What is statistics? – Percentiles, Quartiles, Quantiles, and Deciles

The Cake!
You have Birthdays and your mum bought you a BIG cake! You are expecting to have 10 Birthday guests. Your mom cut your cake without taking your advice in ten pieces.

Order!
We can suggest that each piece of Cake is exactly the same —let’s say— in length. However, the number of cookies that each piece includes differ. Let’s say that while you are waiting until your guests arrive, you have ordered your cake pieces in an ascending fashion based on the number of cookies that they have. Therefore, your first guest will receive the piece of cake with the least number of cookies and your last guest will receive this cake piece that is richest in cookies.

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Positions!
You are now thinking how you will equally share your cake pieces based on the number of guests you will receive:

If you are expecting to have only TWO guests: You must find THIS ONE position that will indicate EQUAL number of Cake pieces for each guest. Therefore, each guest must receive Five cake pieces.

If you are expecting to have only FIVE guests: You must find these FOUR positions that will indicate EQUAL number of Cake pieces for each guest. Therefore, each guest must receive TWO -equal in length- cake pieces.

Let’s say that you are expecting to have only TEN guests: You must find these NINE positions that will indicate EQUAL number of Cake pieces for each guest. Therefore, each guest must receive one cake piece.

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Percentiles and Cake example
Percentiles DO care only about positions that will result “Equal Pieces” in an Arithmetic Series. That is, positions that produce “Arithmetic pieces” that will include the same “Quantity” of numbers AFTER a dataset has been ordered (ascended). Percentiles “cut” an arithmetic set in “equal pieces”. Each piece will include the same quantity of numbers.

What is Percentile: Theoretical Definition
The Percentile results 100 equal pieces by implying 99 dividing positions for an Arithmetic Series. Then, it can tell you what percentage of your data can be before or after a specific position, AFTER it has been ordered in an asceding fashion. And then, this position can indicate a specific number. Percentile itself does not directly indicate to you this number, only a position that splits your dataset in some prerequisited quantity!

What is Percentile: Differences with Percentages
The percentile is not the same with the Percentages. The percentages do not require Ordering and they produce an indicator value themselves, based on some data. However, Percentiles are an indicator of position, -then, this position can indicate a value- and Require ordering.

A slight detail: Percentage has an “inherit” order or “implies” an inherit order. However, percentiles do not imply this. They explicity asks you to do this ordering, as we will see in some actual examples.

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For example: The minimum and maximum score were from 0 to 100 on a school test that included e.g. 10 questions. You received a score of 60. Then, we know that you had 60% of the answers right while the 40% of your answers were wrong. No ordering is required, it is implied! However, we cannot know in what Percentile your performance is placed, in relation to the performance of the your classmates. In order to find this information, we must know the test performance of all your classmates, and then to place these numbers in an ascending order.

What is Quartile: Theoretical Definition
What is Quartile:

i) While Percentile results 100 equal pieces by implying 99 dividing positions, Quartile results four (4) equal pieces by implying 3 dividing positions.
ii) Therefore, the position that is indicated by the first Quartile (Q1) implies that the 25% of the information or data are before this position —while the 75% is after this position.
iii) Another 25% of this information will be between the position that is indicated by Q1 and the position that is indicated by Q2.
iv) Another 25% of this information will be between the position that is indicated by Q2 and the position that is indicated by Q3.
v) The final 25% of information or data will be after the position that is indicated by Q3. That is: 4*25%=100%.

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vi) The 50% of the information or data is included between the position that is indicated by the first Quartile (Q1) and the position that is indicated by the Third Quartile (Q3).
vii) The rest of 50% (25% + 25%) of the information or data is i) before Q1 and ii) after Q3.
viii) The Q2 position splits an ordered dataset exactly in Half.
ix) Therefore, Q2 position always indicates the Median value.

What is -tiles: Theoretical Definition
You can define whatever -iles you want e.g.:

i) Tertiles occupies 2 positions that result 3 equal pieces. Each piece includes 33% of the total information or data.
ii) Quartiles occupies 3 positions that result 4 equal pieces. Each piece includes 25% of the total information or data.
iii) Deciles occupies 9 positions that result 10 equal pieces. Each piece includes 10% of the total information or data.
iv) Percentiles occupies 99 positions that result 100 equal pieces. Each piece includes 1% of the total information or data.
v) Permilles occupies 999 positions that result 1000 equal pieces. Each piece includes 1‰ or 0.1% of the total information or data.

You must also have in mind that these -iles exist: centile or percentile (99), vigintile (19), duodecile (11), decile (9), nonile (8), octile (7), septile (6), sextile(5), quintile (4), quartile (3), and tercile or tertile (2). Not all these -tiles are convenient to be used in statistics.

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The Relation of -iles to Percentiles
Here, we must say that -iles have a similar relation as millimeters have to centimeters, and centimeters to meters. That is:

The First Tertile occupies the same position as the ~33.34th Percentile (approximately): T_{1}=P_{~33.34}
The Second Tertile occupies the same position as the ~66.68th Percentile (approximately): T_{2}=P_{~33.34}

First Quartile occupies the same position as the 25th Percentile: Q_{1}=P_{25}
Second Quartile occupies the same position as the 50th Percentile: Q_{2}=P_{50}
Third Quartile occupies the same position as the 75th Percentile: Q_{3}=P_{75}

The First Decile occupies the same position as the 10th Percentile: D_{1}=P_{10}
The Second Decile occupies the same position as the 20th Percentile: D_{2}=P_{20}
The Sixth Decile occupies the same position as the 60th Percentile: D_{6}=P_{60}
The Ninth Decile occupies the same position as the 90th Percentile: D_{9}=P_{90}

What is Percentile: Statistical Definition
The Statistical formula for Percentile is this one:

P=\frac{k*(n+1)}{100}

The k is the specific Percentile that we are searching to find a specific position e.g. the 25th Percentile is a specific position for an asceding ordered dataset and thus the number that exists in that position (25th) divides or splits this dataset in such a way that 25% of the data will be before this position and 75% of the data will be after this position: k=25.

The n is the total size of a dataset
The 100 exists in the formula because Percentile results 100 equal pieces.

What is Quartile: Statistical Definition
The Statistical Formula for calculating the Quartile is this one:

Q=\frac{k*(n+1)}{4}

The k is the specific Quartile that we are searching to find a specific position e.g. the 2nd Quartile is a specific position for an asceding ordered dataset and thus the number that exists in that position (2th) divides or splits this dataset in such a way that 50% of the data will be before this position and 50% of the data will b after this position. Note that there are only three (3) Quartiles, and thus k takes only these values: 1, 2, and 3.

The n is the size of a dataset
The 4 exists in the formula because Quartile results 4 equal parts.

What is -tiles: Statistical Definition
Generally, the formula for -iles can be generalized as below:

t=\frac{(k)*(n+1)}{i_{m}}

The t is the -ile position that we are searching to find and thus, it can indicate a specific number in that position that will result some percentage (%) information of your dataset to be before of this value and the rest percentage (%) of this information to be after this value.

The i_{m} is the -ile which we have defined and is replaced by how many equal “iles” or “pieces are produced by the -ile concept we will choose to implement. Therefore, i_{m}=100 if the Percentile concept is used which results 100 equal pieces in a dataset, i_{m}=4 if the Quartile concept is used which results 4 equal pieces in a dataset, and i_{m}=10 if the Decile concept is used which results 10 equal pieces in a dataset.

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The k value shows the number of “cut positions” that a -tile concept applies and NOT the number of produced pieces from this “cut”. For example: Percentile has 99 “cut positions” in a dataset which results 100 equal pieces, while quartile has 3 “cut positions” which results 4 equal pieces, and the Decile has 9 “cut positions” which results 10 equal pieces. The k is replaced by the “cut position” we are searching for e.g. by 2 if we are searching to find what is the “cut position” of the 2nd Percentile in a dataset.

Statistical Example Ι
In a class of 10 students, a Physics test was given. The maximum possible score was 20. The results are the following:  15, 13, 11, 7, 18, 5, 15, 10, 10, 13.

The first step is to place these results in an ascending order:

5, 7, 10, 10, 11, 13, 13,  15,  15,  18.

Let’s say that we are searching to find the position that this Quartile lies: Q_{2}, this Decile: D_{5}, and this Percentile: P_{50} in a dataset. By applying the appropriate formula each time for n=10, then we get:

The positions that 2nd Quartile, 5th Decile, and 50th Percentile lies

t=\frac{(2)*(10+1)}{4}=\frac{22}{4}=5.5,
therefore, the exact position is 5.5
t=\frac{(5)*(10+1)}{10}=\frac{55}{10}=5.5,
therefore, the exact position is 5.5
t=\frac{(50)*(10+1)}{100}=\frac{550}{100}=5.5,
therefore, the exact position is 5.5

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Statistical Example and Median value, what the heck?

By summarizing these results, we get these equalities: Q_{2}=D_{5}=P_{50}=5.5

What If we are interested in to find also what is the Median value of this dataset? We have a dataset that includes 10 values. In order to find the Median, we must find the middle position. therefore, we divide the 10 pieces of information by 2 which results the 5th position. The 5th position from the left is occupied by the 11 while from the right, is occupied from the number 13. Finally, in order to find the Median, we must get the mean value of these two values, which is:

\frac{11+13}{2}=\frac{24}{2}=12

When we applied the appropriate formula for the Quartile, Decile, and Percentile, we had as results what? Which position these results highlighted? The 5.5 position. That is, the position that exists between the 5th and 6th number in this dataset. Which numbers occupy these two positions? But of course the values of 11 and 13! How can we find what exact value is in the 5.5 position? But we have found it! Do you remember? It is the Median value! Therefore:

Q_{2}=D_{5}=P_{50}=Md=5.5 and X=12

Statistical result for the Cake!
Remember that you have palced your Cake Pieces in an Ascending order according to the number of cookies that they have on them. Let’s suggest that each value of the Arithmetic Series of the previous Example is the corresponding number of cookies of each Cake Piece, in the same (ascending) order. Now, you would like to find what is this Cake Position that will divide your Cake Pieces exactly in Half.

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Which is this position? Who else! The position that is between the Cake Pieces that have 11 cookies and 13 cookies on them!

Note, the Percentiles are not “FAIR” about the quantity of cookies that each Guest will receive BUT they are fair about the quantity of Cake Pieces / Arithmetic pieces! And this is very useful in many cases / Cakes!