We have worked with a stem and leaf plot of the distribution of estimates of a population based on 100 random samples of size 10. The display is reasonably bell-shaped, with estimates occurring on both sides of 500 (the actual total number of penguins). There is a concentration of estimates around 500, with fewer estimates occurring as you move farther away from 500. Note 7
We can think of these estimates as "typical" of what you would get if you were to select another 100 samples of size 10. That is, you would generate a similar (but not exactly the same) distribution. The stem and leaf plot would also be similar, and you would expect about the same proportions of estimates to fall into the intervals we identified earlier.
Under normal circumstances, if you were asked to estimate the size of a population, you wouldn't already know the population size -- otherwise, you wouldn't need to estimate it! Also, you would not repeatedly select samples as we did in this session. In practice, you take only one sample to make your estimate based on the results in your sample.
How can you predict how accurate that one sample is likely to be? For our problem of counting penguins, we can use probability to make that prediction, using the "typical" distribution we found for the 100 samples:
Let's say that the one sample you found yielded 360 for your estimate. This is not a very good estimate, since the actual population size is 500. But since only two of our samples produced this estimate, the probability of coming up with that estimate is only about 2/100.
On the other hand, your sample might generate an estimate of 500, right on target! Your probability for this is approximately 8/100, because eight of the samples produced an estimate of 500.