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Decile

A decile is a quantile that divides a sorted dataset into ten equal parts, each containing 10% of the observations, with the nine decile points corresponding to the 10th, 20th, 30th, 40th, 50th, 60th, 70th, 80th, and 90th percentiles.^[1] This measure extends the concept of quartiles (which divide data into four parts) and quintiles (five parts) by providing finer granularity for analyzing distributions, particularly in large datasets where understanding segmented ranges is essential.^[2] Deciles are calculated by first ordering the data from lowest to highest and then identifying positions using the formula for the k-th decile: L_{D_k} = \frac{k(n+1)}{10}, where n is the number of data points and k ranges from 1 to 9; if the position falls between integers, interpolation is typically applied.^[3] For grouped or continuous data, an adjusted formula incorporates cumulative frequencies and class intervals to estimate the decile value within the relevant interval.^[4] In practice, deciles are widely applied in economics, finance, and social sciences to summarize income, wealth, and earnings distributions, revealing patterns of inequality and variability across population segments.^[2] For instance, the U.S. Bureau of Labor Statistics routinely publishes deciles alongside quartiles to describe usual weekly earnings for full-time workers, aiding policymakers in assessing labor market trends.^[5] Similarly, deciles help in educational and health research to categorize outcomes by socioeconomic groups, such as identifying mortality differentials across earnings deciles.^[6]

Definition and Fundamentals

Definition of Decile

A decile is any of the nine values that divide a sorted dataset into ten equal-frequency subsets, with each subset containing 10% of the data points.^[7] These values mark the boundaries where the cumulative frequency reaches 10%, 20%, up to 90% of the total observations.^[8] The term "decile" derives from the Latin word decem, meaning "ten," reflecting its role in partitioning data into tenths.^[9] In statistical contexts, the concept was first introduced in 1882 by Francis Galton, who used it to describe divisions in anthropometric data distributions.^[10] Deciles are typically denoted as D_k for the k-th decile, where k = 1 to $9, representing the lower deciles that separate the subsets. Deciles represent specific instances within the broader framework of percentiles, which generalize such divisions to any percentage.^[8]

Relation to Percentiles and Quartiles

Deciles represent specific instances of percentiles, dividing a dataset or probability distribution into ten equal parts, each comprising 10% of the data. The k-th decile corresponds precisely to the (10k)-th percentile, such that the first decile (D1) is the 10th percentile, the second decile (D2) is the 20th percentile, and so on, up to the ninth decile (D9) as the 90th percentile.^[11]^[12] In comparison to quartiles, which partition data into four equal segments of 25% each—denoted as the first quartile (Q1 at the 25th percentile), second quartile (Q2 at the 50th percentile), third quartile (Q3 at the 75th percentile), and with the median serving as Q2—deciles offer a more subdivided view by creating ten segments of 10% each. Notably, the median aligns as both the second quartile (Q2) and the fifth decile (D5), providing a common reference point across these measures.^[13]^[14] Visually, deciles appear as points along the cumulative distribution function (CDF) of a random variable, marking the values where the CDF reaches 0.1, 0.2, ..., 0.9, thereby illustrating the progressive accumulation of probability mass in the distribution. This positioning on the CDF highlights how deciles capture the quantiles at these intervals, offering a stepwise depiction of the distribution's shape.^[15]^[16] Deciles provide advantages over quartiles by delivering finer granularity, which is particularly beneficial for analyzing skewed distributions where additional division points better reveal asymmetries and tail behaviors that coarser quartiles might obscure.^[14]^[17]

Calculation and Computation

Empirical Method for Sample Data

To compute deciles from a finite sample dataset, begin by sorting the data in ascending order to obtain the ordered sample x_1 \leq x_2 \leq \cdots \leq x_n, where n is the sample size.^[18] The k-th decile D_k (for k = 1, 2, \dots, 9) divides the data such that approximately 10k% of the observations lie at or below it.^[18] The position of the k-th decile in the ordered sample is given by the formula
i_k = \frac{k}{10} (n + 1).
If i_k is an integer i, then D_k = x_i.^[18] This formula applies regardless of whether n is even or odd, as the addition of 1 ensures consistent positioning across sample sizes.^[18] If i_k is not an integer, express it as i_k = i + f, where i = \lfloor i_k \rfloor is the integer part and $0 < f < 1 is the fractional part. Linear interpolation yields
D_k = x_i + f (x_{i+1} - x_i).
This approach provides a smooth estimate between adjacent ordered values.^[18] In the presence of ties (repeated values in the dataset), sort the data as usual, placing tied observations consecutively in the ordered list; the position formula and interpolation proceed unchanged, using the tied values directly, which naturally averages across equal observations when the fraction f spans them.^[18] For exact integer positions falling on tied values, the decile takes that shared value; if interpolation requires averaging adjacent tied values (e.g., f = 0.5 between identical x_i and x_{i+1}), the result remains the tied value itself.^[18] Consider a small example dataset of 10 test scores: 55, 62, 67, 71, 74, 78, 82, 85, 89, 95 (n = 10). The ordered data are x = [55, 62, 67, 71, 74, 78, 82, 85, 89, 95]. Positions are i_k = (k/10) \times 11.

For D_1: i_1 = 1.1, so D_1 = 55 + 0.1(62 - 55) = 55 + 0.7 = 55.7.
For D_2: i_2 = 2.2, so D_2 = 62 + 0.2(67 - 62) = 62 + 1 = 63.
For D_3: i_3 = 3.3, so D_3 = 67 + 0.3(71 - 67) = 67 + 1.2 = 68.2.
For D_4: i_4 = 4.4, so D_4 = 71 + 0.4(74 - 71) = 71 + 1.2 = 72.2.
For D_5: i_5 = 5.5, so D_5 = 74 + 0.5(78 - 74) = 74 + 2 = 76.
For D_6: i_6 = 6.6, so D_6 = 78 + 0.6(82 - 78) = 78 + 2.4 = 80.4.
For D_7: i_7 = 7.7, so D_7 = 82 + 0.7(85 - 82) = 82 + 2.1 = 84.1.
For D_8: i_8 = 8.8, so D_8 = 85 + 0.8(89 - 85) = 85 + 3.2 = 88.2.
For D_9: i_9 = 9.9, so D_9 = 89 + 0.9(95 - 89) = 89 + 5.4 = 94.4.

This computation divides the sample into 10 equal parts, each containing 10% of the data.^[18]

Theoretical Deciles in Distributions

In probability theory, the k-th decile D_k of a random variable X with cumulative distribution function (CDF) F is defined as the value satisfying P(X \leq D_k) = k/10, for k = 1, 2, \dots, 9.^[19] This places D_k at the (k/10)-quantile of the distribution, dividing the probability mass into ten equal parts below and above it.^[19] The theoretical decile is computed using the quantile function, the inverse of the CDF, given by D_k = F^{-1}(k/10).^[19] For continuous distributions where F is strictly increasing, this inverse exists uniquely; for general cases, it is defined as the generalized inverse \inf\{x : F(x) \geq k/10\}.^[19] This probabilistic approach contrasts with empirical methods by relying on the population distribution rather than observed data. For the normal distribution with mean \mu and standard deviation \sigma, the deciles are derived from the standard normal quantile function using z-scores. Specifically, the first decile D_1 corresponds to a z-score of approximately -1.28, so D_1 \approx \mu - 1.28\sigma.^[20] Higher deciles follow similarly, with z-scores increasing toward positive values (e.g., D_9 \approx \mu + 1.28\sigma).^[20] In the uniform distribution on the interval [a, b], the CDF is F(x) = (x - a)/(b - a) for a \leq x \leq b, yielding the quantile function D_k = a + (k/10)(b - a)./03%3A_Distributions/3.06%3A_Distribution_and_Quantile_Functions) This linear form evenly spaces the deciles across the interval, reflecting the constant density. As sample sizes grow large, empirical deciles—computed from sorted sample data—converge uniformly almost surely to their theoretical counterparts, as established by the Glivenko-Cantelli theorem, which guarantees that the empirical CDF converges to the true CDF.^[21] This asymptotic property ensures that sample-based approximations reliably approach the population deciles for sufficiently large datasets.^[21]

Applications and Uses

In Descriptive Statistics

In descriptive statistics, deciles serve as key tools for summarizing data distributions during exploratory data analysis by dividing ordered data into ten equal parts, each encompassing 10% of the observations, thereby providing a more granular view of spread and variability than quartiles alone.^[7] This finer partitioning allows analysts to identify patterns in data concentration and dispersion that might be obscured by coarser summaries. Deciles relate to percentiles as subsets, where each decile corresponds to a 10% interval (e.g., the first decile aligns with the 10th percentile), enabling broader quantile-based overviews of the dataset. Deciles enhance visualizations such as box plots and histograms by extending the interquartile range (IQR) to reveal more detailed aspects of the distribution's spread. In extended box plot variants or quantile-based diagrams, decile markers can delineate multiple intervals beyond the IQR, facilitating the detection of outliers that lie above the ninth decile (D9) or below the first (D1), where values exceed typical variation thresholds.^[22] Similarly, in histograms modified using deciles—known as decile histograms—bins are constructed with equal frequencies (each containing exactly 10% of the data), which highlights skewness, multimodality, and tail behaviors more effectively than equal-width bins, as demonstrated in analyses of marathon finishing times showing right-skewed distributions.^[23] For assessing skewness, decile ranges offer a robust, non-parametric approach by comparing the widths of intervals in the lower and upper tails; for instance, a wider span between the median and D9 versus D1 and the median indicates positive skewness, reflecting a longer right tail without relying on moments like Pearson's coefficient.^[24] This method is particularly useful in exploratory settings with potential outliers, as it leverages central deciles for symmetry evaluation. Decile-based frequency tables approximate categorical representations of continuous data by grouping observations into ten equiprobable classes defined by decile boundaries, which simplifies analysis of large datasets and reveals proportional distributions across ordered categories. Interpretation of deciles emphasizes their role in conveying relative positioning; for example, if 80% of values fall below the eighth decile (D8), it signals a concentration in lower values, suggesting potential floor effects or inequality in the distribution that warrants further investigation.^[7] Such insights guide decisions in data exploration, like identifying subgroups for deeper scrutiny. In economics, decile ratios such as the D9/D1 ratio—comparing the average income of the ninth decile (80th to 90th percentile) to the first decile (0th to 10th percentile)—serve as key metrics for assessing income inequality, offering a simpler alternative to the Gini coefficient by highlighting disparities between upper-middle and lower income groups.^[25] The World Bank has incorporated decile-based measures, including shares of income held by each decile and ratios like the 90/10 equivalent, into its poverty and inequality analyses since the 1990s, using them in global reports to track shared prosperity and distributional changes across countries.^[26] For instance, these ratios help quantify how growth benefits the poor versus the affluent, with applications in evaluating pro-poor policies in emerging economies.^[27] In finance, deciles are employed to rank asset returns and construct factor models for performance evaluation. The Fama-French three-factor model, introduced in 1993, sorts stocks into decile portfolios based on market capitalization (size) and book-to-market ratios, revealing that small-cap and value stocks (often in lower or higher deciles, respectively) exhibit higher average returns, which informs investment strategies and risk premia estimation.^[28] This decile-based sorting extends to portfolios formed on operating profitability and investment factors in updated models, enabling analysts to benchmark asset performance against market benchmarks.^[29] Additionally, in risk management, the first decile (D1) of historical return distributions approximates Value at Risk (VaR) at the 90% confidence level under the historical simulation method, where past losses are ranked to estimate potential downside without assuming normality, providing a non-parametric bound for portfolio tail risk. In the social sciences, particularly education policy, deciles facilitate analysis of attainment and performance disparities. The OECD's Programme for International Student Assessment (PISA) reports group student outcomes by socio-economic deciles using the PISA index of economic, social, and cultural status (ESCS), revealing how performance in mathematics, reading, and science varies across income bands to inform equity-focused reforms.^[30] For example, PISA 2022 data showed that in many countries, students in the top socio-economic decile outperformed those in the bottom by over 90 score points in mathematics, guiding policies on access and resource allocation.^[31] This decile framework also extends to cross-country comparisons, where nations are analyzed in performance bands to identify systemic strengths and gaps. A notable case study is the trend in U.S. household income deciles from Census Bureau data, which illustrates widening inequality post-2008 recession. Between 2007 and 2016, the mean income of the top decile (90th-100th percentile) grew faster than the bottom decile, with the 90/10 income ratio rising from approximately 9.5 to 10.2, reflecting slower recovery for lower-income households amid wage stagnation and job losses. This D9-D1 gap expansion, driven by factors like financial sector gains benefiting higher deciles, underscores broader economic polarization, as upper-decile households captured a larger share of post-recession growth.^[32] By 2018, the disparity had further intensified, with the top decile's mean income exceeding $200,000 compared to under $20,000 for the bottom. However, more recent data from the U.S. Census Bureau indicate a reversal, with income inequality decreasing in 2022—the first decline since 2007—as the Gini index fell from 0.410 in 2021 to 0.397 in 2022, amid broader income gains across the distribution.^[33]

Special Concepts and Variants

Decile Mean

The decile mean refers to the arithmetic mean computed separately for the observations within each decile group of a sorted dataset, where the data are divided into ten equal-sized segments representing 10% of the total observations each. This technique segments the distribution to highlight central tendencies at different points along the range, offering a granular view of how averages vary across the data spectrum. Unlike the overall arithmetic mean, which aggregates all values and can be disproportionately skewed by outliers in the tails, decile means confine the influence of extremes to their specific groups, thereby providing a more balanced representation within bounded intervals.^[34] The formula for the mean of the j-th decile group, where j = 1, 2, \dots, 10, is given by

\bar{x}_j = \frac{\sum_{i \in D_j} x_i}{|D_j|},

where D_j is the set of observations in the j-th decile, x_i are the data values, and |D_j| = n/10 for a dataset of size n (assuming n is divisible by 10 for simplicity; interpolation may be used otherwise). An aggregated overall decile mean can then be formed as the weighted average

\bar{x} = \sum_{j=1}^{10} \left( \frac{1}{10} \right) \bar{x}_j,

which equates to the dataset's global arithmetic mean but underscores the contribution of each segment. This computation is particularly useful in analyses requiring decomposition of the distribution, as it facilitates the examination of subgroup-specific summaries without the distortion from the full range.^[35] In practice, decile means have been employed in distributional analyses, such as income studies, to reveal intra-group patterns that the overall mean obscures. For instance, analyses of U.S. family income distributions show that means for lower deciles are substantially lower than those for upper deciles, illustrating skewness and the impact of high-end values on overall averages. This approach enhances robustness in descriptive contexts by focusing on localized averages, making it valuable for identifying equitable resource allocation or inequality trends without overemphasizing extremes.^[35]

Decile Ranks and Bands

Decile ranking assigns ordinal scores to data points by sorting the dataset in ascending order and dividing it into ten equal-frequency groups, where the lowest 10% receive rank 1 (bottom decile) and the highest 10% receive rank 10 (top decile).^[8] This approach provides a standardized way to categorize observations based on their relative position within the distribution, facilitating comparisons across datasets.^[8] The specific rank score for an observation can be calculated using the formula \text{decile rank} = \left( \frac{\text{rank} - 1}{n - 1} \right) \times 10, where rank is the ordered position (from 1 to n) and n is the total sample size; this yields a continuous value ranging from 0 to 10, typically rounded or adjusted to the nearest integer from 1 to 10 for categorization.^[36] Decile bands extend this ranking by aggregating individual deciles into broader intervals, which simplifies visualization and highlights patterns in large datasets. For instance, data may be grouped into low bands (D1–D3, representing the bottom 30%), middle bands (D4–D7, the central 40%), and high bands (D8–D10, the top 30%), allowing for clearer representation in charts or reports without overwhelming detail.^[37] Such banding is particularly useful in exploratory data analysis, where it reduces complexity while preserving the ordinal structure of the deciles.^[37] In applications like scoring models, decile bands have been employed in credit scoring systems to delineate risk tiers since the 1980s, when computational advances enabled widespread adoption of automated risk assessment.^[38] Lenders segment applicants into these bands based on predicted default probabilities, with lower deciles indicating higher risk and guiding decisions on interest rates or approval thresholds; for example, the bottom decile might represent the highest-risk tier requiring additional scrutiny.^[38] This practice emerged prominently with the development of models like FICO in 1989, enhancing efficiency in retail credit evaluation.^[38] A key limitation of decile ranks and bands is the loss of granularity within each group, as all values in a band are treated uniformly despite potential internal variations.^[39] For example, consider a dataset of 100 student test scores sorted from 0 to 100; the bottom decile (ranks 1–10, scores 0–25) might include values ranging from 0 to 25, but assigning all to rank 1 or the low band obscures differences, such as distinguishing a score of 5 from 24, which could affect nuanced interpretations of performance.^[39] This aggregation can lead to oversimplification in ordinal analysis, particularly in smaller samples where band widths amplify the issue.^[39]

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