HOW TO SOLVE QUANTILE PROBLEMS

   QUANTILES

          A statistical term. Quantiles is a generic term for cut-off points. A universal picture telling us how we split the string. If we are talking one straight line for example which measures 100 feet, we can cut it off by 10's, 20's, or by 25's, etc. When we are cutting the said straight line by 10's we can say we are cutting it in :decile". Or if cut it off by 25's we are doing quartile-cut. We can also do cutting foot by foot to make it 100 cuts for percentile.

         Quantiles has three types. The quartiles, which is 25% per size, decile which 10%  per size, and percentile which 1% per size. Quantiles is used either in "Grouped data" or "Ungrouped Data".

         We use quantiles in Statistics if we are resizing the data in the desired size we want. We use quartiles if we need to separate data in four equal parts, decile and percentile for 10's and 1's respectively. We have formulae for that as to resize. The formula for "Ungrouped data" is different from "Grouped data". 

Here is the formula for "Ungrouped data"

For Percentile:

For Quartile:

For Decile:

Illustrative examples:

Given the following data.

4, 2, 5, 6, 9, 10, 3, 7, 9, 12, 8, 15

Solve for the following Percentiles:

(In Percentile problem, we can assign values for k from 1 - 100). In this problem. we will use only 5 problem for Percentile. We will use 20th, 34th, 67th, 50th,25th Percentile. In some cases, you can use any kth percentile. But mostly, we do not solve for the 100th Percentile.

1. 20th Percentile

Formula: P = 20 (n + 1) / 100

Given: 
  
k = 20 (desired size)
n = 12 (number of data we have)

Required: P20

Solution:

P = k (n + 1) / 100

P = 20 (12 + 1) /100

P = 20 (13) / 100

P = 260 / 100

P = 2.6 Answer

2. 34th Percentile

Formula: P = 34 (n + 1) / 100

Given: 
  
k = 34 (desired size)
n = 12 (number of data we have)

Required: P34

Solution:

P = k (n + 1) / 100

P = 34 (12 + 1) /100

P = 34 (13) / 100

P = 442 / 100

P = 4.42 Answer

3. 67th Percentile

Formula: P = 67 (n + 1) / 100

Given: 
  
k = 67 (desired size)
n = 12 (number of data we have)

Required: P67

Solution:

P = k (n + 1) / 100

P = 67 (12 + 1) /100

P = 67 (13) / 100

P = 871 / 100

P = 8.71 Answer

4. 50th Percentile

Formula: P = 50 (n + 1) / 100

Given: 
  
k = 50 (desired size)
n = 12 (number of data we have)

Required: P50

Solution:

P = k (n + 1) / 100

P = 50 (12 + 1) /100

P = 50 (13) / 100

P = 650 / 100

P = 6.5 Answer

5. 25th Percentile

Formula: P = 25(n + 1) / 100

Given: 
  
k = 25 (desired size)
n = 12 (number of data we have)

Required: P25

Solution:

P = k (n + 1) / 100

P = 25 (12 + 1) /100

P = 25 (13) / 100

P = 325 / 100

P = 3.25 Answer

Solve for the following Quartiles:

(In Quartile problem, we can assign values for k from 1 - 4). But mostly, we do not solve for the 4th quartile.

1. 1st Quartile

Formula: Q = 1(n + 1) / 4

Given: 
  
k = 1 (desired size)
n = 12 (number of data we have)

Required: Q1

Solution:

Q = k (n + 1) / 4

Q = 1 (12 + 1) / 4

Q = 1 (13) / 4

Q = 13 / 4

P = 3.25 Answer

2. 2nd Quartile

Formula: Q = 2(n + 1) / 4

Given: 
  
k = 2 (desired size)
n = 12 (number of data we have)

Required: Q2

Solution:

Q = k (n + 1) / 4

Q = 2 (12 + 1) / 4

Q = 2 (13) / 4

Q = 26 / 4

P = 6.5 Answer

3. 3rd Quartile

Formula: Q = 3(n + 1) / 4

Given: 
  
k = 3 (desired size)
n = 12 (number of data we have)

Required: Q3

Solution:

Q = k (n + 1) / 4

Q = 3 (12 + 1) / 4

Q = 3 (13) / 4

Q = 39 / 4

P = 9.75 Answer

4. 4th Quartile (No need to solve usually)

Formula: Q = 4(n + 1) / 4

Given: 
  
k = 4 (desired size)
n = 12 (number of data we have)

Required: Q4

Solution:

Q = k (n + 1) / 4

Q = 4 (12 + 1) / 4

Q = 4 (13) / 4

Q = 52 / 4

Q = 13 Answer (this is by 12 + 1)

Solve for the following Deciles:

(In Decile problem, we can assign values for k from 1 - 10). In this problem, we will use only 5 problem for Decile. We will use 2nd, 4th, 5th, 7th,9th Decile. In some cases, you can use any kth decile. But mostly, we do not solve for the 10th Decile.

1. 2nd Decile

Formula: D = 2(n + 1) / 10

Given: 
  
k = 2 (desired size)
n = 12 (number of data we have)

Required: D2

Solution:

D = 2 (n + 1) / 10

D = 2 (12 + 1) / 10

D = 2 (13) / 10

D = 26 / 10

P = 2.6 Answer

2. 4th Decile

Formula: D = 4(n + 1) / 10

Given: 
  
k = 4 (desired size)
n = 12 (number of data we have)

Required: D4

Solution:

D = 4 (n + 1) / 10

D = 4 (12 + 1) / 10

D = 4 (13) / 10

D = 52 / 10

P = 5.2 Answer

3. 5th Decile

Formula: D =  5(n + 1) / 10

Given: 
  
k = 5 (desired size)
n = 12 (number of data we have)

Required: D5

Solution:

D = 5(n + 1) / 10

D = 5 (12 + 1) / 10

D = 5 (13) / 10

D = 65 / 10

P = 6.5 Answer

4. 7th Decile

Formula: D = 7(n + 1) / 10

Given: 
  
k = 7 (desired size)
n = 12 (number of data we have)

Required: D7

Solution:

D = 7(n + 1) / 10

D = 7(12 + 1) / 10

D = 7(13) / 10

D = 91 / 10

P = 9.1 Answer

5. 9th Decile

Formula: D = 9(n + 1) / 10

Given: 
  
k = 9 (desired size)
n = 12 (number of data we have)

Required: D9

Solution:

D = 9(n + 1) / 10

D = 9 (12 + 1) / 10

D = 9 (13) / 10

D = 117 / 10

P = 11.7 Answer

Question:

1. What have you noticed about the result of P50, D5, and Q2?

2. Why? 

3. How could it be?

Reply on the "Post a Comment"  Box below and we will discuss about it.

Try these yourself!

Given data:

12, 20, 19, 34, 18, 10

Solve for the following:

1. 2nd quartile

2. 50th percentile

3. 5th decile

Post your answer in the "Post a Comment"  Box below.