Decile – Definition, Formula, Rank, Examples

Decile is a statistical measure that divides a dataset into ten equal parts, and each part contains 10% of the data. It is different from percentage as in percentage there are 100 parts. In this article, we will learn in detail about what is decile, the formulas of decile (grouped data and ungrouped data)

A decile is a quantitative method of dividing data into 10 equal parts. Suppose there is a list of numbers, then the data in the list can be put in order from smallest to largest, and then split into 10 groups with an equal number of numbers in each group. This method is often used in finance and economics studies to help understand and analyze data.

Deciles are different from percentiles, and quintiles quartiles, which divide data into 100 parts, 4 parts, and 5 parts respectively.

Definition of Decile

Splitting a group of data into 10 equal parts is defined as a decile. There are 9 points to divide the data and make sections that each represent 1/10th of the whole. Decile helps arrange a large amount of data in either ascending or descending order.

This arrangement of data is done on a scale of 1 to 10, where each step represents an increase of 10 percentage points.

Decile has two ways to be calculated based on the type of data.

The two formulas to calculate decile are:

• Decile Formula for Ungrouped Data
• Decile Formula for Grouped Data

Decile Formula for Ungrouped Data

To find the decile for ungrouped data, first, the data needs to be arranged in ascending order. Then, the following formula is applied:

Decile_n = (n/10) × (N + 1)

where,

• Decile_n represents the nth decile.
• n is the decile number (e.g., 1st decile, 2nd decile, etc.).
• N is the total number of data points.

Decile Formula for Grouped Data

To calculate the decile of a grouped data, cumulative frequency is used. The formula to find the decile of a grouped date is as follows:

Decile_n = L + [Tex]\left( \frac{\left(\frac{n}{10} \times N \right) – F}{f} \right) \times c[/Tex]

where,

• Decile_n represents the nth decile.
• n is the decile number (e.g., 1st decile, 2nd decile, etc.).
• L is the lower boundary of the decile class.
• N is the total frequency.
• F is the cumulative frequency of the class preceding the decile class.
• f is the frequency of the decile class.
• c is the class width.

Decile class rank refers to a method of ranking students based on their academic performance relative to their peers within their class. The class is divided into ten equal parts or “deciles,” with each decile representing 10% of the class.

Calculate the decile of any given set of data following the steps given below:

Sort the Data: Arrange the data points in ascending order from smallest to largest.

Identify Decile Number: Decide which decile you want to find (e.g., 1st decile, 2nd decile, etc.).

Calculate Position: For ungrouped data, use the formula:

Decile_n = ( n/10 )×(N+1)

Where:

• Decile_n is the nth decile.
• n is the decile number.
• N is the total number of data points.

NOTE: For grouped data, use cumulative frequency to find the position of the decile.

Find the Value: Once you have the position, locate the corresponding value in the sorted data set. This value represents the decile.

For example, Suppose we have a data set consisting of the following numbers representing the scores of students in a math test: 65, 72, 84, 57, 68, 75, 80, 92, 88, 78, 60, 70, 85, 95, 62, 73, 79, 83, 90, 77.

Step 1: Arrange the data in increasing order. This gives us: 57, 60, 62, 65, 68, 70, 72, 73, 75, 77, 78, 79, 80, 83, 84, 85, 88, 90, 92, 95.

Step 2: Identify the total number of points. Here, n=20.

Step 3: Apply the decile formula to calculate the position of the required data point.

D(1) = (n+1) / 10 = 21/10 = 2.1

This implies the value of the 2.1st data point has to be determined. This will lie between the scores in the 2nd and 3rd positions. In other words, the 2.1st data point is 0.1 of the way between the scores 60 and 62.

Step 4: The value of the decile can be determined as lower score+(distance)(higher score−lower score). This is given as 60 + 0.1 × (62−60)

= 60 + 0.1 × 2

= 60 + 0.2

= 60.2

Step 5: Apply steps 3 and 4 to determine the rest of the deciles.

D(2) = 2 × (n+1) / 10 = 4.2^th data between digits 4 and 5.

Thus, 62 + 0.2 × (65-62)

= 62 + 0.2 × 3 = 62 + 0.6 = 62.6

Example 1: Given the following grouped data representing the scores of students in a class:

Find the first decile for this data.

Solution:

First we will calculate the cumulative Frequency

Total number of observations (N) = 5 + 8 + 12 + 15 + 10 = 50

Position of Decile D₁ = N/10

= 50/10 = 5

Now, cumulative frequency just greater than D₁ is corresponding to the score range 11-20

Now we know, that for ungrouped data

D(n) = L + [Tex]\left( \frac{\left(\frac{n}{10} \times N \right) – F}{f} \right) \times c[/Tex]

where,

• L is Lower boundary of the score range containing D₁ (11 in this case)
• F is Cumulative frequency of the score range before D₁ (5)
• f is Frequency of the score range containing D₁ (8)
• w is Width of the score range (10 – 0 = 10)

Putting the values in the formula we get,

D₁ = 11 + [Tex]\left( \frac{5 – 5}{8} \right)[/Tex] × 10

D₁ = 11 + (0/8) × 10

D₁ = 11 + 0 × 10

∴ D₁ = 11

Example 2: Given the following ungrouped data representing the ages of ten individuals: 22,25,30,33,35,40,45,50,55,60. Find the deciles for this data.

Solution:

First, we will arrange the given data in ascending order: 22, 25, 30, 33, 35, 40, 45, 50, 55, 60.

Now using the formula of calculating decile; D(n) = N × (n +1) /10 we will find the deciles. Where, N is the total frequency, and n ranges from 1 to 9

First Decile D(1) = 1 × (9 +1) /10 = 10/10 =1

Second Decile D(2) = 2 × (9 +1) /10 = 20/10 = 2

Third Decile D(3) = 3 × (9 +1) /10 = 30/10 = 3

Fourth Decile D(4) = 4 × (9 +1) /10 = 40/10 = 4

Fifth Decile D(5) = 5 × (9 +1) /10 = 50/10 = 5

Sixth Decile D(6) = 6 × (9 +1) /10 = 60/10 = 6

Seventh Decile D(7) = 7 × (9 +1) /10 = 70/10 = 7

Eighth Decile D(8) = 8 × (9 +1) /10 = 80/10 = 8

Ninth Decile D(9) = 9 × (9 +1) /10 = 90/10 = 9

Q1. Calculate the value of the 3rd decile for the following set of data: 12, 15, 18, 20, 22, 25, 28, 30, 32, 35.

Q2. Given the cumulative frequency distribution below, determine the value of the 7th decile:

Q3. Determine the 5th decile for the following set of data 18, 22, 25, 30, 35, 40, 45, 50, 55, 60.

Q4. The following data represents the marks obtained by 60 students in a test:

Determine the value of the 9th decile.

What is Decile Rank?

Decile rank is a method to divide a group of data into 10 equal parts, where each part contains 10% of the data.

What is a first decile and second decile?

First decile is the value below which 10% of the data falls, and the second decile is the value below which 20% of the data falls.

What is the difference between decile and percentage?

Decile divides data into 10 equal parts, while percentage represents parts out of 100.

What does 90% decile mean?

90% decile means the value below which 90% of the data falls, indicating that only 10% of the data lies above this value.