A decile is a type of quantile (a value that splits ordered data into equal-sized groups). Specifically, deciles divide a ranked dataset into ten equal portions. Each decile point marks the cutoff such that a fixed percentage of the observations lie at or below it (10% per decile point).
Key terms
– Quantile: a cutoff value that partitions data into intervals with equal probability or count.
– Decile rank: the category number from 1 to 10 assigned to a value according to which tenth of the distribution it falls into.
– Interpolation: when the decile position falls between two actual data points, you estimate the decile value by proportionally weighting the two neighboring observations.
When to use deciles
Deciles help summarize large datasets and compare items relative to a group. Common uses include:
– Ranking firms by valuation multiples (e.g., price-to-earnings) to find the cheapest and most expensive deciles.
– Grouping mutual funds or portfolios to compare performance percentiles.
– Measuring income distribution (e.g., share of total income received by the top decile) to study inequality.
How to calculate deciles (step-by-step)
1. Sort the data in ascending order (smallest to largest).
2. Let n be the number of observations.
3. For the kth decile (k = 1, 2, …, 9) compute the position index: P = k*(n + 1) / 10.
4. If P is an integer, the decile value is the data point at position P.
5. If P is not an integer, let m = floor(P). Interpolate between the mth and (m+1)th data values:
Decile_k = value_m + (P − m) × (value_{m+1} − value_m).
6. D5 (k = 5) corresponds to the median using this formula.
Checklist for computing and interpreting deciles
– [ ] Data are sorted consistently (ascending or descending).
– [ ] Count n and verify n + 1 used in the formula if using the interpolation method shown.
– [ ] Decide whether to use interpolation or a rank-based method before computing (be consistent).
– [ ] When interpreting: D1 ≈ 10% cutoff, D2 ≈ 20% cutoff, …, D9 ≈ 90% cutoff.
– [ ] Remember D5 ≈ 50% cutoff (median) for central tendency comparisons.
Worked numeric example
We have 30 exam scores (already sorted, ascending):
48, 52, 55, 57, 58, 60, 61, 64, 65, 66, 69, 72, 73, 75, 76, 78, 81, 82, 84, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97, 99
n = 30, so n + 1 = 31.
1) First decile (D1), k = 1:
P = 1 × 31 / 10 = 3.1 → falls between 3rd and 4th values.
3rd value = 55; 4th value = 57.
D1 = 55 + (0.1) × (57 − 55) = 55 + 0.2 = 55.2
Interpretation: about 10% of scores are at or below approximately 55.2.
2) Third decile (D3), k = 3:
P = 3 × 31 / 10 = 9.3 → between 9th and 10th values.
9th = 65; 10th = 66.
D3 = 65 + (0.3) × (66 − 65) = 65 + 0.3 = 65.3
Interpretation: about 30% of scores are at or below ~65.3.
3) Fifth decile (D5, median), k = 5:
P = 5 × 31 / 10 =
P = 5 × 31 / 10 = 15.5 → falls between the 15th and 16th values.
15th value = 72; 16th value = 74.
D5 = 72 + (0.5) × (74 − 72) = 72 + 1 = 73.
Interpretation: about 50% of scores are at or below approximately 73 (D5 is the median).
4) Seventh decile (D7), k = 7:
P = 7 × 31 / 10 = 21.7 → between the 21st and 22nd values.
21st = 82; 22nd = 83.
D7 = 82 + (0.7) × (83 − 82) = 82 + 0.7 = 82.7.
Interpretation: about 70% of scores are at or below ~82.7.
5) Ninth decile (D9), k = 9:
P = 9 × 31 / 10 = 27.9 → between the 27th and 28th values.
27th = 90; 28th = 92.
D9 = 90 + (0.9) × (92 − 90) = 90 + 1.8 = 91.8.
Interpretation: about 90% of scores are at or below ~91.8.
Quick checklist — how to compute a decile (kth decile, k = 1..9)
– Sort the data in ascending order.
– Compute position P = k × n / 10 (this example uses the convention P = k·
·n/10). – If P is an integer, the kth decile is the value at position P. – If P is not an integer, let L = floor(P) and U = ceil(P). Let f = P − L (the fractional part). Then interpolate:
Decile k = value at position L + f × (value at position U − value at position L).
– Report which convention you used (P = k·n/10 versus alternatives) and whether you interpolated.
Worked micro‑example
– Data (sorted): [12, 15, 18, 21]. n = 4.
– Want D3 (k = 3): P = 3 × 4 / 10 = 1.2 → between positions 1 and 2.
– L = 1 (value = 12), U = 2 (value = 15), f = 0.2.
– D3 = 12 + 0.2 × (15 − 12) = 12 + 0.6 = 12.6.
Common alternative conventions
– Some sources use P = k·(n + 1)/10; others apply different interpolation algorithms (many statistical packages implement one of several types). These choices matter most when n is small. Always state the method when publishing results.
Practical checklist before you report deciles
1. Sort the data and confirm n.
2. Choose and document the position rule (k·n/10 or k·(n+1)/10) and interpolation method.
3. Compute P and apply integer or interpolation rule.
4. Check edge cases: k = 1..9, ties, repeated values, and very small n.
5. For grouped/frequency data use cumulative frequencies to locate P, then interpolate inside the group.
6. Explain interpretation: “About k×10% of observations are at or below Dk” (approximate—depends on method).
When to use deciles (and when not)
– Use deciles to summarize distribution shape or to compare relative standing in a population without overloading with detail.
– Avoid deciles for tiny samples (few observations) or when a smooth parametric model is more appropriate. For finer resolution use percentiles; for coarser use quartiles or tertiles.
Software quick reference
– Excel: PERCENTILE.INC(array, k/10) returns the kth decile using the inclusive percentile convention. Some users prefer PERCENTILE.EXC(array, k/10); read Excel help to understand the difference.
– Python / NumPy: numpy.percentile(data, 10*k, interpolation=’linear’) or in newer NumPy versions numpy.percentile(data, 10*k, method=’linear’). Example: numpy.percentile([12,15,18,21], 30) → 12.6 for D3.
– R: quantile(x, probs = k/10, type = 7) — R supports multiple types; type = 7 is the default and corresponds to one common interpolation rule.
Pitfalls and notes
– Different software/packages use different interpolation rules; this changes deciles slightly for finite samples.
– For discrete or binned data, interpolation produces values that may not occur in the raw data—state this when interpreting results.
– Deciles summarize relative position, not probability of future outcomes. They are descriptive, not predictive.
Reputable references
– Investopedia — Decile:
– NIST/SEMATECH e-Handbook of Statistical Methods — Percentiles and Quantiles:
– Microsoft Support — PERCENTILE.INC/PERCENTILE.EXC:
– NumPy documentation — numpy.percentile:
– R documentation — quantile
Educational disclaimer
This explanation is for educational purposes only and does not constitute individualized investment advice or recommendations. Methods and interpretations may vary; verify choices (position rule, interpolation) before using deciles in analysis.