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Topic: Q1 and Q3 in Minitab y
Replies: 1   Last Post: Aug 31, 1996 9:35 AM

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Bob Hayden

Posts: 2,384
Registered: 12/6/04
Q1 and Q3 in Minitab y
Posted: Aug 30, 1996 9:40 PM
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----- Forwarded message from John Burnette -----

As we progress through M&M's first chapter we are noticing a difference in
terminology between Minitab and the text.

Not that it makes much difference but if Q1 doesn't lie on a data point it
interpolates between the two it is between. This isn't a lot different
from the way that Moore defines Q1.

To make things more confusing, the boxplots ARE the same as M&M defines
however minitab calls those values "Upper and lower hinges" HL and HU, as
they call the values, seem to be calculated by the process Moore uses to
find Q1 and Q3.

----- End of forwarded message from John Burnette -----

That's only the tip of the iceberg! I have seen more than a dozen
different ways of defining these points which give slightly different
results. In some cases, two methods give the same results for some
values of n and different results for other values of n. What
Minitab does (or did in the older versions I'm most familiar with) for
Q1 and Q3 is part of a general approach to quantiles that is older
than boxplots. It can involve some messy interpolation in the
general case, so when Tukey invented the boxplot (remember, a lot of
his inventions were meant to be done while flying on a plane in the
days before laptops) he used approximate quartiles which he called
"hinges". Elementary textbooks tend to blur this distinction and
give one definition and call the result a "quartile". For example,
Siegel uses Tukey's hinges and calls them "quartiles". Later a
different approximate quartile was adopted for the QLP materials. It
was subsequently adopted in the texts by David Moore and by the TI-82
and TI-83, and is pretty much standard in K-12 as a result. These
materials also blur the distinction Minitab is (correctly) making.

Those are my general comments, and this sentence is a note to move on
if you do not want more detail on

1. the problems of using non-standard terminology
2. what the actual different definitions are

I think the QLP materials are wonderful -- better than 99% of the
stuff being used to teach statistics in the colleges. But I do wish
they were a little less "creative" in their terminology. Hinges and
quartiles were already well established before QLP came along, so I
think that would be a good reason to go with one or the other. I'm
not sure why they went with a third alternative (and called it a
quartile). I also note that they called the things that most people
(and Minitab) call "dotplots" "lineplots". Whatever the reasons,
using nonstandard terminology does have the price of confusion sooner
or later. I've even been criticized or "corrected" by high school
teachers because I used standard terminolgy for things rather than the
nonstandard terminology they were accustomed to. (Was it McCauley who
said, "Beware the man who's read but one book"?)

TI did their homework and asked around to find out which of the many
definitions of quartiles/hinges they would implement on their
calculators. However, they were more interested with usage in high
school classrooms rather than in the statistics profession.

Speaking of TI and standard terminology, the TI-82 implemented what I
would call "quick" boxplots. These are boxplots without any flagging
of outliers. Since these lack one of the two main reasons for
doing boxplots in the first place, I was disappointed. I was happier
when I saw that the TI-83 does plain old real boxplots as well. I was
not so happy when I saw that the manual called the quick boxplots
"regular baxplots" and the regular boxplots "modified boxplots", as if
the limitations of the 82 were the standard of regularity, and a real
boxplot was some kind of aberration.

Here are three flavors of quartiles. Everybody agrees you need to
sort the data first. I'll just talk about the first quartile. To get
the third, sort your data in the wrong direction and then follow the
steps below.


The first quartile has rank (n+1)/4. Note that everyone agrees that
the median has rank (n+1)/2 and Minitab is just extending the pattern.
It does the same sort of thing to get deciles or percentiles. If the
rank is not a whole number, Minitab uses linear interpolation between
two adjacent data values. Note that in the case of quartiles this may
put you one-fourth or three-fourths of the way between two data


Find the median. Then find the median of the data values whose ranks
are LESS THAN OR EQUAL TO the rank of the median. This will be a data
value or it will be half way between two data values.


Find the median. Then find the median of the data values whose ranks
are STRICTLY LESS THAN the rank of the median. This will be a data
value or it will be half way between two data values.

Note that for SOME values of n, SOME of these methods give the same
results. No two of them give the same results for all n.

| | Robert W. Hayden
| | Department of Mathematics
/ | Plymouth State College
| | Plymouth, New Hampshire 03264 USA
| * | Rural Route 1, Box 10
/ | Ashland, NH 03217-9702
| ) (603) 968-9914 (home)
fax (603) 535-2943 (work)

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