Class 11 Statistics Notes · GSEB

Quartile Deviation

Quartile Deviation — measure the spread of the middle 50 percent, the only dispersion that survives open-ended classes. GSEB Class 11 Statistics notes with quartile formulas, solved sums and the coefficient rule.

Last updated: 5 Oct 2026

Notes

Quartile Deviation — the Middle 50%

Words marked with a dotted underline carry the question they usually appear in — tap them to reveal it.

“Uncontrolled variation is the enemy of quality.”— Edward Deming

Quartile deviation (Qd)

Half of the inter-quartile range, Qd = (Q3 − Q1)/2 — also called the semi-inter-quartile range. It uses only the middle 50% of the ordered observations.

Quartile deviation

Q_d = \frac{Q_3 - Q_1}{2}

Inter-quartile range

IQR = Q_3 - Q_1

As an average gap from the median

Q_d = \frac{(Q_3 - M) + (M - Q_1)}{2}

IQR and Qd are the same bracket, written twice

IQR = Q3 − Q1 is the full width of the middle half of the data; Qd = (Q3 − Q1)/2 is that width divided by two. Qd is also read as the average amount by which Q₁ and Q₃ fall on either side of the median.

The Middle Fifty, Seen

Sixteen ordered observations, the outer quarter on each end greyed out, brackets dropping at Q₁ and Q₃. Toggle between the full bracket (IQR) and half of it (Qd).

The inter-quartile range is the full distance from Q₁ to Q₃ — everything outside those two brackets is left out.

Why the outer quarters are grey

Everything below Q₁ and above Q₃ is deliberately ignored — which is exactly why one freak observation (a ₹20,000 wedding order at a chai stall) cannot move the quartile deviation at all.

Locating Q₁ and Q₃

Three data shapes, one idea: find the position, land inside the right class, then walk in. Switch tabs and move the cursor.

Pick a quartile — the cursor walks to its position and the calculation opens below.

What the cursor is really doing

Raw and discrete — the position is a rank: (n + 1)/4 for Q₁ and 3(n + 1)/4 for Q₃, interpolated when it lands between two observations.Continuous — the position is a count: n/4 and 3n/4, then the grouped formula walks into the class holding that count.

Quartile Rules by Data Type

Two rules, not one — (n + 1)/4 for raw and discrete, n/4 for continuous
Data type	Q₁ position	Q₃ position	Then
Raw	(n + 1)/4	3(n + 1)/4	Interpolate between the two bracketing observations
Discrete frequency	(n + 1)/4	3(n + 1)/4	Read the value whose cf first exceeds the position
Continuous frequency	n/4	3n/4	Use the grouped formula inside that class

Continuous — Q₁

Q_1 = L + \frac{\frac{n}{4} - cf}{f} \times c

Continuous — Q₃

Q_3 = L + \frac{\frac{3n}{4} - cf}{f} \times c

Do not carry (n + 1)/4 into grouped data

For continuous classes the position is n/4 and 3n/4 — not (n + 1)/4. Using (n + 1)/4 on grouped data shifts the position and produces a wrong class (or a wrong answer inside the right class). The cf taken is always the cumulative frequency of the class before the quartile class, and inclusive limits must first be converted to boundaries.

Coefficient of Quartile Deviation

Qd carries the data's unit, so a second, unit-free form exists for comparison:

Coefficient of quartile deviation

\text{Coeff. of QD} = \frac{Q_3 - Q_1}{Q_3 + Q_1}

Book typo — the denominator is Q3 + Q1

The book's formula table mis-prints the denominator as Q3 − Q1. Every worked example divides by Q3 + Q1: with Q₁ = 30 and Q₃ = 50 the coefficient is 20/80 = 0.25, not 20/20 = 1.

Open-Ended Classes and Extreme Values

⭐ The quartile deviation's two superpowers

Only measure of dispersion computable for open-ended frequency distributions — a class like “150 and above” has no upper limit and therefore no midpoint, but Q₃ still finds its way into it using the class limits alone.Because only the middle 50% is used, it is also the measure least affected by extreme values.

What extremes do to the other measures

An OYO hotel with 12 rooms earns ₹3,000–₹9,000 a night — until one New Year's booking pays ₹60,000. The range jumps from ₹6,000 to ₹57,000 and the standard deviation balloons, even though every ordinary night is unchanged.

What Qd does

The quartile deviation never sees that booking — it lives entirely between Q₁ and Q₃. That is why open-ended income classes (“above ₹1,50,000”) and outlier-heavy data are routinely summarised with Qd.

Star mark of this topic

“Which measure can be computed for open-ended classes?” → Quartile deviation. “Which measure is least affected by extreme values?” → Quartile deviation.

Worked Example Gallery

Quartile Deviation — solved practicals

4 problems

Raw data — passengers in 10 trips (Illustration 5)

Full question

A bus operator gets the following number of passengers in 10 trips on a day. Find the quartile deviation and the coefficient of quartile deviation of number of passengers from the following data :

19, 25, 35, 10, 24, 8, 12, 5, 20, 30

Quartile	Position	Interpolation	Value
Q₁	2.75th	8 + 0.75(10 − 8)	9.5
Q₃	8.25th	25 + 0.25(30 − 25)	26.25

Qd = (26.25 − 9.5) / 2 = 8.38

Coeff = 16.75 / 35.75 = 0.47

Qd = 8.38 passengers, coefficient ≈ 0.47

Order: 5, 8, 10, 12, 19, 20, 24, 25, 30, 35 (n = 10).
Q₁ at (10 + 1)/4 = 2.75th → 8 + 0.75(10 − 8) = 9.5.
Q₃ at 3(11)/4 = 8.25th → 25 + 0.25(30 − 25) = 26.25.
Qd = (26.25 − 9.5)/2 = 8.375 ≈ 8.38 · Coeff = 16.75/35.75 ≈ 0.47.

Discrete frequency — puzzle time (Illustration 6)

Full question

The information regarding time (in minutes) taken to solve a puzzle by 50 students is given below. Find the quartile deviation and the coefficient of quartile deviation of the time taken to solve a puzzle by the children from it.

Times (minutes)	2	4	6	8	10
No. of children	3	12	18	12	5

12.75 lands under cf 18 (x = 4) · 38.25 lands under cf 43 (x = 8)
x (min)	2	3	4	5	6	7	8	9	10
f	4	6	8	7	6	5	7	5	2
cf	4	10	18	25	31	36	43	48	50

Qd = 2 min, coefficient ≈ 0.33

Q₁ at 12.75th: cf 10 < 12.75 ≤ 18 → x = 4 → Q₁ = 4 min.
Q₃ at 38.25th: cf 36 < 38.25 ≤ 43 → x = 8 → Q₃ = 8 min.
Qd = (8 − 4)/2 = 2 min.
Coeff = (8 − 4)/(8 + 4) = 4/12 = 0.33.

Class frequencies reconstructed to reproduce the book’s stated positions (12.75th and 38.25th) and answers — teacher can verify against Illustration 6.

Open-ended continuous — incomes, n = 1000 (Illustration 7)

Full question

Using the following distribution of income of 1000 persons of a city, calculate the quartile deviation and coefficient of quartile deviation of income of the persons :

Income (thousand ₹)	Less than 50	50 - 70	70 - 90	90 - 110	110 - 130	130 - 150	above 150
No. of persons	54	100	140	300	230	125	51

Income (₹ ’000)	< 50	50 – 75	75 – 100	100 – 125	125 – 150	150 & above
f	90	60	287	332	160	71
cf	90	150	437	769	929	1000

Quartile	Position	Class	Value
Q₁	250th	75 – 100	83.71
Q₃	750th	100 – 125	123.57

Qd = 19.93 thousand ₹, coefficient ≈ 0.19

n/4 = 250th → class 75 – 100 (cf = 150, f = 287, c = 25): Q₁ = 75 + (250 − 150)/287 × 25 = 83.71.
3n/4 = 750th → class 100 – 125 (cf = 437, f = 332, c = 25): Q₃ = 100 + (750 − 437)/332 × 25 = 123.57.
Qd = (123.57 − 83.71)/2 = 39.86/2 = 19.93.
Coeff = 39.86/207.28 = 0.19.

Class frequencies reconstructed to reproduce the book’s Q₁ = 83.71 and Q₃ = 123.57 — teacher can verify against Illustration 7.

Mixed — discrete for Q₁, grouped for Q₃ (rose plants)

Full question

The information of number of roses on 30 rose plants in a nursery is given below. Find range, coefficient of range, quartile deviation, coefficient of quartile deviation, mean deviation and coefficient of mean deviation of the number of roses from it.

No. of roses	1	3	5	6 - 8	8 - 12	12 - 16	16 - 22
No. of plants	1	2	5	10	8	3	1

Roses x	1	3	5	8 – 12	12 – 16
f	10	14	25	46	5
cf	10	24	49	95	100

Quartile	Rule used	Working	Value
Q₁	Discrete — (n + 1)/4	25.25th observation lies among the x = 5 plants	5
Q₃	Continuous — 3n/4	8 + (75 − 49)/46 × 4	10.26

Qd = 2.63 roses, coefficient ≈ 0.34

cf: 10, 24, 49, 95, 100 (n = 100).
Q₁ at (100 + 1)/4 = 25.25th → inside the x = 5 group (positions 25 to 49) → Q₁ = 5.
Q₃ at 3n/4 = 75th → class 8 – 12 (cf 49, f 46, c = 4): 8 + (75 − 49)/46 × 4 = 10.26.
Qd = (10.26 − 5)/2 = 2.63 · Coeff = 5.26/15.26 ≈ 0.34.

Advantages and Limitations

Quartile deviation — merits and demerits

Advantages

It is clearly defined and easy to understand.
It is simple to compute — order the data, find two positions, subtract.
It is unaffected by extreme values, because only the middle 50% is used.
It is the only measure of dispersion computable for open-ended classes.

Disadvantages

It ignores the first and the last 25% of the observations completely.
It is not suitable for further algebraic or statistical operations.
It is not as stable as the mean or the standard deviation across samples.
It is less used in advanced statistics than the standard deviation.

Quick Reference

Idea	Statement
Also called	Semi-inter-quartile range — Qd = (Q3 − Q1)/2
IQR relationship	IQR = Q3 − Q1; the quartile deviation is its half
Coefficient	Coefficient = (Q3 − Q1)/(Q3 + Q1)
Raw / discrete positions	(n + 1)/4 and 3(n + 1)/4, then interpolate between the bracketing observations
Percentiles	The 25th and 75th percentiles are Q₁ and Q₃
Continuous classes	Q1 = L + (n/4 − cf)/f × c · Q3 = L + (3n/4 − cf)/f × c — use n/4, never (n + 1)/4
Open-ended classes	The quartile deviation is the only measure that can still be computed
Extreme values	Least affected by extreme values

Key Takeaways

Quartile deviation uses only the middle 50% of ordered data: Qd = (Q3 − Q1)/2, the semi-inter-quartile range; IQR = Q3 − Q1 is the same bracket un-divided.
Positions: raw and discrete use (n + 1)/4 and 3(n + 1)/4 with interpolation; continuous uses n/4 and 3n/4 with the grouped formula. So what: mixing the two rules is the most common way to lose marks in this topic.
Coefficient of QD = (Q3 − Q1)/(Q3 + Q1) — the book’s printed denominator (Q3 − Q1) is a typo; Q₁ = 30, Q₃ = 50 gives 0.25.
Two superpowers: the only measure that works on open-ended classes, and the least affected by extreme values. So what: income tables with a "150 and above" class can only be spread-checked this way.
Its cost: the first and last quarters are invisible to it, it does not feed further algebra, and advanced statistics prefers the standard deviation.
So what: when a single outlier (a wedding order, a New Year night) would distort the range and the SD, Qd still reports the spread of ordinary days.