Class 11 Statistics Notes · GSEB

Geometric Mean

Geometric Mean — find the true average growth rate of prices, population and investments where the arithmetic mean misleads. GSEB Class 11 Statistics notes with the compounding comparison, root tricks and solved sums.

Last updated: 5 Oct 2026

Notes

When GM Beats the Arithmetic Mean

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

Suppose a variable changes over time — a price, a population, the value of a machine. If the question is “what was the average rate of change?”, plugging numbers into the arithmetic mean is not appropriate, because change over time compounds: each rate applies to the already-changed value. What you need is the average of rates of change over time.

Increase of p%

value = (100 + p)

e.g. +50% → 150, +25% → 125

Decrease of p%

value = (100 − p)

e.g. −20% → 80 (the next month starts from 80)

The one hard condition

GM cannot be found if any observation ≤ 0. A single zero makes the whole product zero, and a negative product has no real even root — so for data with losses or nil values, fall back to the mean or the median.

The ₹200 Story — One Visual, Whole Case for GM

An item costs ₹200. Its price rises 50% in the first month and 25% in the second. The real path is 200 → 300 → ₹375. Now watch what happens when you average the two percentages:

The same ₹200 item, three ways. Each row applies its average rate twice — only one of them lands on the real price.

What really happens

Two actual months of change — +50% then +25%

The truth

₹200

start

× 1.50

+50%

₹300

after month 1

× 1.25

+25%

₹375

after month 2

Average with AM = 137.5%

(150 + 125) / 2 = 137.5, applied twice

Wrong — overshoots

₹200

start

× 1.375

AM rate

₹275

after month 1

× 1.375

AM rate

₹378.18

after month 2

Average with GM = 136.93%

G = √(150 × 125) = 136.93, applied twice

Correct — matches the truth

₹200

start

× 1.3693

GM rate

₹273.86

after month 1

× 1.3693

GM rate

₹375

after month 2

Why GM lands on ₹375

Growth multiplies: 1.50 × 1.25 = 1.875. The GM of 150 and 125 is √(150 × 125) = 136.93, and applying it twice gives 1.3693 × 1.3693 = 1.875 — the exact compound factor. AM applied twice gives 1.375² = 1.8906, which is why it lands at ₹378.18 instead of ₹375.

Wrong: averaging the swing

A trader says “I lost 50% and then gained 50% — I am back to even.” That is the AM of 50 and 150 = 100, applied to a ₹10,000 portfolio.

Claims: ₹10,000 → ₹10,000

Right: compounding the swing

10,000 → ×0.5 → ₹5,000 → ×1.5 → ₹7,500. The GM path (√(50 × 150) = 86.6, applied twice) reaches the same ₹7,500.

Truth: a 25% net loss

The Formula

Geometric Mean — n observations

G = \sqrt[n]{x_1 \times x_2 \times \ldots \times x_n}

Two numbers

G = \sqrt{x_1 \times x_2}

Read it as: multiply everything, then take the n-th root. It is called the geometric mean because it is the side of the n-dimensional box whose volume equals the product of the numbers — for two numbers that is just a rectangle (previous section).

Root trick — no scientific calculator needed

A 4th root is a square root taken twice (√√), and an 8th root is a square root taken three times (√√√). In Illustration 14, ⁿ√184009200 was found as √184009200 = 13564.9991, then √13564.9991 = 116.47.

G of 4 and 9 = √(4 × 9) = √36 = 6

Two numbers, one square root — the whole sum is one line long.

Product vs Midpoint — Why x̄ ≥ G

Take any two positive numbers. The arithmetic mean is the midpoint of the segment joining them; the geometric mean is the side of the square with the same area as the rectangle they form. The midpoint can never sit below that side.

x̄ = 12.5 ≥ G = 12

The arithmetic mean never falls below the geometric mean — for 9 and 16 it sits exactly 0.5 higher.

The inequality you can quote in one line

For any set of positive observations, the arithmetic mean is always equal to or greater than the geometric mean — and they are equal only when every observation has the same value.

Where Geometric Mean Is Used

Population growth

A town grows 15%, 18%, 13% and 20% over four years — one average rate for all four.

Prices & inflation

Onion prices jump 40% in one month and fall 10% next — the average must compound, not add.

Machine value

A packaging machine worth ₹5 lakh depreciates 10%, 7%, 5%, 2% across four years.

Investment / startup growth

Three years of growth rates on a Zerodha-style user base have to be chained, not averaged.

Startup growth, done honestly

Suppose an Indian startup’s order volume grows 30%, 50% and 20% in three successive years. Its headline growth cannot be AM = 33.33%; it is GM = ³√(130 × 150 × 120) = 132.76 → 32.76% per year, the rate that actually chains the three years together. Investors ask for exactly this number.

Worked Examples

Geometric Mean — Solved Practicals

3 problems

Average growth rate — Illustration 14

Full question

The population of an area increased by 15%, 18%, 13%, 20% respectively for four years. Find the average rate of increase in the population.

Year	Increase	Value (100 + p)
1	15%	115
2	18%	118
3	13%	113
4	20%	120
Product	—	115 × 118 × 113 × 120 = 184009200

G = ⁴√184009200 = √(√184009200) = √13564.9991 = 116.47

G = 116.47 ⇒ average rate of growth = 16.47%

Convert each percentage increase into its (100 + p) value: 115, 118, 113, 120.
G = ⁴√(115 × 118 × 113 × 120) = ⁴√184009200.
Take the square root twice: √184009200 = 13564.9991, then √13564.9991 = 116.4689 ≈ 116.47.
Average growth = 116.47 − 100 = 16.47%.

The root trick: 4th root = square root of the square root.

Two numbers, a relationship — Illustration 15

Full question

The geometric mean of two numbers is 2. If one number is 4 times the other number, find the numbers.

2 = √(a × 4a) = √(4a²) = 2a ⇒ a = 1

The numbers are 1 and 4

Check: √(1 × 4) = √4 = 2 = G ✓

Let the smaller number be a; then the larger number is 4a.
G = √(a × 4a) = √(4a²) = 2a.
2a = 2 ⇒ a = 1, so the numbers are a = 1 and 4a = 4.

Verify x̄ ≥ G — Illustration 16

Full question

Find the arithmetic mean and geometric mean of two numbers 9 and 16 and verify that x̄ > G.

x̄ = Σx / n = (9 + 16) / 2 = 12.5

G = √(x₁ × x₂) = √(9 × 16) = √144 = 12

x̄ = 12.5 > G = 12 — verified

x̄ = (9 + 16) / 2 = 25 / 2 = 12.5
G = √(9 × 16) = √144 = 12
Compare: 12.5 > 12, so x̄ > G is verified.

Equality would need both numbers to be the same value.

Advantages and Limitations

Advantages

It is rigidly defined.
It is based on all observations.
It is suitable for algebraic calculations.
It is comparatively less affected by too large or too small values.

Disadvantages

It can be found only if all the observations have positive values.
It is difficult to calculate.
Its calculation becomes difficult if the number of observations is large.

Quick Reference

Idea	Statement
Formula	G = ⁿ√(x₁ × x₂ × … × xₙ); for two numbers G = √(x₁ × x₂)
Percentage convention	Increase p% → (100 + p) · Decrease p% → (100 − p)
AM vs GM	For every set of observations the arithmetic mean is at least the geometric mean — x̄ ≥ G, with equality only when all values are equal.
When to use GM	Rates and percentages that compound over time — growth, depreciation, returns. Never when any value is zero or negative.
Root trick	4th root = √√ (twice) · 8th root = √√√ (three times)

Key Takeaways

Geometric mean is the n-th root of the product of n positive observations; for two numbers it is one square root — so two-number sums finish in a single line.
Convert percentages first: increase p% → (100 + p), decrease p% → (100 − p). Never average raw percentages.
Compounding is multiplication — AM of 150 and 125 applied twice gives ₹378.18, while GM gives the true ₹375.
x̄ ≥ G for every set of positive observations; equality holds only when all observations are identical.
GM needs all-positive data — a business with a loss year cannot quote a geometric growth rate from it.
Practical uses: population growth, price changes, machine depreciation and investment or startup growth rates.