Decile

A decile is a statistical measure that divides a dataset into 10 equal parts. These parts contain 10% of the observations when arranged in ascending order. Deciles help in ranking and analyzing distributions. The first decile (D1) represents the lowest 10% data, while the 9th decile (D9) represents the highest 90%. The deciles split data into ten equal parts to simplify analysis.

They help to compare values across different sections of the dataset. The common applications of decile include income distribution, stock market analysis, and academic performance. Also, before calculating deciles, data must be sorted. Deciles can be calculated for both ungrouped and grouped data.

The decile formula can be used to calculate the deciles for grouped and ungrouped data. When data is in its raw form, it's known as ungrouped data. When this data is organized, it becomes grouped data. The formulas are given below for both types of data:

For Ungrouped Data:

The formula used to calculate the deciles for ungrouped data is:

D(x) = (n + 1) × x / 10

where, 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 dataset.

For Grouped data:

The formula used to calculate the deciles for grouped data is:

D(x) = L + w / f(Nx / 10 - C)

Where, L is the lower boundary of the class containing the decile given by (x x cf)/10

cf is the cumulative frequency of the entire dataset

w is the size of the class 

N is the total frequency 

C is the cumulative frequency of the preceding class
 

To calculate a decile, we have to follow the following steps:

Step 1: Arrange the given dataset in both ascending and descending order. For instance, when arranging the data in ascending order, start with the smallest number and list the values in increasing order. 

Step 2: Then we have to use the formula:

    D_k = k(n + 1) / 10

Where k is the decile number (1 to 9), and n is the total number of data points. This formula gives the position of the decile in the data set. If the position is a whole number, take the corresponding data value. If the position is a decimal, apply interpolation by averaging the two nearest values. For grouped data, use the formula:

    D_k = L + ((kN / 10) - F / f) x h

Where L is the lower boundary of the decile group, N is the total frequency, F is the cumulative frequency before the decile group, f is the frequency of the decile group, and h is the class width. This method helps analyze large datasets effectively.
 

There are many uses of deciles in our day-to-day life. Let us now see the various fields and applications we use deciles:

Economics and Income Distribution:

Deciles are commonly used in economics and income distribution analysis. The government utilizes them to analyze the income distribution across a population. For example, the lowest decile represents the poorest 10% of the population and the highest represents the richest 10%. Policymakers also use the decile to assess the standard of living and determine the eligibility for subsidies or financial aid.

Finance and Investment:

We use deciles in finance and investments to help us classify stocks based on returns, volatility, or risk factors. Banks use them to segment borrowers based on their credit score.

Education:

We use deciles in education, where schools use them to analyze student test scores, and also help them in ranking applicants based on their academic performance or entrance test scores.