sci.math.num-analysis #43376 (19 + 365 more)                          [1]+-[1]
From: "Bean" <thitt@igateway.com>                                        |-[1]
[1] Mean, Median, Average ?                                              |-[1]
Date: Wed Jul 15 09:40:51 EDT 1998                                       |-[1]

Could someone explain the difference between
Mean, Median, and Average ?

Thank you,                     Bean, thitt@igateway.com

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sci.math.num-analysis #43378 (18 + 365 more)                          (1)+-[1]
From: user923005@aol.com (User923005)                                    |-[1]
[1] Re: Mean, Median, Average ?                                          |-[1]
Date: Wed Jul 15 11:24:27 EDT 1998                                       \-[1]

Mean and average are usually defining the same thing, the arithmetic mean.
this is the sum of all items divided by the count of all items.  Median is the
middle item in a sorted list.  If the number of elements is even it is the
average of the middle two.
Look here for precise definitions:
http://www.astro.virginia.edu/~eww6n/math/math0.html
--
C-FAQ ftp sites: ftp://ftp.eskimo.com ftp://rtfm.mit.edu
Hypertext C-FAQ: http://www.eskimo.com/~scs/C-faq/top.html
C-FAQ Book: ISBN 0-201-84519-9.
Want Software?  Algorithms?  Pubs? http://www.infoseek.com

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sci.math.num-analysis #43384 (17 + 365 more)                          (1)+-(1)
From: Lynn Killingbeck <killbeck@phoenix.net>                            |-[1]
Reply-To: killbeck@phoenix.net                                           |-[1]
[1] Re: Mean, Median, Average ?                                          |-[1]

Mean (arithmetic): add up all the values, and divide by the number of
items. The arithmetic mean of 1,2,3,4,1000 is (1010/5)=202.

Mean (geometric): multiply the magnitudes of all the values, and take
the n_th root. The geometric mean of 1,2,3,4,1000 is 24000^(1/5),
about 7.516... The signs generally get lost, if they are not all the
same (e.g., the geometric mean of -1,+1 is just 1).

Mean (there are others, such as the harmonic mean).

Median: the middle value. For an even number of items, it is usual to
take the average of the middle two items. The median of 1,2,3,4,1000 is 3.

Average: sometimes, implicitly, the arithmetic mean. More ofter, it is
purposely left vague by the user, with the intent of confusing and/or
misleading the reader. Beware if that's the word used! See, for example,
'How to Lie With Statistics' by Huff, and '200% of Nothing' by
Dewdney. Note that any of 202, 7.5, and 3 are legitimately called an
'average' in this example of 1,2,3,4,1000.

The arithmetic mean tends to be unduly influenced by extreme values. The
geometric mean tends to be unduly influenced by very small values (just
throw a zero into the example!). The median tends to be the 'best'
indication of the overall behavior of a group of numbers, simply
throwing away any extreme points. All have their uses.

Hope this didn't just add to any confusion.

Lynn Killingbeck
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sci.math.num-analysis #43387 (15 + 365 more)                          (1)+-(1)
From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)                |-(1)
[1] Re: Mean, Median, Average ?                                          |-(1)
Date: Wed Jul 15 14:39:22 EDT 1998                                       |-[1]

These terms have been explained in other responses, and I will just add to
this:

The arithmetical mean, or average,  X of x_i, i=1,...,N (equally weighted)
turns out to minimize

 sum[i=1 to N] (x_i - X)^2

(Sometimes we attach "weights" to measurements, but you will find the
modification in appropriate textbooks.)

In contrast, the median M of the same collection of numbers turns out to
minimize

 sum[i=1 to N] abs(x_i - M)

(ties broken by taking a midpoint)

and this is less influenced by the "outliers" (e.g.  measurements of the
vibrations of a delicate piece of equipment when an unannounced heavy
truck rolls by).

Hope it helps, ZVK
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sci.math.num-analysis #43394 (14 + 365 more)                          (1)+-(1)
From: bruck@math.usc.edu (Ronald Bruck)                                  |-(1)
[1] Re: Mean, Median, Average ?                                          |-(1)
Date: Wed Jul 15 20:33:14 EDT 1998                                       |-(1)

Mean (arithmetic mean) is the same as average; you take the sum of the
values, say a1, a2, ..., an, and divide by n:

  mean = (a1 + a2 + ... + an)/n.

Median is the number which has as many values greater than it as less than
it in the dataset; that is, it's smack-dab in the middle.  If the dataset
has an even number of members, it's usually taken to be the midpoint of the
two values at the very middle.

Finally:  although you didn't ask, "mode" is the MOST FREQUENT observation.

These quantities have the following variational interpretations:

Form the quantity f[p] = Abs[x-a1]^p + Abs[x-a2]^p + ... + Abs[x-an]^p,
and find a value xp which minimizes it.

  When p = 2, the minimizer is the mean.
  When p = 1, the minimizer is (a) median;
  When p = 0, this is the mode.

This last one has to be interpreted rather carefully; you should think of
it as letting xp be (a) minimizer of fp, and then take the limit of xp as
p decreases to 0.  (Otherwise we run into the problem of 0^0.)

When p = 1 and the dataset contains an even number of elements, then any
number between the two middle elements will minimize fp.  That's why I say
it's (a) median.

--Ron Bruck

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