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Mean and Standard Deviation of the Sample Mean for Beginning Statistics

By — McGraw-Hill Professional
Updated on Aug 12, 2011

Practice problems for these concepts can be found at:

Since the sample mean has a distribution, it is a random variable and has a mean and a standard deviation. The mean of the sample mean is represented by the symbol and the standard deviation of the sample mean is represented by . The standard deviation of the sample mean is referred to as the standard error of the mean. Example 7.9 illustrates how to find the mean of the sample mean and the standard error of the mean.

EXAMPLE 7.9   In Example 7.7, the sampling distribution of the mean shown in Table 7.7 was obtained.

The mean of the sample mean is found as follows:

    = 33 × .1 + 34 × .1 + 35 × .2 + 36 × .2 + 37 × .2 + 38 × .1 + 39 × .1 = 36

The variance of the sample mean is found as follows:

.

The relationship between the mean of the sample mean and the population mean is expressed by

    μx = μ             (7.2)

The relationship between the variance of the sample mean and the population variance is expressed by formula (7.3), where N is the population size and n is the sample size.

    .

EXAMPLE 7.10   In Example 7.7, the population consisting of the five cities with the most African- American-owned businesses was introduced. The population mean, μ, is equal to 36 and the variance, σ2, is equal to 18. In Example 7.9, the mean of the sample mean, μx , was shown to equal 36 and the variance of the sample mean, , was shown to equal 3. It is seen that μ = μx = 36, illustrating formula (7.2). To illustrate formula (7.3), note that

    .

The standard error of the mean is found taking the square root of both sides of formula (7.3), and is given by

    .

The term is called the finite population correction factor. If the sample size n is less than 5% of the population size, i.e., n < .05N, the finite population correction factor is very near one and is omitted in formula (7.4). If n < .05N, the standard error of the mean is given by

    .

EXAMPLE 7.11   The mean cost per county in the United States to maintain county roads is $785 thousand per year and the standard deviation is $55 thousand. Approximately 4% of the counties are randomly selected and the mean cost for the sample is computed. The number of counties is 3143 and the sample size is 125. The standard error of the mean using formula (7.4) is:

    .

The standard error of the mean using formula (7.5) is:

    .

Ignoring the finite population correction factor in this case changes the standard error by a small amount.

Practice problems for these concepts can be found at:

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