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Quartile deviation calculator

Quartile deviation measures the spread of the middle half of a distribution. Enter a list, an (x, f) table or class intervals; the calculator finds Q₁ and Q₃, then the quartile 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.

Quartile deviation (Q₃ − Q₁)/2 9.47917
Coefficient of QD0.3285
Q₁19.375
Q₃38.3333
Interquartile range Q₃ − Q₁18.9583
DataN = 50 (6 classes)
MethodClass-interval formula
ClassfcfContains
0-1055
10-20813Q₁
20-301225
30-401540Q₃
40-50646
50-60450
N50
010203040506001020304050 Q₁Q₃ upper class boundary cumulative frequency

Less-than ogive. The marked points are read off at cumulative frequencies 12.5, 37.5.

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. Q₁ class: N/4 = 50/4 = 12.5. The first cumulative frequency ≥ 12.5 is 13, so the Q₁ class is 10–20.

    Q₁ = L + ((N/4 − cf) / f) × h = 10 + ((12.5 − 5) / 8) × 10 = 19.375

  3. Q₃ class: 3N/4 = 3 × 50/4 = 37.5. The first cumulative frequency ≥ 37.5 is 40, so the Q₃ class is 30–40.

    Q₃ = L + ((3N/4 − cf) / f) × h = 30 + ((37.5 − 25) / 15) × 10 = 38.3333

  4. Quartile deviation: half the interquartile range.

    QD = (Q₃ − Q₁)/2 = (38.3333 − 19.375)/2 = 18.9583/2 = 9.47917

  5. Coefficient of quartile deviation: a unit-free ratio for comparing series.

    (Q₃ − Q₁)/(Q₃ + Q₁) = 18.9583/57.7083 = 0.3285

L is the lower boundary of the quartile class, cf the cumulative frequency of the class before it, f the frequency of the quartile class and h its width.

The formulas

Quartile deviation QD = (Q₃ − Q₁) / 2 Coefficient of QD = (Q₃ − Q₁) / (Q₃ + Q₁)

The quartiles themselves come from the usual rules: the k(n + 1)/4th item for a list or a discrete series, and Qₖ = L + ((kN/4 − cf)/f) × h for class intervals. The quartile calculator explains each rule in detail.

Worked example: continuous series

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

Classfcf
0-1055
10-20813
20-301225
30-401540
40-50646
50-60450
  1. Q₁: N/4 = 12.5 falls in 10-20 (cf 13). Q₁ = 10 + ((12.5 − 5)/8) × 10 = 19.375.
  2. Q₃: 3N/4 = 37.5 falls in 30-40 (cf 40). Q₃ = 30 + ((37.5 − 25)/15) × 10 = 38.333.
  3. QD = (38.333 − 19.375)/2 = 18.958/2 = 9.479.
  4. Coefficient of QD = 18.958/(38.333 + 19.375) = 18.958/57.708 = 0.3285.

Worked example: individual series

Switch to an individual series and the default list is 23, 15, 31, 8, 26, 12, 38, 17, 29, 21. Sorted, Q₁ is the 2.75th item, 14.25, and Q₃ is the 8.25th item, 29.5. So QD = (29.5 − 14.25)/2 = 7.625, and the coefficient is 15.25/43.75 = 0.3486. With Excel's QUARTILE.INC convention the quartiles would be 15.5 and 28.25, giving QD = 6.375; the method list lets you match whichever your course uses.

When to use quartile deviation

Quartile deviation is the natural partner of the median. Use it when the data are skewed, contain outliers, or come in a table with open-ended classes such as “below 10” or “60 and above”, where a mean or standard deviation cannot be calculated honestly. Because it only uses Q₁ and Q₃, a single extreme value has no effect on it.

For symmetric, well-behaved data the standard deviation is more informative, because it uses every observation. For a normal distribution, QD is about 0.6745 times the standard deviation, so the two can be compared roughly: a QD far below two-thirds of the SD suggests heavy tails.

Absolute and relative measures

Absolute measureRelative measure (coefficient)
Range L − S(L − S)/(L + S)
Quartile deviation (Q₃ − Q₁)/2(Q₃ − Q₁)/(Q₃ + Q₁)
Mean deviationMD / mean (or median)
Standard deviation σσ / x̄ (× 100 gives the coefficient of variation)

The absolute measures are in the units of the data. The coefficients are pure numbers, so use them whenever you compare two series in different units or with very different averages.

Common mistakes

  • Reporting the interquartile range as the quartile deviation. QD is half of it.
  • Dividing by 2 in the coefficient. The coefficient is (Q₃ − Q₁)/(Q₃ + Q₁); the halves cancel.
  • Using class limits rather than boundaries for inclusive classes (10-19, 20-29). The calculator converts them for you.

Common questions

What is the formula for quartile deviation?

QD = (Q₃ − Q₁) / 2. It is half the interquartile range, which is why it is also called the semi-interquartile range. Its relative form is the coefficient of quartile deviation, (Q₃ − Q₁) / (Q₃ + Q₁).

What does the coefficient of quartile deviation tell you?

It is the spread of the middle half of the data relative to the size of the quartiles, with no units. That lets you compare the variability of, say, wages in rupees with heights in centimetres, or two series with very different averages. A higher coefficient means relatively more spread.

Is quartile deviation the same as the interquartile range?

No, it is half of it. For the grouped example on this page the interquartile range is 38.33 − 19.375 = 18.96, and the quartile deviation is 9.48.

What are the merits and limitations of quartile deviation?

It is easy to calculate, is not affected by extreme values, and works with open-ended classes because it only needs the middle half of the data. On the other hand it ignores the lowest 25% and highest 25% of observations entirely, and it does not lend itself to further algebra the way the standard deviation does.

Can the coefficient of quartile deviation be more than 1?

Not for data that are all positive: Q₃ − Q₁ is always smaller than Q₃ + Q₁ when Q₁ is above zero. If Q₁ is negative the ratio can exceed 1 or become meaningless, so the coefficient is only used for positive quantities.

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