1. 7
Chapter
The sampling distribution of the sample mean
Chapter outline
1. Sampling error; the need for sampling distributions
2. The mean and standard deviation of the sample mean
3. The sampling distribution of the sample mean
The sampling distribution of a statistic is the distribution of the statistic.
1. Sampling error; the need for sampling distributions
A sample to acquire information about a population is often preferable to conducting a
census.
Sampling is less costly and can be done more quickly than a census; it is often the only
practical way to gather information.
A sample provides data for only a portion of an entire population.
We cannot expect the sample to yield perfectly accurate information about the population.
A certain amount of error is known as sampling error.
Sampling error:
Sampling error is error resulting from a using to estimate a population characteristic.
The distribution of a statistic
All of the possible observations of the statistics for samples of a given size is called the
sampling distribution of the statistic.
The sampling distribution of the sample mean, that is x.
Sampling distribution of the sample mean
For a variable x and a given sample size, the distribution of the variable x is called the
sampling distribution of the sample mean.
The following terms and phrases are synonymous.
Sampling distribution of the sample mean

2. Distribution of the variable x
Distribution of all possible sample means of a given sample size
The sampling distribution of the sample mean with an example that is both realistic and
concrete is difficult because even for moderately large populations the number of possible
samples is enormous, thus prohibiting an actual listing of the possibilities.
Sample size and sampling error
The larger the sample size, the smaller the sampling error tends to be in estimating a
population mean, µ, by a sample mean, x.
2. The mean and standard deviation of the sample mean
Mean of the sample mean
For samples of size n, the mean of the variable x equals the mean of the variable under
consideration. In symbols, µ x = µ.
Standard deviation of the sample mean
For samples of size n, the standard deviation of the variable x equals the standard deviation
of the variable under consideration divided by the square root of the sample size. In
symbols, σ x = σ / n