Deciles

What Is Deciles?

Deciles are statistical measures that divide a set of ordered data points into ten equal parts. Each decile represents 10% of the data distribution, ranging from the first decile (the 10th percentile) to the ninth decile (the 90th percentile). In the realm of statistical analysis and financial metrics, deciles provide a clear way to understand the spread and concentration of values within a dataset, such as incomes, wealth, or asset returns. They are particularly useful for examining trends in income inequality and wealth distribution among populations.

History and Origin

The concept of dividing a dataset into equal parts, known broadly as quantiles, has roots in statistical theory, evolving to help researchers and analysts better understand distributions. While specific historical figures are not solely credited with the invention of deciles, the broader framework of quantiles gained prominence with statisticians like Francis Galton in the late 19th century, who utilized similar concepts such as quartiles to analyze data sets. Deciles, along with other quantiles like percentiles and quartiles, became standard tools for descriptive statistics as the field matured, offering a standardized approach to breaking down and interpreting large volumes of information. Quantiles help split data or distributions into equal parts to better understand the data's position.⁴

Key Takeaways

• Deciles divide an ordered dataset into ten equal groups, with each group representing 10% of the data.
• They are used to understand the distribution of values, such as income, wealth, or test scores, within a population.
• The first decile is the value below which 10% of the data falls, the second decile covers up to 20%, and so on, up to the ninth decile at 90%.
• Deciles help identify disparities and concentrations, providing insights into various socioeconomic and financial trends.
• They are a form of order statistics, requiring data to be sorted before calculation.

Formula and Calculation

Calculating deciles involves ordering the dataset from the smallest to the largest value and then identifying the values that mark the boundaries of each 10% segment. For a dataset with (N) data points, the position of the (k)-th decile ((D_k)) can be approximated by the formula:

Where:

• (P) = the position of the decile in the ordered dataset.
• (k) = the decile number (1 for the 1st decile, 2 for the 2nd decile, ..., 9 for the 9th decile).
• (N) = the total number of data points in the dataset.

If (P) is an integer, the (k)-th decile is typically the average of the value at position (P) and the value at position (P+1). If (P) is not an integer, the (k)-th decile is the value at the next whole number position. This method ensures that the data is divided into approximately ten equal parts.

Interpreting the Deciles

Interpreting deciles provides insight into where a particular value stands relative to the rest of the dataset. For instance, if an individual's income falls within the 7th decile, it indicates that their income is higher than 70% of the population within that specific income data distribution. Conversely, it also means that 30% of the population earns more. Deciles are often used in socioeconomic studies to illustrate income inequality or to categorize segments of a population based on specific characteristics, such as financial performance of companies or the distribution of credit scores.

Hypothetical Example

Consider a hypothetical dataset of annual incomes (in USD) for 20 individuals:

$30,000, $32,000, $35,000, $38,000, $40,000, $42,000, $45,000, $48,000, $50,000, $55,000, $60,000, $62,000, $65,000, $70,000, $75,000, $80,000, $85,000, $90,000, $95,000, $100,000

To find the first decile ((D_1)), we calculate its position:
(P = (1 \times 20) / 10 = 2)
The first decile is the average of the 2nd and 3rd values: ($32,000 + $35,000) / 2 = $33,500. This means 10% of individuals earn $33,500 or less.

To find the fifth decile ((D_5)), which is also the median:
(P = (5 \times 20) / 10 = 10)
The fifth decile is the average of the 10th and 11th values: ($55,000 + $60,000) / 2 = $57,500. This indicates that 50% of individuals earn $57,500 or less.

To find the ninth decile ((D_9)):
(P = (9 \times 20) / 10 = 18)
The ninth decile is the average of the 18th and 19th values: ($90,000 + $95,000) / 2 = $92,500. This means 90% of individuals earn $92,500 or less.

These deciles provide a quick snapshot of income distribution within this hypothetical group, offering insights into segments for market segmentation or policy analysis.

Practical Applications

Deciles are widely used across various financial and economic fields. In economics, they are fundamental for analyzing income inequality and wealth distribution within a country or globally. Organizations like the Federal Reserve and the Pew Research Center frequently publish reports detailing how household incomes and wealth are distributed across deciles to track socioeconomic trends.³,²

In finance, deciles can be applied in portfolio management to categorize investment performance. For instance, a fund manager might compare their fund's returns against a universe of similar funds, categorizing them into deciles to see where their fund ranks (e.g., in the top decile for performance). This helps in evaluating financial performance and refining investment strategies. Analysts also use deciles for risk assessment, segmenting populations by credit scores or default rates to understand risk profiles across different groups.

Limitations and Criticisms

While deciles offer a straightforward way to categorize data, they have limitations. Like all quantile measures, deciles simplify complex distributions into discrete segments, potentially obscuring nuances within each 10% group. For example, two individuals in the same decile might have significantly different values if the data within that decile is widely spread. Furthermore, deciles only indicate relative position, not absolute value. A person in the top decile of wealth in a developing country might still have less wealth than someone in a lower decile in a highly developed economy.

Critics also point out that focusing solely on deciles for income inequality can sometimes oversimplify the underlying structural factors contributing to the distribution. For example, the Brookings Institution highlights the complexities in comparing distributional data across countries and cautions that focusing solely on growth incidence curves can be an "overburdened and inaccurate depiction".¹ Such analyses need to consider factors like household composition, age, and regional cost of living, which deciles alone do not account for. This can impact discussions around economic mobility and policy interventions.

Deciles vs. Quartiles

Deciles and quartiles are both types of quantiles used to divide datasets, but they differ in the number of segments they create. Quartiles divide data into four equal parts, or quarters, marked by three cutoff points: the first quartile (Q1) at the 25th percentile, the second quartile (Q2) at the 50th percentile (which is also the median), and the third quartile (Q3) at the 75th percentile.

The confusion often arises because both provide insights into the spread of data. However, deciles offer a more granular view of the data distribution by breaking it down into smaller, 10% increments, whereas quartiles offer a broader, quarter-based perspective. Choosing between deciles and quartiles depends on the level of detail required for the statistical analysis.

FAQs

How are deciles different from percentiles?

Deciles are a specific type of percentiles. A percentile divides data into 100 equal parts. So, the first decile is equivalent to the 10th percentile, the second decile is the 20th percentile, and so on, up to the ninth decile which is the 90th percentile. Deciles simplify the percentile concept into broader, more manageable segments.

Why are deciles used in financial reporting?

Deciles are used in financial reporting to segment and analyze various financial metrics, such as investment returns, bond yields, or credit scores. This allows analysts to compare performance or risk profiles across different groups. For example, a mutual fund's financial performance might be ranked by decile against its peer group, indicating whether it falls within the top 10% or bottom 10%.

Can deciles be used for qualitative data?

Deciles are primarily used for quantitative, numerical data points that can be ordered from lowest to highest. While qualitative data can sometimes be assigned numerical ranks (e.g., survey responses on a Likert scale), true decile analysis requires a measurable, continuous, or ordinal scale.

Do deciles account for population size?

Deciles are based on the relative position within an ordered dataset, meaning they inherently reflect the distribution across the entire population being analyzed, regardless of its absolute size. When comparing different populations, it's important to ensure the data is normalized or adjusted for relevant factors like population size or household composition to ensure meaningful comparisons regarding economic indicators.