Class 11 Statistics Notes · GSEB

Range

Range — compute the simplest spread in one step and judge it with the coefficient of range. GSEB Class 11 Statistics notes with grouped-class rules, solved sums and the quality-control use.

Last updated: 5 Oct 2026

Notes

Range — Definition and Formula

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

Range

Range = difference between the highest and lowest observation of the data. It is the simplest of all measures of dispersion — two readings and one subtraction.

Range

R = x_{H} - x_{L}

The bracket alone is the range — unit-bound. Divide it by x_H + x_L and it becomes a pure number.

Coefficient of range = (x_H − x_L) ÷ (x_H + x_L) = 24 ÷ 118 = 0.20

Only about 20% of (x_H + x_L) is “spread” — a small coefficient means low variability.

Frequency Plays No Role

⭐ What the book insists on

Frequency does not play any role in determining the range, even for grouped data.Whether the middle class holds 2 students or 200, the range does not move — it never looks at anything except the top of the highest class and the bottom of the lowest class.

Grouped data — take the limits, never the mid-values

x_H = upper limit of the highest class · x_L = lower limit of the lowest class
Class	10 – 15	15 – 20	20 – 25	25 – 30	30 – 35
Reading	x_L = 10	—	—	—	x_H = 35

R = 35 − 10 = 25. Using mid-values (32.5 − 12.5 = 20) would be wrong — the limits are what the class intervals actually span.

Coefficient of Range

The range carries the data's unit, so it cannot compare two datasets. Its paired relative measure removes the unit: Coefficient of range = (x_H − x_L)/(x_H + x_L).

Coefficient of range

\text{Coeff. of range} = \frac{x_{H} - x_{L}}{x_{H} + x_{L}}

A typo in the book's formula table

The formula table mis-prints the denominator as x_H − x_L. Every worked example in the book divides by x_H + x_L — e.g. the runs data gives 75/111 ≈ 0.68, not 75/75 = 1. Always take the sum in the denominator.

Reading the coefficient

Small coefficient → less variability · large coefficient → more variability. Because it is unit-free, it can be quoted alongside coefficients from datasets measured in entirely different units.

Where the Range Is Used

Statistical quality control

Control charts plot the range of a sample over time. A packaging line filling 500 g pouches is checked by weighing a handful each hour — the spread of that handful, not its average, tells the operator whether the machine has drifted.

Money, exchange and share prices

A share that moved between ₹80 and ₹115 over 8 weeks has a range of ₹35— a one-line answer to “how volatile has this stock been?”

Range alone

A Zomato delivery partner is judged on his best and worst run of the week. Two riders with the same range look identical — yet one is steady every other day and the other swings wildly. The range never looked at those days.

Range + a deviation-based measure

Quote the range for a quick first look, then confirm with standard deviation (or at least the quartile deviation) before making the call — that is the habit this chapter is building.

Solved Example — Illustration 1

Solved Example

Problem

Runs scored in 10 innings: 48, 75, 37, 52, 93, 81, 25, 72, 18, 60. Find the range and the coefficient of range.

Solution

R = 75 runs · coefficient ≈ 0.68

Worked Example Gallery

Range — solved practicals

5 problems

Raw data — Illustration 1 (runs)

Full question

The runs scored by a batsman in his last 10 innings of cricket matches are 48, 75, 37, 52, 93, 81, 25, 72, 18 and 60. Find the range and the coefficient of range of his runs.

x_H	x_L	R = x_H − x_L	x_H + x_L	Coefficient
93	18	75 runs	111	75/111 ≈ 0.68

R = 75 runs, coefficient ≈ 0.68

x_H = 93, x_L = 18.
R = 93 − 18 = 75 runs.
Coeff = 75/(93 + 18) = 75/111 = 0.6757 ≈ 0.68.

Discrete frequency distribution — Illustration 2 (salaries)

Full question

From the following information of monthly salary (in ₹) of workers of a factory, find the range and the coefficient of range of the salary.

Monthly Salary (₹)	3500	4000	5000	7500	10,000	12,000
No. of workers	3	21	30	19	6	5

f is printed for completeness — the range ignores it completely
Salary (₹)	3,500	5,000	6,500	8,000	9,500	12,000
Workers (f)	3	4	5	5	4	3

R = ₹8,500, coefficient ≈ 0.55

Highest salary x_H = 12,000, lowest salary x_L = 3,500 — frequencies never enter.
R = 12,000 − 3,500 = ₹8,500.
Coefficient = 8,500/(12,000 + 3,500) = 8,500/15,500 = 0.548 ≈ 0.55.

Continuous classes — weights

Full question

The items produced in a factory are packed into different boxes according to their weight. Using the following information, find the range and the relative range of weight of boxes :

Weight (kg)	10 - 15	15 - 20	20 - 25	25 - 30	30 - 35
No. of boxes	8	15	26	47	4

x_H	x_L	R	x_H + x_L	Coefficient
35	10	25 kg	45	25/45 ≈ 0.56

R = 25 kg, coefficient ≈ 0.56

Mid-values would have given 32.5 − 12.5 = 20 — wrong for range.

Take limits, not mid-values: x_H = 35 (upper limit of 30 – 35), x_L = 10 (lower limit of 10 – 15).
R = 35 − 10 = 25 kg.
Coefficient = 25/(35 + 10) = 25/45 = 0.555… ≈ 0.56.

Inclusive classes — marks

Full question

Find the range and the coefficient of range of the marks from the following frequency distribution of marks scored by 50 students of a school in a certain examination.

Marks	50 - 59	60 - 69	70 - 79	80 - 89	90 - 99
No. of Students	2	15	23	6	4

x_H	x_L	R	x_H + x_L	Coefficient
99	50	49 marks	149	49/149 ≈ 0.33

R = 49 marks, coefficient ≈ 0.33

x_H = 99 (upper limit of 90 – 99), x_L = 50 (lower limit of 50 – 59).
R = 99 − 50 = 49 marks.
Coefficient = 49/(99 + 50) = 49/149 = 0.3289 ≈ 0.33.

Grouped, exercise — pipe diameters

Full question

For a company making pipes, the following information of diameter (in cm) of pipes is obtained. Find the range of the diameter of the pipes.

Diameter (cm)	20 - 40	40 - 60	60 - 80	80 - 100	100 - 120
No. of pipes	15	40	75	20	11

Range = upper limit of the last class − lower limit of the first = 120 − 20 = 100 cm.

R = 100 cm

Grouped rule: upper limit of the last class − lower limit of the first class.
120 − 20 = 100 cm.

Advantages and Limitations

Range — merits and demerits

Advantages

It is clearly defined — every student gets the same answer from the same data.
Its computation is the simplest of all measures: identify two values, subtract.
It is useful when the variability in the data is known to be small — a quick, cheap first look.

Disadvantages

It ignores every observation except the two extremes.
It is very sensitive to sampling fluctuations — one unusual item changes it completely.
It is not suitable for further algebraic or statistical operations.
It cannot be computed for open-ended frequency distributions (for example "100 and above").

Quick Reference

Idea	Statement
Definition	Range = difference between the highest and lowest observation of the data
Formula	R = x_H − x_L
Grouped data	Upper limit of the highest class − lower limit of the lowest class (take limits, not mid-values)
Frequencies	Irrelevant — frequency does not play any role in determining the range
Coefficient	(x_H − x_L)/(x_H + x_L) — the book table prints x_H − x_L as the denominator; worked examples use the sum
Type	Range is an absolute measure — it carries the data unit
Open-ended classes	Cannot be computed — no upper limit to read off

Key Takeaways

Range = x_H − x_L — the difference between the highest and the lowest observation, and nothing else.
For grouped data take the limits (upper limit of the last class − lower limit of the first), never the mid-values.
Frequency does not play any role in determining the range, even for grouped data. So what: a class with 2 students and a class with 200 students give the identical range.
Coefficient of range = (x_H − x_L)/(x_H + x_L); small means little variability, large means a lot — and the book's printed denominator (x_H − x_L) is a typo.
Range is trivial to compute but blind to everything between the extremes, unstable under sampling, and impossible for open-ended classes — so it is a first look, never the final word.
So what: control charts on a factory floor, a share's ₹80–₹115 swing, and a rider's best/worst run all start with the range — and all of them need a second measure before a decision is made.