The Mean, Standard Deviation, and Sampling Distribution of the Sample Mean
This section gives several concrete examples of calculating the exact distributions of the sample mean. The corresponding means and standard deviations are computed for demonstration based on these distributions. Next, it discusses sampling distributions of sample means when the sample size is large. It also considers the case when the population is normal. Finally, it uses the central limit theorem for large sample approximations.
The Sampling Distribution of the Sample Mean
- To learn what the sampling distribution of is when the sample size is large.
- To learn what the sampling distribution of is when the population is normal.
The Central Limit Theorem
In Note 6.5 "Example 1" in Section 6.1 "The Mean and Standard Deviation of the Sample Mean" we constructed the probability distribution of the sample mean for samples of size two drawn from the population of four rowers. The probability distribution is:
Figure 6.1 "Distribution of a Population and a Sample Mean" shows a side-by-side comparison of a histogram for the original population and a histogram for this distribution. Whereas the distribution of the population is uniform, the sampling distribution of the mean has a shape approaching the shape of the familiar bell curve. This phenomenon of the sampling distribution of the mean taking on a bell shape even though the population distribution is not bell-shaped happens in general. Here is a somewhat more realistic example.
Figure 6.1 Distribution of a Population and a Sample Mean
Suppose we take samples of size, , , or from a population that consists entirely of the numbers and , half the population , half , so that the population mean is . The sampling distributions are:
Histograms illustrating these distributions are shown in Figure 6.2 "Distributions of the Sample Mean".
Figure 6.2 Distributions of the Sample Mean
What we are seeing in these examples does not depend on the particular population distributions involved. In general, one may start with any distribution and the sampling distribution of the sample mean will increasingly resemble the bell-shaped normal curve as the sample size increases. This is the content of the Central Limit Theorem.
The Central Limit Theorem
For samples of sizeor more, the sample mean is approximately normally distributed, with mean and standard deviation , where is the sample size. The larger the sample size, the better the approximation.
The Central Limit Theorem is illustrated for several common population distributions in Figure 6.3 "Distribution of Populations and Sample Means".
Figure 6.3 Distribution of Populations and Sample Means
The dashed vertical lines in the figures locate the population mean. Regardless of the distribution of the population, as the sample size is increased the shape of the sampling distribution of the sample mean becomes increasingly bell-shaped, centered on the population mean. Typically by the time the sample size is
The importance of the Central Limit Theorem is that it allows us to make probability statements about the sample mean, specifically in relation to its value in comparison to the population mean, as we will see in the examples. But to use the result properly we must first realize that there are two separate random variables (and therefore two probability distributions) at play:
- , the measurement of a single element selected at random from the population; the distribution of is the distribution of the population, with mean the population mean and standard deviation the population standard deviation ;
- , the mean of the measurements in a sample of size ; the distribution of is its sampling distribution, with mean and standard deviation .
Letbe the mean of a random sample of size drawn from a population with mean and standard deviation .
a. Find the mean and standard deviation of
b. Find the probability that assumes a value between and .
c. Find the probability that assumes a value greater than .
a. By the formulas in the previous section
b. Since the sample size is at least, the Central Limit Theorem applies: is approximately normally distributed. We compute probabilities using Figure 12.2 "Cumulative Normal Probability" in the usual way, just being careful to use and not when we standardize:
Note that if in Note 6.11 "Example 3" we had been asked to compute the probability that the value of a single randomly selected element of the population exceeds
The numerical population of grade point averages at a college has meanand standard deviation . If a random sample of size is taken from the population, what is the probability that the sample mean will be between and ?
The sample meanhas mean and standard deviation , so
Normally Distributed Populations
The Central Limit Theorem says that no matter what the distribution of the population is, as long as the sample is "large," meaning of sizeor more, the sample mean is approximately normally distributed. If the population is normal to begin with then the sample mean also has a normal distribution, regardless of the sample size.
For samples of any size drawn from a normally distributed population, the sample mean is normally distributed, with mean
The effect of increasing the sample size is shown in Figure 6.4 "Distribution of Sample Means for a Normal Population".
Figure 6.4 Distribution of Sample Means for a Normal Population
A prototype automotive tire has a design life ofmiles with a standard deviation of miles. Five such tires are manufactured and tested. On the assumption that the actual population mean is miles and the actual population standard deviation is miles, find the probability that the sample mean will be less than miles. Assume that the distribution of lifetimes of such tires is normal.
For simplicity we use units of thousands of miles. Then the sample meanhas mean and standard deviation . Since the population is normally distributed, so is , hence
That is, if the tires perform as designed, there is only about a
An automobile battery manufacturer claims that its midgrade battery has a mean life ofmonths with a standard deviation of months. Suppose the distribution of battery lives of this particular brand is approximately normal.
a. On the assumption that the manufacturer's claims are true, find the probability that a randomly selected battery of this type will last less than
b. On the same assumption, find the probability that the mean of a random sample of such batteries will be less than months.
a. Since the population is known to have a normal distribution
b. The sample mean has meanand standard deviation . Thus