Why are there only 3 quartiles when dividing data into 4 equal parts?

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Discussion Overview

The discussion revolves around the concept of quartiles in statistics, specifically addressing why there are only three quartiles when dividing data into four equal parts. Participants explore the definitions and implications of quantiles, intervals, and cuts, as well as the confusion surrounding these terms.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants assert that dividing data into four equal parts results in three quartiles, as there is one less quantile than the number of divisions (Post 1).
  • Others argue that the fourth quartile exists as it represents the highest range of data, suggesting a misunderstanding of the terminology (Post 3).
  • One participant emphasizes that the distinction between intervals and cuts is crucial, stating that making n cuts results in n intervals (Post 4).
  • Another participant clarifies that a quartile is defined as a specific value (cut) rather than an interval, which may lead to confusion in understanding quartiles (Post 6).
  • There is a discussion about the terminology used, with some participants correcting each other on the definitions of quartiles and quantiles (Post 7, Post 8).

Areas of Agreement / Disagreement

Participants express differing views on the existence and definition of the fourth quartile, with some asserting it is merely a conceptual misunderstanding while others maintain that it is a valid classification. The discussion remains unresolved regarding the clarity of terminology and definitions.

Contextual Notes

There is a noted confusion between the concepts of intervals and cuts, as well as the specific definitions of quartiles versus quantiles. The discussion highlights the need for clarity in statistical terminology.

15123
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My professor stated the following:

"Dividing in pieces is called 'quantiles'. In almost all cases quartiles are used, where n=4. We divide it into four equal pieces and we are going to check where the margin values are.
Pay close attention: if I split my data into 4 equal pieces, I will have 3 quartiles. If I have 10 equal pieces, I have 9 deciles. You always have one less quantile than the n parts."

I don't understand why there are always n-1 quantiles. Why is the 4th quartile never mentioned, not even in boxplots? Some people say this is because the 4th quartile is the supremum (<=100).
So why does he say there are only 3 quartiles, when you have a 4th quartile as well?
e.g.: if a student scores 95%, he will belong in the 4th quartile because I cannot classify him in the 3rd quartile (75%), because 95% is between 75% and 100% (75<95<100). If a quartile is 25%, then why do we only have 3 quartiles (3*25%=75%. What about the remaining 25%)?
 
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That has nothing to do with probability or statistics but simple arithmetic. If you divide a rope or a stick into "n" pieces you will have to make "n-1" cuts.
 
There is still a fourth quartile. If a score on a test is 22%, I put it in the first quartile, if the score is 33%, I put it under the Median and if the score is 70%, I put it under the third quartile. What do you do with a value that is 95%? You should put it in the fourth quartile, so to me there is a fourth quartile. My professor says there are only three quartiles.

I am still lost on this.
 
There seems to be a confusion between the number of intervals (4) and the number of cuts (3). It sounds to me that the professor is confused or you didn't understand him.
 
mathman said:
There seems to be a confusion between the number of intervals (4) and the number of cuts (3). It sounds to me that the professor is confused or you didn't understand him.
I probably didn't understand him. I think he simply meant cuts=intervals-1.
 
A quartile, or any quantile for that matter, is a number, not an interval. The 25th quartile is the single value which 25% of the data falls below. It is not the interval, it is the cut. There certainly is an interval containing the largest 25% of data but that is not, by definition, a quartile. You are confusing intervals with their endpoints.
 
alan2 said:
A quartile, or any quantile for that matter, is a number, not an interval. The 25th quartile is the single value which 25% of the data falls below. It is not the interval, it is the cut. There certainly is an interval containing the largest 25% of data but that is not, by definition, a quartile. You are confusing intervals with their endpoints.
25th quartile? Don't you mean quantile? However, thanks for the explanation. Someone mentioned this to me, but I think he is incorrect:
someperson said:
Q1 - the lower 25% of the total probability

Q2 - the area encompassing 25%-75% of the total probability - Q2 is actually an area of 50% probability centered at the median.

Q3 - the area from 75%-100% of the total probability.
 
No, I meant quartile because you referred to 25%. It's just terminology but median, tercile, quartile, quintile, and percentile are special names for the quantiles that divide your interval into 2, 3, 4, 5, and 100 sub-intervals respectively. Whoever mentioned that quote to you was incorrect but I think you've got it now.
 

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