The Quintile Rank

Similar to quartiles, quintiles, which are denoted |R_{5},| are used to divide a dataset into |5| generally equal parts, and then to locate a data point within one of those parts.​

Unlike percentiles, a data point with a low quintile rank, e.g., the 1st, will be part of the data points with the highest value in the distribution.

1. Place the data in descending order.

2. Count the number of data points.

3. Divide the data into 5 groups.

4. Separate the data into 5 groups.

5. Assign a quintile rank to each group.

​Here are the heights (in cm) of some of the teachers in a school.||165,168,156,180,175,170,175,181,176,174,163,152,179,177,171,182||What is the quintile rank of |177\ \text{cm}?|

1. Place the data in descending order||182,181,180,179,177,176,175,175,174,171,170,168,165,163,156,152||

2. Count the number of data points
   There are |16| data points in this distribution.

3. Divide the data into 5 groups
   If we divide |16| by |5,| we get |3| remainder |1.| Therefore, we should have |4| groups of |3| data points and |1| group of |4| data points

4. Separate the data into 5 groups||\color{#3b87cd}{182,181,180},\color{#ec0000}{179,177,176},\color{#3a9a38}{175,175,174},171,170,168,\color{#fa7921}{165,163,156,152}||

5. Assign a quintile rank to each group
   According to the definition, the group with the data of the greatest value will be the 1^st quintile; the following one will be the 2^nd quintile, and so on. Thus, a person who is |177\ \text{cm}| tall would be part of the 2^nd group, and therefore be assigned to the 2^nd quintile.

1. Form groups with a similar amount of data
   In the previous example, |4| groups of |3| and |1| group of |4| were formed. However, it would have been incorrect to make |2| groups of |2,| |1| group of |3,| and |2| groups of |4.| Keep as a reference the result of the division made in step 3 of the example to approximate the amount of data in each of the quintiles.

2. Two data points of the same value must be part of the same quintile
   In the example above, the data point |175| appears twice. Because it has the same value, it is necessary to ensure the data points are found in the same quintile.

||R_{5}(x)=\dfrac{\text{Amount of data greater than}\ x+\dfrac{\text{Amount of data equal to}\ x}{2}}{\text{Total amount of data}}\times 5||If the result is not a whole number, it must be rounded up to the next whole number.​

Here are the heights (in cm) of some of the teachers in a school.||165,168,156,180,175,170,175,181,176,174,163,152,179,177,171,182||What is the quintile rank of |177\ \text{cm}?|

1. Place the data in descending order
   Since it is relevant to the formula, it is preferable to put the distribution in descending order.||152,156,163,165,168,170,171,174,175,175,176,177,179,180,181,182||

2. Apply the formula||\begin{align}R_{5}(177)&=\dfrac{4+\dfrac{1}{2}}{16}\times5\\&\approx1.41\end{align}||By rounding up to the next whole number, we get |R_{5}(177)=2,| the same result obtained by applying the definition.