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When the mean is not an accurate estimate of the Median , the list of numbers, or Frequency Distribution , is said to be Skewed . INTRODUCTION The symbol μ (Greek: mu) is used to denote the arithmetic mean of a population. If we denote a list of data by ''X'' = (''x''1, ''x''2, ..., ''x''''n''), then the sample mean is typically denoted with a horizontal bar over the variable ('''', generally enunciated "''x'' bar"). In practice, the difference between μ and ' is that μ is typically unobservable because one observes only a sample rather than the whole population, and if the sample is drawn randomly, then one may treat ', but not μ, as a Random Variable , attributing a Probability Distribution to it (the Sampling Distribution of the mean). Both are computed in the same way: : If ''X'' is a Random Variable , then the Expected Value of ''X'' can be seen as the long-term arithmetic mean that occurs on repeated measurements of ''X''. This is the content of the Law Of Large Numbers . As a result, the sample mean is used to estimate unknown expected values. Note that several other "means" have been defined, including the Generalized Mean , the Generalized F-mean , the Harmonic Mean , the Arithmetic-geometric Mean , and various Weighted Mean s. Examples
PROBLEMS WITH THE MEAN While the mean is often used to report Central Tendency , it may not be appropriate for describing Skewed Distribution s, because it is easily misinterpreted. The arithmetic mean is greatly influenced by Outlier s. These distortions can occur when the mean is different from the median. When this happens the Median may be a better description of central tendency. A classic example is Average Income . The arithmetic mean may be misinterpreted to imply that most people's incomes are higher than is in fact the case. When presented with an "average" one may be led to believe that ''most'' people's incomes are near this number. This "average" (arithmetic mean) income ''is'' higher than most people's incomes, because high income outliers skew the result higher (in contrast, the median income "resists" such skew). However, this "average" says nothing about the number of people near the median income (nor does it say anything about the modal income that most people are near). Nevertheless, because one might carelessly relate "average" and "most people" one might incorrectly assume that most people's incomes would be higher (nearer this inflated "average") than they are. For instance, reporting the "average" Net Worth in Redmond, Washington as the arithmetic mean of all annual net worths would yield a surprisingly high number because of Bill Gates . Consider the scores (1, 2, 2, 2, 3, 9). The arithmetic mean is 3.17, but five out of six scores are below this! In certain situations, the arithmetic mean is the wrong measure of central tendency altogether. For example, if a stock fell 10% in the first year, and rose 30% in the second year, then it would be incorrect to report its "average" increase per year over this two year period as the arithmetic mean (−10% + 30%)/2 = 10%; the correct average in this case is the Geometric Mean which yields an average increase per year of only 8.1%. The reason for this is that each of those percents have different starting points. If the stock starts at $30 and falls 10%, it is now at $27. If the stock then rises 30%, it is now $35.1. The arithmetic mean of those rises is 10%, but since the stock rose by $5.1 in 2 years, an Average of 8.1% would result in the final $35.1 figure = $30(1+8.1%)(1+8.1%) = $35.1 . If one used the arithmetic mean 10% in the same way, you would not get the actual increase = $36.3 . SEE ALSO
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