| 
  • If you are citizen of an European Union member nation, you may not use this service unless you are at least 16 years old.

  • You already know Dokkio is an AI-powered assistant to organize & manage your digital files & messages. Very soon, Dokkio will support Outlook as well as One Drive. Check it out today!

View
 

arithmetic mean

Page history last edited by PBworks 15 years, 8 months ago

arithmetic mean ="Average"

 

 

The arithmetic mean is what students are taught very early to call the "average".

 

 

In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list.

 

 

Problems with some uses of the mean

While the mean is often used to report central tendency, it may not be appropriate for describing skewed distributions, because it is easily misinterpreted. The arithmetic mean is greatly influenced by outliers. 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 Medina, 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 8.2 %. 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 8.2 % would result in the final $35.1 figure [$30(1-10 %)(1+30 %) = $30(1+8.2 %)(1+8.2 %) = $35.1]. If one used the arithmetic mean 10 % in the same way, you would not get the actual increase [$30(1+10 %)(1+10 %) = $36.3].

 

Particular care must be taken when using cyclic data such as phases or angles. Taking the arithmetic mean of 1 degree and 359 degrees yields a result of 180 degrees, whereas 1 and 359 are both adjacent to 360 degrees which may be a more correct average value. In general application such an oversight will lead to the average value artificially moving towards the middle of the numerical range.

 

 

 

See Also

geometric mean

 

 

External links

*Calculations and comparisons between arithmetic and geometric mean between two numbers

*Mean or Average

*Rational Mean

Comments (0)

You don't have permission to comment on this page.