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It is important to keep in mind that every statistic, not just the mean has a sampling distribution. For example, shows all possible outcomes for the range of two numbers (larger number minus the smaller number). shows the frequencies for each of the possible ranges and shows the sampling distribution of the range.
Outcome | Ball 1 | Ball 2 | Range |
---|---|---|---|
1 | 1 | 1 | 0 |
2 | 1 | 2 | 1 |
3 | 1 | 3 | 2 |
4 | 2 | 1 | 1 |
5 | 2 | 2 | 0 |
6 | 2 | 3 | 1 |
7 | 3 | 1 | 2 |
8 | 3 | 2 | 1 |
9 | 3 | 3 | 0 |
Range | Frequency | Relative Frequency |
---|---|---|
0 | 3 | 0.333 |
1 | 4 | 0.444 |
2 | 2 | 0.222 |
It is also important to keep in mind that there is a sampling distribution for various sample sizes. For simplicity, we havebeen using $N=2$ . The sampling distribution of the range for $N=3$ is shown in .
In the previous section, the population consisted of three pool balls. Now we will consider sampling distributions whenthe population distribution is continuous. What if we had a thousand pool balls with numbers ranging from 0.001 to 1.000in equal steps. (Although this distribution is not really continuous, it is close enough to be considered continuous forpractical purposes.) As before, we are interested in the distribution of means we would get if we sampled two balls andcomputed the mean of these two. In the previous example, we started by computing the mean for each of the nine possibleoutcomes. This would get a bit tedious for this problem since there are 1,000,000 possible outcomes (1,000 for the firstball x 1,000 for the second.) Therefore, it is more convenient to use our second conceptualization of sampling distributionswhich conceives of sampling distributions in terms of relative frequency distributions. Specifically, the relative frequencydistribution that would occur if samples of two balls were repeatedly taken and the mean of each sample computed.
When we have a truly continuous distribution, it is not only impractical but actually impossible to enumerate all possible outcomes. Moreover, in continuous distributions, theprobability of obtaining any single value is zero. Therefore, as discussed in our introduction to Distributions , these values are called probability densities rather than probabilities.
As we stated in the beginning of this chapter, sampling distributions are important for inferential statistics. In theexamples given so far, a population was specified and the sampling distribution of the mean and the range weredetermined. In practice, the process proceeds the other way: you collect sample data and, from these data, you estimateparameters of the sampling distribution. This knowledge of the sampling distribution can be very useful. For example, knowingthe degree to which means from different samples would differ from each other and from the population mean would give you asense of how close your particular sample mean is likely to be to the population mean. Fortunately, this information isdirectly available from a sampling distribution: The most common measure of how much sample means differ from each otheris the standard deviation of the sampling distribution of the mean. This standard deviation is called the standard error of the mean . If all the sample means were very close to the population mean, then the standard error of themean would be small. On the other hand, if the sample means varied considerably, then the standard error of the mean wouldbe large.
To be specific, assume your sample mean were 125 and you estimated that the standard error of the mean were 5 (using a methodshown in a later section). If you had a normal distribution, then it would be likely that your sample mean would be within10 units of the population mean since most of a normal distribution is within two standard deviations of the mean.
Keep in mind that all statistics have sampling distributions, not just the mean. In later sections we will be discussing the sampling distribution of the variance , the sampling distribution of the difference between means , and the sampling distribution of Pearson's correlation , among others.
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