Part 1: Variability of Sample Means in Finite versus Infinite Populations
This part may not help you very much if you havenít learned about this formula yet: .
But if you have learned about this formula, then let me tell you that itís false if the population is finite. It was derived under the assumption that the population is infinite. When do mere mortals ever encounter genuinely infinite populations? WHAT GOOD IS THIS FORMULA? Well, we have said elsewhere that if the population is large enough (compared to the sample) then it may as well be infinite. Letís see why, with an example or two.
Suppose you have a population of 200 objects, and youíve sampled 25 of them. You measure something on all 25 objects and find the mean of the 25 measurements. Letís say the true population standard deviation is 10 (just to have a number). Question: What is the standard deviation of sample means, when the sample size is 25?
The formula at the top of this page says the standard deviation of sample means when is .
But this is not the right formula to use when the population is finite. Our population has only 200 objects in itódefinitely not infinitely many!
The right formula to use turns out to be , where , the population size.
So the true standard deviation of sample means (for this particular population, when ) is , not 2!
Now, some of you will say, ďWho cares? 1.875 is not very different than 2.Ē But in some situations (say, in chapters 6, 7, 8, etc.) the difference will be big enough to matter. Big enough to influence some decision you make. Big enough to possibly lead you to a costly mistake, such as killing someone with a drug that should not have been used, or whatever.
Letís do another example. This time, the population size is . Letís use a true standard deviation of 10 and a sample size of 25 measurements, as we did in the previous example. The true standard deviation of sample means (for this particular population, when ) is
, close enough to 2 to be good enough. The point here is,
In other words, if the population is large enough, it may as well be infinite, where these calculations are concerned.
The bad news is, we will continue to use the formula in this class, even when it does not actually apply.
The good news is, those of you that go on to take research methods classes will use, in those classes (and in your professions!), the right formulas when your populations are finite. The better news is that most of the populations you will use will be so large they might as well be infinite, so thereís no need to worry (most of the time). And which formula is easier to use, or ?
You see, the arithmetic is simpler if we assume the population is infinite! Weird, huh?
Part 2: Population Size, Sample Size, and Random Sampling
Suppose your company receives a shipment of 50 widgets, of which 10 are defective. (Thatís 10 / 50 = 20% defective. Thatís a lot!) You select three widgets at random to test. What is the probability that all three widgets are defective?
Small population, sample without replacement: Well, it depends. Letís assume you sample without replacement, that is, once youíve selected a widget, you donít put it back in the box (replace it) before selecting the next widget. We need the probability that all three widgets are defective, that is, the first one is defective AND the second one is defective AND the third one is defective. I emphasize the word ďandĒ here, because it tells us what arithmetic to use. Probability Rule #5 says that ďandĒ means ďmultiply,Ē when events are independent. Sampling without replacement qualifies. SoÖ
The probability that the first widget we pick is defective, is 10/50. Having selected a defective widget, there are only 9 defectives left to pick, and only 49 widgets total to pick them from, so the probability of the second widget being defective is 9/49. Having selected two defective widgets, there are 8 defectives left to pick, and 48 widgets total remain. So the probability of the third one being defective is 8/48. ďAndĒ means ďmultiplyĒ so we do, and get , give or take.
Small population, sample with replacement: Now letís assume that (for whatever reason) you replace each widget you test, before selecting the next widget. This is called sampling with replacement. Again, we need the probability that all three widgets are defective, that is, the first one is defective AND the second one is defective AND the third one is defective. I emphasize the word ďandĒ here, because it tells us to multiply (as opposed to adding or something). SoÖ
The probability that the first widget we pick is defective, is 10/50. Having selected and tested a defective widget, we put it back in the box. That means there are 10 defective widgets in the box, out of a total of 50 widgets to choose from. So the probability of the second widget being defective is also 10/50. We test the second widget, replace it, and pick a third widget. Again, 10 of the 50 widgets in the box are defective, so the probability of the third widget being defective is also 10/50. As before, we multiply and get , which is not ; itís about 31% higher than 0.00612.
(You may not be impressed by this, but the effect becomes more dramatic as the number of widgets tested becomes larger. For example, if you select 10 widgets to test, and select them without replacement, the probability that all 10 are defective is 9.73 x 10-11, but if you sample with replacement, the probability of all ten being defective is about 1000 times higher: 1.024 x 10-7.)
Large population, sample without replacement: Now letís say there 5000 widgets in the shipment, and youíre going to select 3 to be tested. To make a fair comparison with the two previous examples, letís say 20% of the widgets are defective. 20% 0f 5000 is 1000, so 1000 out of 5000 widgets are defective. Sample the three without replacement. The probability that all three are defective is .
Large population, sample with replacement: If you sample your three widgets with replacement, when 1000 out of 5000 are defective, the probability that all three are defective is .
Amazing! Both times we sampled with replacement, we got the same probability! But wait! This time, the difference between sampling with replacement and sampling without replacement is much smaller than before: 0.00800 is only about 0.25% higher than 0.00798.
Why is the difference so much less than it was when there were only 50 widgets? Because a population of 5000 is large compared to a sample of 3. (3 x 100 = 300; by our rule of thumb, a population of 300 is large compared to a sample of size 3!) But a population of 50 is not large compared to a sample of size 3.
As the size of the population gets bigger compared to the size of the sample, the difference between sampling with replacement and sampling without replacement gets smaller and smaller. So if the population is infinite and the sample is finite, the difference between sampling with replacement and sampling without replacement is approximately ZERO. In other words,
And you might as well compute your probabilities as though you were sampling with replacement (independently) even though you sample without replacement (dependently) in practice. Once again, the arithmetic is simpler if we assume the population is infinite!
(Note: The difference still becomes more dramatic as the number of widgets tested becomes larger. For example, if 1000 out of 5000 widgets are defective, and you select 10 widgets to test, and select them without replacement, the probability that all 10 are defective is 9.88 x 10-8, but if you sample with replacement, the probability of all ten being defective is still 1.024 x 10-7, which is about 3.7% higher, as opposed to 0.25% higher.)
Are there any truly, actually infinite populations? I donít know. I doubt any mortal knows. But for what itís worth, I believe that there is a huge variety of potentially infinite populations. So we might as well assume they are infinite. Or something.
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