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### Finding the Median with Ties

```Date: 06/16/2009 at 10:54:13
From: Henrik
Subject: To find the median when there are ties in the set

Find the median for the set {2,3,5,5,5,10}.

I understand how to calculate the median when there are odd or even
number of elements in a set.  However, I am confused about situations
when there are ties.  For the set given, if I use the traditional
method, it would be 5.  But 5 would not be a correct median since only
one value (10) is above 5, and two values are below 5 (2 and 3).  It
is therefore not a true central tendency.  Is there an alternative way
to calculate the correct median in these instances?  Thanks.

```

```
Date: 06/16/2009 at 12:53:47
From: Doctor Peterson
Subject: Re: To find the median when there are ties in the set

Hi, Henrik.

This is a very perceptive question!

With small data sets, there are cases, like this one, where there is
no number you could choose that fits the common, naive definition of
median, "the value that divides the ordered data into two equal
halves".  This forces us to make a more careful definition of the
term:

The median of a data set is a value such that at most half the
population have values less than the median, and at most half
have values greater than the median.

Note two things: First, it is _A_ value fitting the condition; we
commonly take it as the average of two middle values, but really any
number between them would work!  Second, we don't say that EXACTLY
half are on each side, but only that AT MOST half are on each side.

In your example, 2, 3, 5, 5, 5, 10, there are 2 values less than 5
and 1 value greater than 5, which fits the definition: no more than
3 are in either part.  If we chose anything greater than 5, more than
half the data would be less than our "median", and if we chose
anything less than 5, more than half would be greater than that.  So the
only possible choice is 5.  The traditional method is in fact an
efficient way to find a median that fits the definition.

Note also that the meaning of "measure of central tendency" is far
more broad than you are imagining.  All that means is "any statistic
that is guaranteed to be between the lowest and highest values"!  It
does not have to be in the "exact center" in any sense; the mean,
median, and midrange all define "center" in different ways, and the
mode can't really be said to have anything to do with the center.
Yet they are all SOMEWHERE in the middle, and that is enough.

For more on these ideas, see

A Closer Look at the Definition of Median
http://mathforum.org/library/drmath/view/72726.html

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Statistics
Middle School Statistics

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