’s start with the Raina and Peter problem from 2012. Not all the problem, just the parts C and D. I have put the important info on the problem below.

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Notice the adjustment to the mean given in the formula? That is a really important thing to realize. When you stratify, you can use the known properties of the population to adjust the mean and standard deviation to get more accurate results than if you didn’t adjust.

For instance, look at the problem as it is presented. Did either learner, Peter or Rania, sample more than the other? Did either one do more work? Did either one need more copies or more time interviewing people? No.

So as it stands, both people did exactly the same amount of work to get their results. Did they get equivalent results? Well, their means are in different places, but it seems okay so far, that could be due to sampling variation.

But if we look at their standard deviation something interesting happens, Rania’s SD is found through the formula: clip_image004= 0.1979. Ok, things just changed radically.

Rania’s standard deviation is much, much smaller than Peter’s and it is ALL due to the fact she stratified the sample. This is what stratification gets you. Stratifying your sample lowers the SD of the stratified sample.

Let’s put that another way. You have two different samples; one is not stratified and one is stratified, which one has more POWER (yes, 1-beta, that power.)

If we use what we know from the AP curriculum, we know that increasing the sample size also increases power because it reduces the standard deviation. Wait, what’s that? Reducing the standard deviation of the sample increases power, so therefore stratifying a sample increases power also!

Oh snap. Things just got real.

Why is stratification part of the core of statistics? Because the practice of stratification increases the power of a sample, (and here is the big part) WITHOUT INCREASING THE SAMPLE SIZE. We don’t have to do more work to get the increase in power; we just have to think more.

But let’s extend this a little bit.

Rania chose the 60 females and 40 males in the problem because that matches the known proportion of the population. Is that the ideal sample to take to minimize the Standard Deviation?

In essence we want to graph the equation, clip_image005and find the minimum of the function. At that point we will have the best value for how many men and women Rania should have asked to maximize power and minimize the SD.

A little help from our Desmos friends and we end up with:

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And there we go. 55 women and 45 men will minimize the standard deviation and simultaneously maximize the power of the test to not make Type II errors. All for no more work than Peter did with his sample of 100 random people.

Stratification is good, has real results and is not just a word to teach as vocab. It has real value in understanding what we are doing in statistics.

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We need to have this caveat though: this only works when the characteristics of the population we are stratifying on and sampling from have different standard deviations. If the characteristics of the strata are the same, then we end up with what Peter did. We have to think hard about the independence condition part of the conditions check before we will KNOW it works.


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