1) Mean and Median Discussions
Date: Thu, 23 Jan 1997 22:41:21 -0400 (EDT)
From: "Ralph A. Raimi"  rarm@db1.cc.rochester.edu
Subject: Re: Help teaching "Median"
To: Debbie Bloom 
MIME-version: 1.0
Status: RO
X-Status: A

On Thu, 23 Jan 1997, Debbie Bloom wrote:

> I would like some help with a section of our eighth grade statistics unit.
> My students can recognize the mode of a set of data, and they comprehend
> the usefulness of such information. They also can understand the advantage
> of determining the range and mean. But the concept of determining, or even
> the necessity of finding, the median totally eludes them. Beyond the
> standard mathematical calculation of this, I admit that I have been at a
> loss to find any supplemental materials to add to my lessons.
> 
> Any suggestions you can offer would be greatly appreciated!
> 
> Debbie Bloom 
> 
	I am a little puzzled at the messages that came into math-teach this 
evening, two of them from you. For one thing, I am replying to a 
Swarthmore address, not the one at the bottom of your message, and I 
don't know why your message, like several of the others, does this.  Do 
you forward your requests through Swarthmore?  What is UCBOE, anyway?
	Now about median.  Answering your question is difficult because I 
don't know what your textbook says on the subject, and on allied 
subjects.  Especially when you say your students "Know the advantage of" 
computing the mean.  Why the mean and not the median?  To me the median 
is a more obvious sort of thing than the mean.
	Here is the point of view to take:  You are presented with a list 
of numbers and you want to describe what they say in less space than it 
takes to list them all, especially when the list is long.  For example, 
if we measure the weights of everyone in a school we might get 500 
numbers, and certainly the best way to tell about the weights of all 
these people is to merely list the 500 numbers.  But sometimes you don't 
need that much information.  If you know everyone weighs between 50 and 
160 pounds, i.e. the range of the data, you already know something 
interesting.  What do you do with such information?  Well, if you want to 
buy swings for the playground, this is exactly what you need.
	But there is a difference between this question and the question 
of how much milk to buy for the lunchroom.  Kids tend to eat and drink in 
proportion to their weights, so the average (mean) weight will tell you 
about that.  If the average weight is 100 pounds, the school weighs a 
total of 500X100, or 50,000 pounds, and a dietitician will know how much 
milk goes into 50,000 pounds of kids.  Just knowing the list of 500 
weights would make a more complicated calculation, for you'd have to 
figure out how much milk each child is likely to drink, child by child, 
a long job.  You don't need it.
	The median is not as useful for statistical purposes, in general,
but is easier to understand, and does help give a picture.  This is a
question of rank.  If you lined up all 500 kids with the smallest
(lightest in weight) on the left and the heaviest on the right, a long
line, the median child would be the one in the middle.  Well, with 500, an
even number, two children share the middle spot.  Just imagine a child in
the middle spot, whose weight is halfway between the weights of those two: 
That child has the "median" weight. 
	The median is often close to the average, and takes less 
arithmetic to find.  You don't have to add all 500 numbers and divide by 
500, you just have to line them up and look for the middle one.  This is 
probably the main reason people talk about the median:  ease of 
calculation.  But you can easily invent a list where the median is pretty 
far from the average, for example, 50,50,51,51,51,150,150,150,150.  The 
median is 51, i.e. the 'middle child' weighs 51 pounds, and there are 9 
children with weight range from 50 to 150 pounds.  But the average weight 
is almost 95 pounds, and if you were to order only enough milk for 9 
children each of 51 pounds you would run short pretty soon.  But this is 
an artificial example, and you will find that in many cases the numbers 
you use are spread out pretty evenly; in such cases the median is close 
to the average.
	There is not too much more to say about the median, except that 
where ranking people is important we use more than the median; we also 
mention the "quartiles", which are the measurements 1/4, 1/2, 3/4 from 
bottom to top. (the 1/2 way number is called the median, of course, but it 
is also called the 'second quartile'.)  So if our list of numbers were 
pretty long we might get some idea of how they are spread out by nameing 
not only the median but the ones halfway from the bottom to the median 
(i.e. the first quartile) and halfway from the median to the top (the 
third quartile).  In the list 50,50,51,51,51,150,150,150,150 we have 51 
at the first quartile, 51 at the median or second quartile, and 150 at 
the third quartile.  Just this much information is enough to warn us that 
the 51 median is a bad approximation to the mean, because it is so close 
to the first quartile number that we clearly have not much spread to the 
left but a lot to the right.
	You might ask your students to invent examples where the median 
and mean are equal, and where they are far apart.  And it is good to talk 
over what the median is when you have an even number of entries, and what 
the "quartiles" are when the number of entries doesn't split into four 
equal parts.  Of course these things are unimportant when your list of 
entries runs into the millions, as they do in census data, income tax 
returns, etc.  You can ask, which is the better measure of a nation's 
wealth, the average income or the median income?
	Here is an exercise for your class:  Have them invent a set of
measurements where the median is 50 and the average (mean) is 40.  Kids
can play with that quite a bit, but don't let them try to make their lists
too long. 

Ralph A. Raimi            Tel. 716 275 4429, or (home) 716 244 9368
University of Rochester   Fax  716 244 6631
Rochester, NY 14627       Homepage: http://www.math.rochester.edu/u/rarm 


2) Percentile in Standardized testing. Box-and-Whisper plots.
Date: Fri, 24 Jan 1997 16:03:52 -0400
To: uc@mathforum.org (Debbie Bloom)
From: paddyb@true.net (Jennifer Kaplan)
Subject: Re: Help teaching "Median"

Debbie,

I don't know how meaningful this is to 8th graders, but the median is the
score at the 50th percentile. If any of them have undergone standardized
testing, they have been introduced to percentile, but may not really
understand them.  This is a good extension of the median concept.

Also, you can set up problems in which data is best compared by
box-and-whisker plots. To create a b-a-w, you must find the meadian and the
two quartiles. This can be used to compare rainfall or temperatures in two
different cities, for example.

I realize that this is sketchy, but let me know if it's also
incomprehensible. Hope it helps.

Jennifer

>I would like some help with a section of our eighth grade statistics unit.
>My students can recognize the mode of a set of data, and they comprehend
>the usefulness of such information. They also can understand the advantage
>of determining the range and mean. But the concept of determining, or even
>the necessity of finding, the median totally eludes them. Beyond the
>standard mathematical calculation of this, I admit that I have been at a
>loss to find any supplemental materials to add to my lessons.
>
>Any suggestions you can offer would be greatly appreciated!
>
>Debbie Bloom 

Jennifer Kaplan
Colegio Internacional de Caracas                email: paddyb@true.net
CCS3048,PO Box 025323                           phone: (582) 51 95 55
Miami, FL 33102-5323                            fax:   (582) 93 05 33



A SUREFIRE WAY TO MAKE A PERSON FALL IN LOVE WITH YOU:

"Take the girl out to eat.  Make sure it's something she likes to eat.
French fries usually works for me."
                                     Bart, age 9     


3) Family Math Activity
Date: Fri, 24 Jan 1997 21:39:26 GMT+5
From: Debbie Bass 
To: Debbie Bloom 
Subject: Re: Help teaching "Median"

There is a nice activity in "Family Math" from Lawrence Hall of
Science (Univ of Cal) Berkeley..(ISBN 0-912511-06-0)
It's entitled:
"How Long is a Name?"  For median the students make a list of family
members (relatives, friends) write one letter of the person's name on
1" squares example:  [M]  [A]   [R]   [Y]

Line them up from longest to shortest (one name per row)
Then put a number card next to each row that tells the number of
letters in the person's name
Example:  4   [M]  [A]  [R]  [Y]

Remove the number cards and place them in numerical order.
Find the center number...the median
Hope this is a fresh approach for your students.  :)
debbie

Deborah Bass, Ph.D.
School of Education, Box 28
University of South Carolina Aiken
171 University Parkway
Aiken, South Carolina  29801
debbieb@aiken.sc.edu
voice (803) 641-3202         fax   (803) 641-3698

4) Number Sense Video

Date: Fri, 24 Jan 1997 20:43:41 -0500 (EST)
From: "Francis M. Fennell" 
To: Debbie Bloom 
Subject: Re: Help teaching "Median"

Debbie,

I am co-project director of the Numbers Alive Project!  We have produced
10 15-minute TV shows for middle grade kids about Number Sense.  One of
our shows deals with median.  The project is available via Public
(Instructional) TV in many states.  It may also be purchased through
Silver Burdett Ginn (commercial publisher).

You could learn more about this project, by accessing the Maryland Public
TV website (sorry, I forgot the number).

Hope this helps.

Skip Fennell

5) Height line up. Salaries.

Date: Sat, 25 Jan 1997 06:58:04 +0800
From: Wong Khoon Yoong 
To: Debbie Bloom 
Subject: Re: Help teaching "Median"

We do this: ask pupils to line up in ascending order of height. Who is the
"middle" pupil in terms of height. Similar activity on other criterion.
Then we talk about salary. It is embarrassing to use pupils' parent salary
so we just make up some values of labourers, teachers, doctors, politicians
etc. Get data from local source.

With best wishes!
------------------------------------------------
Wong Khoon Yoong
Department of Science and Mathematics Education
Sultan Hassanal Bolkiah Institute of Education
Universiti Brunei Darussalam
Bandar Seri Begawan 2028
BRUNEI DARUSSALAM

6) Local Housing Prices

Date: Sat, 25 Jan 1997 09:32:19 -0800
From: Laurel Drake 
To: uc@mathforum.org
Subject: medians

How about house prices in their city?  The high- priced houses really
inflate the average house price in most such statistics reported.  The
median price of houses (along with a range of house prices) gives a much
better indication of what a house may cost in their city.  Maybe your
city's dept. of something-or-other will give you the actual data (since
it is public information) to work with.

Good luck!

Laurel Drake