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Deciles

Deciles

The 10th and 90th percentiles of a set of data.

 

 

See also

Quartiles, quintiles

Key Formula

Dk=value at position k(n+1)10D_k = \text{value at position } \frac{k(n+1)}{10}
Where:
  • DkD_k = The k-th decile, where k ranges from 1 to 9
  • nn = The number of data values in the ordered set
  • kk = The decile number (1 through 9)

Worked Example

Problem: Find the 3rd decile (D3) of the following 20 test scores: 35, 42, 48, 51, 53, 56, 58, 61, 63, 65, 67, 70, 72, 75, 78, 80, 84, 88, 92, 97.
Step 1: The data is already sorted in ascending order, with n = 20 values.
Step 2: Find the position of the 3rd decile using the formula.
Position=3(20+1)10=6310=6.3\text{Position} = \frac{3(20+1)}{10} = \frac{63}{10} = 6.3
Step 3: Position 6.3 falls between the 6th value (56) and the 7th value (58). Interpolate: take the 6th value and add 0.3 of the difference between the 7th and 6th values.
D3=56+0.3×(58−56)=56+0.6=56.6D_3 = 56 + 0.3 \times (58 - 56) = 56 + 0.6 = 56.6
Answer: The 3rd decile is 56.6, meaning approximately 30% of the test scores fall at or below 56.6.

Another Example

Problem: A company records the daily sales (in units) over 10 days: 12, 15, 18, 22, 25, 28, 30, 35, 40, 50. Find D1 and D9.
Step 1: The data is sorted with n = 10. Find the position of D1.
Position of D1=1(10+1)10=1.1\text{Position of } D_1 = \frac{1(10+1)}{10} = 1.1
Step 2: Position 1.1 is between the 1st value (12) and the 2nd value (15). Interpolate.
D1=12+0.1×(15−12)=12+0.3=12.3D_1 = 12 + 0.1 \times (15 - 12) = 12 + 0.3 = 12.3
Step 3: Now find the position of D9.
Position of D9=9(10+1)10=9.9\text{Position of } D_9 = \frac{9(10+1)}{10} = 9.9
Step 4: Position 9.9 is between the 9th value (40) and the 10th value (50). Interpolate.
D9=40+0.9×(50−40)=40+9=49D_9 = 40 + 0.9 \times (50 - 40) = 40 + 9 = 49
Answer: D1 = 12.3 (the bottom 10% boundary) and D9 = 49 (the top 10% boundary).

Frequently Asked Questions

How many deciles are there?
There are nine deciles (D1 through D9), which create ten equal groups. This is similar to how three quartiles create four groups. The fifth decile (D5) is the same as the median.
What is the difference between deciles and percentiles?
Deciles are specific percentiles spaced 10% apart. D1 equals the 10th percentile, D2 equals the 20th percentile, and so on. Percentiles offer finer divisions (99 cut points), while deciles give a broader summary with just 9 cut points.

Deciles vs. Quartiles

Both divide ordered data into equal parts, but at different levels of detail.

Why It Matters

Deciles are widely used in economics and finance to describe income or wealth distribution — for example, reporting what share of total income the top decile earns. Standardized test reports often show your score's decile to give you a quick sense of where you stand relative to other test-takers. They provide a useful middle ground between quartiles (too coarse) and percentiles (too fine) for summarizing how data is spread.

Common Mistakes

Mistake: Thinking there are 10 deciles instead of 9.
Correction: There are 9 decile values (D1 through D9) that create 10 groups. The minimum and maximum of the data set are not deciles — they are the boundaries of the first and last groups.
Mistake: Confusing the decile number with the decile group.
Correction: D3 is the value at the 30th percentile mark. The 'third decile group' refers to all data points between D2 and D3 (i.e., the 20th to 30th percentile range). Be clear whether you mean the cut point or the group.

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