# Central tendency and measures of variat

The most common case is maximum likelihood estimationwhere the maximum likelihood estimate MLE maximizes likelihood minimizes expected surprisalwhich can be interpreted geometrically by using entropy to measure variation: the MLE minimizes cross entropy equivalently, relative entropyKullbackâ€”Leibler divergence.

Bill Gates walks into the bar with a talking parrot perched on his shoulder. One of the median's advantages is that it is not sensitive to outliers. Observations that are significantly larger or smaller than the others in a sample can impact some statistical measures in such a way as to make them highly misleading, but the median is immune to them.

If scored in the 5th percentile on a test, maybe that subject isn't for you. Symmetrical vs. One of the median's advantages is that it is not sensitive to outliers. In other words, it doesn't matter if the biggest number is 20 or 20,; it still only counts as one number. Percentiles A percentile is a number below which a certain percent of the distribution falls. Sample 2 has no variability all scores are exactly the same , whereas Sample 1 has relatively more one case varies substantially from the other four. Unlike the median, however, the mean is sensitive to outliers. This is not so much the case for nominal and ordinal variables. As an example of why the mean might not be the best measure of central tendency for a skewed distribution, consider the following passage from Charles Wheelan's Naked Statistics: Stripping the Dread from the Data : "The mean, or average, turns out to have some problems, namely, that it is prone to distortion by "outliers," which are observations that lie farther from the center. In statistical notation, we would write it out as follows: In that equation, is the mean, X represents the value of each case and N is the total number of cases. The 0-"norm" is not convex hence not a norm. If the mean is greater than the median, the distribution is said to be positively skewed.

In other words, one extraordinarily high or low value in your dataset can dramatically raise or lower the mean. The mean, often shown as an x or a y variable with a line over it pronounced either "x-bar" or "y-bar"is the sum of all the scores divided by the total number of scores.

Thus standard deviation about the mean is lower than standard deviation about any other point, and the maximum deviation about the midrange is lower than the maximum deviation about any other point.

Fifty percent of the observations fall below it.

Information geometry[ edit ] The notion of a "center" as minimizing variation can be generalized in information geometry as a distribution that minimizes divergence a generalized distance from a data set. Unlike the median, however, the mean is sensitive to outliers.

Correspondingly, the mode is not unique â€” for example, in a uniform distribution any point is the mode.

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