A step-by-step sampling distribution example can help to clarify exactly how the calculations are performed and utilized. In this example, researchers are measuring the average height of workers in a factory in order to build ergonomic equipment. They wish to study the probability that the average height of a worker will correctly fit the new equipment.
Collect Data and Begin Calculations
There are 1,000 workers in the factory and the average height of the workers is 68 inches. Researchers determine the standard deviation by first taking the data values, in this case the heights, and finding the average or mean. The mean is determined by adding together all the heights and dividing this number by the number of data points, in this case 1,000 factory workers. Then the difference is calculated between each height and the mean height, and the result is squared, or multiplied by itself. The average is then calculated for these squares and the square root is taken of the average. This is the standard deviation, which in this example is 6 inches.
The researchers decide the sample size will be 50 workers, as this sample size ensures a normal distribution of data, and randomly select these workers by choosing them by anonymous worker identification numbers. This ensures that the sample of the population is without bias. If the sample of workers was chosen by sight, for example, there could be bias as taller or shorter workers may have been chosen either intentionally or unintentionally. A random sample helps verify that the predictions will represent the entire population.
Average and Standard Deviation of the Sampling Distribution
The mean or average of the sample is equal to the mean of the population which in this case is 68 inches. The mean of the entire population can be used because the population is finite. The standard deviation of the population is 6 inches, but the standard deviation of the sample must now be calculated. This standard deviation of the sample is also termed the standard error. Itís determined by dividing the populationís standard deviation by the sample size square root. In this example it would be 6 divided by 7.07, which is the square root of 50. The standard error is therefore 0.84.
After calculating the standard error of the sampling distribution, researchers must determine the confidence intervals for the population. To calculate the confidence interval, researchers must first assume that there is a normal distribution of data, which means that the data has a symmetrical distribution around the mean. Researchers know that the probability is 68% that the height of a random worker is between 68.84 and 67.16 inches; this range is calculated by subtracting and adding the standard error of 0.84 to the mean of 68. The 95% confidence interval is 66.32 to 69.68, which means that it is 95% probable that the average height of a sampled worker would be within that range.
With this information researchers can begin to build the ergonomic machinery in the factory to a height specification between 66.32 and 69.68 inches, knowing there is 95% probability that a worker would be the correct height to utilize it properly. This sampling distribution example is simple but helps to illustrate the basic calculations involved.