How to Calculate the Distribution of the Mean

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The sampling distribution of the mean is an important concept in statistics and is used in several types of statistical analyses. The distribution of the mean is determined by taking several sets of random samples and calculating the mean from each one. This distribution of means does not describe the population itself--it describes the population mean. Thus, even a highly skewed population distribution yields a normal, bell-shaped distribution of the mean.

Take several samples from a population of values. Each sample should have the same number of subjects. Even though each sample contains different values, on average they resemble the underlying population.

Calculate the mean of each sample by taking the sum of the sample values and dividing by the number of values in the sample. For example, the mean of the sample 9, 4 and 5 is (9 + 4 + 5) / 3 = 6. Repeat this process for each of the samples taken. The resulting values are your sample of means. In this example, the sample of means is 6, 8, 7, 9, 5.

Take the average of your sample of means. The average of 6, 8, 7, 9 and 5 is (6 + 8 + 7 + 9 + 5) / 5 = 7.

The distribution of the mean has its peak at the resulting value. This value approaches the true theoretical value of the population mean. The population mean can never be known because it is practically impossible to sample every member of a population.

Calculate the standard deviation of the distribution. Subtract the average of the sample means from each value in the set. Square the result. For example, (6 - 7)^2 = 1 and (8 - 6)^2 = 4. These values are called squared deviations. In the example, the set of squared deviations is 1, 4, 0, 4 and 4.

Add the squared deviations and divide by (n - 1), the number of values in the set minus one. In the example, this is (1 + 4 + 0 + 4 + 4) / (5 - 1) = (14 / 4) = 3.25. To find the standard deviation, take the square root of this value, which equals 1.8. This is the standard deviation of the sampling distribution.

Report the distribution of the mean by including its mean and standard deviation. In the example above, the reported distribution is (7, 1.8). The sampling distribution of the mean always takes a normal, or bell-shaped, distribution.