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EC315 Equations

EC315 Equations - MEASURES OF CENTRAL TENDENCY What is the...

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MEASURES OF CENTRAL TENDENCY What is the measure of center? It is the value of the middle of the data set. How do we find it? In order to answer that question we need to discuss the four different measures of center. 1. The mean . This term for measure is sometimes called the average but please know that all four measures are averages. To determine we add all of the data values in the set and divide by the number of values. The upper case Greek letter sigma, Σ , is used for summation. To readily distinguish between a sample and a population we use the lower case n for the size of a sample set, and the upper case N for the size of a population . We will continue to use the variable x to denote the observation/data values. (Sometimes the sample mean, is referred to as x-bar) The μ symbol for population mean is the lower case Greek letter mu. Wow! That one value really made the mean of the population different from the mean of the sample! Our center measure is greatly affected by extreme values when using the mean to describe data. We call extreme values outliers . They may be correct values reflecting an abnormal characteristic, or they could be observation or recording errors. If the outlier does not affect the statistic, it is said to be robust . The mean is not robust; it is affected by outliers. Because the mean is very sensitive to outliers we sometimes use a trimmed mean . To determine we arrange the data values in ascending order and then delete the bottom 10% (or some other specified percentage) of the values and the same top percentage amount of
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values. We then follow the usual procedure to find the mean of this revised data set. It is imperative that we identify the result as a trimmed mean with the percentage that was trimmed when we report our findings. Example: Find the trimmed mean for the following data set: {3, 5, 5, 9, 13, 301} Since this data set is so small, in order to trim one number from each end the percentage will be 16% (1 out of 6 = 0.16666667), The new trimmed data set will be: {5, 5, 9, 13}. The trimmed mean of the population is much closer to the sample mean, and probably a much better descriptive statistic of the data set than before with the outlier. Notice the medians for both sets are about the same. The extreme value in the second set did not affect the MD. The value of the last number in the data set could be any value > 9 with the same result for MD. This tells us the median is robust; it is not affected by outliers. 3.
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EC315 Equations - MEASURES OF CENTRAL TENDENCY What is the...

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