Sample Size and Statistical theory - Sample Reliability
Sample Size and Statistical theory - Sample Reliability
Of course, all samples will not generate the same value of X (or s). If another simple random sample of size 10 were taken from the population, X might
be 0.3 or 1.2 or 0.4 or whatever. The point is that X will vary from sample to sample.
Intuitively, it is reasonable to believe that the variation in X will be larger as the variance in the population, cr2, is larger. At one extreme, if there is no variation in the population, there will be no variation inX. It also is reasonable to believe that, as the size of the sample increases, the variation in X will decrease. When the sample is small, it takes only one or two extreme scores to substantially affect the sample mean, thus generating a relatively large or small X. As the sample size increases, these extreme values will have less impact when they do appear, because they will be averaged with more values. The variation in X is measured by its standard error, which is
ax 1.49
ov. = standard error of X = —= = .— = .47
Vn V10
(trx can be written simply as a). Note that the standard error of X depends on n, the sample size. If n is altered, the standard error will change accordingly, as Table 12-1 shows.
The variable X has a probability distribution, reflected in Figure 12-1. The sample mean, X, also has a probability distribution. It is customary to assume that the variation of X from sample to sample will follow the normal distribution. Figure 12-3 shows the familiar bell-shaped normal probability distribution. In other words, it indicates that X usually will be close to the population mean (u,) and that it is just as likely to be larger than u. as smaller. The top drawing in Figure 12-3 shows how the area under the normal curve is divided. The area corresponds to probability; that is, the area under the curve between two points is the probability that X will be between those two points. For example, in the middle figure, 95 percent of the area is shown. Thus, the probability that X lies within 2o\j of the population mean (u.) is 0.95. Similarly, the bottom figure shows 90 percent of the
area under the normal curve. Its interpretation is that the probability is 0.90 that X is within % of the population mean (p).5
Thus, the concept of a standard error now can be illustrated in the context of Figure 12-3. There is a 0.95 probability that X will fall within ±2 standard errors of the population mean. In our symphony example from Figure 12-1, suppose we drew 100 different samples of 10 people. About 95 percent of the resulting sample means (X) would be within ±2 standard errors (a^ = .47) of the population mean (p. = 0.3). Figure 12-3 is sometimes called a sampling distribution, since it indicates the probability of getting a particular sample mean.
Table 12-1 illustrates how the standard error of X, (CTx), decreases as the sample size gets larger. Thus, with a large sample, X will tend to be close to p, and the distribution of X will change accordingly. Figure 12-4 shows the effect of a sample-size change from 10 to 40 on the distribution of X. If the sample size were increased further, the X probability distribution would get taller and more narrow.
A source of confusion is the fact that two probability distributions are being discussed. It is very important to keep them separate. The first is the distribution of response over the population, as illustrated by Figure 12-1. The population standard deviation, cr, reflects the dispersion of this distribution. The second is the distribution of X, illustrated by Figures 12-5 and 12-4 (the dispersion of which is reflected by o\j). To conceptualize the X distribution, it is necessary to conceive of many replications of the sample
Of course, all samples will not generate the same value of X (or s). If another simple random sample of size 10 were taken from the population, X might
be 0.3 or 1.2 or 0.4 or whatever. The point is that X will vary from sample to sample.
Intuitively, it is reasonable to believe that the variation in X will be larger as the variance in the population, cr2, is larger. At one extreme, if there is no variation in the population, there will be no variation inX. It also is reasonable to believe that, as the size of the sample increases, the variation in X will decrease. When the sample is small, it takes only one or two extreme scores to substantially affect the sample mean, thus generating a relatively large or small X. As the sample size increases, these extreme values will have less impact when they do appear, because they will be averaged with more values. The variation in X is measured by its standard error, which is
ax 1.49
ov. = standard error of X = —= = .— = .47
Vn V10
(trx can be written simply as a). Note that the standard error of X depends on n, the sample size. If n is altered, the standard error will change accordingly, as Table 12-1 shows.
The variable X has a probability distribution, reflected in Figure 12-1. The sample mean, X, also has a probability distribution. It is customary to assume that the variation of X from sample to sample will follow the normal distribution. Figure 12-3 shows the familiar bell-shaped normal probability distribution. In other words, it indicates that X usually will be close to the population mean (u,) and that it is just as likely to be larger than u. as smaller. The top drawing in Figure 12-3 shows how the area under the normal curve is divided. The area corresponds to probability; that is, the area under the curve between two points is the probability that X will be between those two points. For example, in the middle figure, 95 percent of the area is shown. Thus, the probability that X lies within 2o\j of the population mean (u.) is 0.95. Similarly, the bottom figure shows 90 percent of the
area under the normal curve. Its interpretation is that the probability is 0.90 that X is within % of the population mean (p).5
Thus, the concept of a standard error now can be illustrated in the context of Figure 12-3. There is a 0.95 probability that X will fall within ±2 standard errors of the population mean. In our symphony example from Figure 12-1, suppose we drew 100 different samples of 10 people. About 95 percent of the resulting sample means (X) would be within ±2 standard errors (a^ = .47) of the population mean (p. = 0.3). Figure 12-3 is sometimes called a sampling distribution, since it indicates the probability of getting a particular sample mean.
Table 12-1 illustrates how the standard error of X, (CTx), decreases as the sample size gets larger. Thus, with a large sample, X will tend to be close to p, and the distribution of X will change accordingly. Figure 12-4 shows the effect of a sample-size change from 10 to 40 on the distribution of X. If the sample size were increased further, the X probability distribution would get taller and more narrow.
A source of confusion is the fact that two probability distributions are being discussed. It is very important to keep them separate. The first is the distribution of response over the population, as illustrated by Figure 12-1. The population standard deviation, cr, reflects the dispersion of this distribution. The second is the distribution of X, illustrated by Figures 12-5 and 12-4 (the dispersion of which is reflected by o\j). To conceptualize the X distribution, it is necessary to conceive of many replications of the sample
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