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Decile

Decile is a method that is used to divide a distribution into ten equal parts. When data is divided into deciles a decile rank is assigned to each data point in order to sort the data into ascending or descending order. A decile has 10 categorical buckets while a quartile has 4 and a percentile has 100.

The concept of a decile is used widely in the field of finance and economics to perform the analysis of data. It can be used to check the performance of a portfolio in the field of finance. In this article, we learn more about a decile, its definition, rank, and see associated examples on calculating the decile value.

1.	What is Decile?
2.	Decile Class Rank
3.	Decile Formula
4.	Decile Example
5.	FAQs on Decile

What is Decile?

Decile, percentile, quartile, and quintile are different types of quantiles in statistics. A quantile refers to a value that divides the observations in a sample into equal subsections. There will always be 1 lesser quantile than the number of subsections created.

Decile Definition

Decile is a type of quantile that divides the dataset into 10 equal subsections with the help of 9 data points. Each section of the sorted data represents 1/10 of the original sample or population. Decile helps to order large amounts of data in the increasing or decreasing order. This ordering is done by using a scale from 1 to 10 where each successive value represents an increase by 10 percentage points.

Decile Class Rank

To split the given data and order it according to some specified metric, statisticians use the decile rank also known as decile class rank. Once the given data is divided into deciles then each subsequent data set is assigned a decile rank. Each rank is based on an increase by 10 percentage points and is used to order the deciles in the increasing order. The 5th decile of a distribution will give the value of the median.

Decile Formula

The decile formulas can be used to calculate the deciles for grouped and ungrouped data. When data is in its raw form it is known as ungrouped data. When this data is sorted and organized then it forms grouped data. These are given as follows:

Decile Formula for ungrouped data: D(x) = Value of the \(\frac{x(n+1)}{10}\)th term in the data set.

x is the value of the decile that needs to be calculated and ranges from 1 to 9. n is the total number of observations in that data set.

Decile Formula for grouped data: D(x) = \(l+\frac{w}{f}\left ( \frac{Nx}{10} -C\right )\).

l is the lower boundary of the class containing the decile given by (x × cf) / 10, cf is the cumulative frequency of the entire data set, w is the size of the class, N is the total frequency, C is the cumulative frequency of the preceding class.

The next section will cover the steps for calculating a particular decile.

Decile Example

Suppose a data set consists of the following numbers: 24, 32, 27, 32, 23, 62, 45, 80, 59, 63, 36, 54, 57, 36, 72, 55, 51, 32, 56, 33, 42, 55, 30. The value of the first two deciles has to be calculated. The steps required are as follows:

Step 1: Arrange the data in increasing order. This gives 23, 24, 27, 30, 32, 32, 32, 33, 36, 36, 42, 45, 51, 54, 55, 55, 56, 57, 59, 62, 63, 72, 80.
Step 2: Identify the total number of points. Here, n = 23
Step 3: Apply the decile formula to calculate the position of the required data point. D(1) = \(\frac{(n+1)}{10}\) = 2.4. This implies the value of the 2.4^th data point has to be determined. This will lie between the scores in the 2^nd and 3^rd positions. In other words, the 2.4^th data is 0.4 of the way between the scores 24 and 27
Step 4: The value of the decile can be determined as [lower score + (distance)(higher score - lower score)]. This is given as 24 + 0.4 * (27 – 24) = 25.2
Step 5: Apply steps 3 and 4 to determine the rest of the deciles. D(2) = \(\frac{2(n+1)}{10}\) = 4.8^th data between digit number 4 and 5. Thus, 30 + 0.8 * (32 – 30) = 31.6

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Important Notes on Decile

A decile is a quantile that is used to divide a data set into 10 equal subsections.
The 5^th decile will be the median for the dataset.
The decile formula for ungrouped data is given as \(\frac{x(n+1)}{10}\)th term in the data set.
The decile formula for grouped data is given by \(l+\frac{w}{f}\left ( \frac{Nx}{10} -C\right )\).

Examples on Decile

Example 1: Find the 6th and the 9th decile for the data in the above-mentioned example.
Solution: The arranged data is 23, 24, 27, 30, 32, 32, 32, 33, 36, 36, 42, 45, 51, 54, 55, 55, 56, 57, 59, 62, 63, 72, 80
n = 23
D(6) = \(\frac{6(n+1)}{10}\) = 14.4^th data. This lies between 54 and 55.
D(6) = 54 + 0.4 * (55 – 54) = 54.4
D(9) = \(\frac{9(n+1)}{10}\) = 21.6^th data. This lies between 63 and 72
D(9) = 63 + 0.6 * (72 – 63) = 68.4
Answer: D(5) = 54.4 and D(9) = 68.4
Example 2: Find the median of the following data set using the concept of deciles.
55, 58, 61, 67, 68, 70, 74, 81, 82, 93, 20, 28, 29, 30, 36, 37, 39, 42, 53, 54
Solution: Arranging the data in increasing order 20, 28, 29, 30, 36, 37, 39, 42, 53, 54, 55, 58, 61, 67, 68, 70, 74, 81, 82, 93
The fifth decile is the median of the data set, thus,
n = 20
D(5) = \(\frac{5(n+1)}{10}\) = 10.5^th data. This lies between
D(5) = 54 + 0.5* (55 - 54) = 54.5
Example 3: Find the 7th decile for the following frequency distribution table.

Class Frequency

10 - 20 15

20 - 30 10

30 - 40 12

40 - 50 8

50 - 60 7

60 - 70 18

70 - 80 5

80 - 90 25

Solution:
From the given frequency distribution table, we can have,

Class Frequency Cumulative Frequency (cf)

10 - 20 15 15

20 - 30 10 25

30 - 40 12 37

40 - 50 8 45

50 - 60 7 52

60 - 70 18 70

70 - 80 5 75

80 - 90 25 100

D(7) = \(\frac{7\times 100}{10}\) = 70th data in the cf column

This data lies in the 60 - 70 class

D(7) = \(l+\frac{w}{f}\left ( \frac{Nx}{10} -C\right )\)

= \(60+\frac{10}{18}\left ( \frac{7 \times 100}{10} -52\right )\) = 70

Class	Frequency
10 - 20	15
20 - 30	10
30 - 40	12
40 - 50	8
50 - 60	7
60 - 70	18
70 - 80	5
80 - 90	25

Class	Frequency	Cumulative Frequency (cf)
10 - 20	15	15
20 - 30	10	25
30 - 40	12	37
40 - 50	8	45
50 - 60	7	52
60 - 70	18	70
70 - 80	5	75
80 - 90	25	100

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FAQs on Decile

What is a Decile in Statistics?

A decile in statistics is a method to divide the distribution into 10 equal parts by using 9 data points and assigning decile ranks to each point.

What is a Decile Class Rank?

Once the data set is sorted into deciles then a decile class rank is assigned to each point so as to arrange these deciles into increasing order.

What is the Decile Formula?

The decile formula for ungrouped data is determined by the value of the \(\frac{x(n+1)}{10}\) term. The formula for grouped data is \(l+\frac{w}{f}\left ( \frac{Nx}{10} -C\right )\).

How to Find the Value of the Median Using the Decile Formula?

The value of the 5^th decile represents the median. For ungrouped data the median will be given by D(5) = \(\frac{5(n+1)}{10}\)^th term.

How to Interpret the First Decile?

The first decile is a point such that 90% of the data lies above it and 10% of the data lies below it. Similarly, the 2^nd decile is a point with 20% of data lying below it and 80% lying above it.

How to Calculate the Decile for Ungrouped Data?

The steps to calculate the decile for ungrouped data are as follows:

Arrange the data in increasing order.
Find the position of the decile using the formula D(x) = \(\frac{x(n+1)}{10}\) to check between which scores the decile will be.
Find the value of the decile [lower score + (distance)(higher score - lower score)]

What are the Applications of Decile?

The concept of decile is widely used in the finance and economics industries to assess the performance of a mutual fund or a portfolio. It acts as a comparative number to measure the performance of an asset.

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