standarddeviationcalculator.net

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Decile deviation calculator

A decile-based measure of spread: the distance covered by the middle 80% of the data. Enter a list, an (x, f) table or class intervals, and the calculator finds D₁ and D₉, then the interdecile range, the decile deviation and its coefficient.

Class intervals and frequencies
Class (e.g. 10-20)Frequency fRemove

Exclusive classes (10-20, 20-30) or inclusive classes (10-19, 20-29). Inclusive classes are converted to class boundaries.

Decile deviation (D₉ − D₁)/2 19.1667
Interdecile range D₉ − D₁38.3333
Coefficient (D₉ − D₁)/(D₉ + D₁)0.6571
D₁ (10th percentile)10
D₉ (90th percentile)48.3333
Quartile deviation, for comparison9.47917
DataN = 50 (6 classes)
MethodClass-interval formula
ClassfcfContains
0-1055D₁
10-20813
20-301225
30-401540
40-50646D₉
50-60450
N50
010203040506001020304050 D₁D₉ upper class boundary cumulative frequency

Less-than ogive. The marked points are read off at cumulative frequencies 5, 45.

Show the working, step by step

  1. Add a cumulative frequency (cf) column. N = Σf = 50.

    0–10: f = 5, cf = 5 10–20: f = 8, cf = 13 20–30: f = 12, cf = 25 30–40: f = 15, cf = 40 40–50: f = 6, cf = 46 50–60: f = 4, cf = 50

  2. D₁ class: N/10 = 50/10 = 5. The first cumulative frequency ≥ 5 is 5, so the D₁ class is 0–10.

    D₁ = L + ((N/10 − cf) / f) × h = 0 + ((5 − 0) / 5) × 10 = 10

  3. D₉ class: 9N/10 = 9 × 50/10 = 45. The first cumulative frequency ≥ 45 is 46, so the D₉ class is 40–50.

    D₉ = L + ((9N/10 − cf) / f) × h = 40 + ((45 − 40) / 6) × 10 = 48.3333

  4. Interdecile range: the spread of the middle 80% of the data.

    D₉ − D₁ = 48.3333 − 10 = 38.3333

  5. Decile deviation (semi-interdecile range): half of it.

    (D₉ − D₁)/2 = 38.3333/2 = 19.1667

  6. Coefficient of decile deviation.

    (D₉ − D₁)/(D₉ + D₁) = 38.3333/58.3333 = 0.6571

D₉ − D₁ is the same as the 10–90 percentile range P₉₀ − P₁₀.

Definitions used on this page

Textbooks vary in what they call “decile deviation”, so here are the three measures the calculator reports:

Interdecile range (10–90 range) = D₉ − D₁ = P₉₀ − P₁₀ Decile deviation (semi-interdecile range) = (D₉ − D₁) / 2 Coefficient of decile deviation = (D₉ − D₁) / (D₉ + D₁)

These mirror the quartile measures exactly: the interquartile range Q₃ − Q₁, the quartile deviation (Q₃ − Q₁)/2 and its coefficient (Q₃ − Q₁)/(Q₃ + Q₁), with the first and ninth deciles in place of the quartiles. D₁ and D₉ are found with the usual rules: the k(n + 1)/10th item for listed values, and Dₖ = L + ((kN/10 − cf)/f) × h for class intervals.

Worked example: continuous series

The calculator opens on this distribution (N = 50):

Classfcf
0-1055
10-20813
20-301225
30-401540
40-50646
50-60450
  1. D₁: N/10 = 5, reached exactly by the first class. D₁ = 0 + ((5 − 0)/5) × 10 = 10.
  2. D₉: 9N/10 = 45 falls in 40-50 (cf 46). D₉ = 40 + ((45 − 40)/6) × 10 = 48.333.
  3. Interdecile range = 48.333 − 10 = 38.333.
  4. Decile deviation = 38.333/2 = 19.167.
  5. Coefficient = 38.333/(48.333 + 10) = 38.333/58.333 = 0.6571.

For comparison, the quartile deviation of the same data is 9.479. The decile deviation is about twice as large because it reaches further into the tails.

Worked example: individual series

For 23, 15, 31, 8, 26, 12, 38, 17, 29, 21 (sorted 8, 12, …, 38), D₁ is the 1.1th item, 8 + 0.1 × (12 − 8) = 8.4, and D₉ is the 9.9th item, 31 + 0.9 × (38 − 31) = 37.3. The decile deviation is (37.3 − 8.4)/2 = 14.45, with coefficient 28.9/45.7 = 0.6324. With only ten values the deciles sit next to the minimum and maximum, so this measure is best kept for larger data sets.

Reading the result

For a normal distribution, D₁ and D₉ lie 1.2816 standard deviations either side of the mean, so the decile deviation is about 1.28σ and the interdecile range about 2.56σ. If the decile deviation of your data is well below 1.28 times its standard deviation, heavy tails are pulling the standard deviation up; well above it, the data are flatter than a normal curve.

MeasureShare of data coveredNormal-curve value
Quartile deviation (Q₃ − Q₁)/2Middle 50%0.674σ
Decile deviation (D₉ − D₁)/2Middle 80%1.282σ
Half the rangeAll of itGrows with n

Common mistakes

  • Reporting D₉ − D₁ when the question asks for the decile deviation, or the other way round. Check which definition your book uses.
  • Using k(N + 1)/10 with class intervals; grouped data uses kN/10.
  • Using class limits for inclusive classes. The calculator converts 10-19, 20-29 to boundaries 9.5-19.5, 19.5-29.5.

Common questions

What is decile deviation?

Half the distance between the ninth and first deciles: (D₉ − D₁)/2. It is also called the semi-interdecile range. It measures the spread of the middle 80% of the data, in the same way that the quartile deviation measures the spread of the middle 50%.

What is the coefficient of decile deviation?

(D₉ − D₁)/(D₉ + D₁). Like the coefficient of quartile deviation it has no units, so it can compare series with different units or averages. For the default grouped data it is 38.33/58.33 = 0.6571.

Is the interdecile range the same as the 10–90 percentile range?

Yes. D₁ is the 10th percentile and D₉ the 90th, so D₉ − D₁ = P₉₀ − P₁₀. Some books call it the percentile range or the 10–90 range. Halving it gives the decile deviation.

Why use deciles instead of quartiles for dispersion?

The interdecile range covers 80% of the observations instead of 50%, so it says more about the spread of the bulk of the data while still ignoring the most extreme 10% at each end. It is common in income statistics, where the extremes are very far out.

Do all textbooks define decile deviation the same way?

Not quite. Most use (D₉ − D₁)/2 and call D₉ − D₁ the interdecile range; a few quote the range D₉ − D₁ itself as the measure. This page reports both, with the definition printed next to each number, so you can use whichever your book asks for.

Written and reviewed by our editorial team. Last updated . Method and sources: how these numbers are computed.