Quantitative Aptitude · Measures of Central Tendency and Dispersion

Geometric Mean and Harmonic Mean for CA Foundation

Updated 1 October 2026

Geometric mean (GM) is the nth root of the product of n positive values. Harmonic mean (HM) is n divided by the sum of the reciprocals. Use GM for growth rates and ratios, and HM for averaging rates such as speed over equal distances. For positive values, AM ≥ GM ≥ HM.

Understand Geometric Mean and Harmonic Mean

The arithmetic mean adds values. That works when values are added together, like marks or heights. It fails when values multiply (growth) or when they are rates over a fixed quantity (speed over a fixed distance). GM and HM fix these two cases.

Geometric mean is the value that, if used in every period, gives the same final product. If a price grows by factors 2 and 8 in two years, the GM is √(2 × 8) = 4. A steady factor of 4 each year gives the same total growth of 16 times. So GM is the right average for growth factors, index numbers and ratios.

Harmonic mean is the reciprocal of the average of reciprocals. Take a trip where you cover equal distances at two speeds. Time is distance ÷ speed, so the speeds must be averaged through their reciprocals. That gives HM. If instead you travel for equal times at different speeds, the plain AM is correct.

For positive values, the three means always follow AM ≥ GM ≥ HM. They are all equal only when every value is the same. For exactly two positive numbers, GM² = AM × HM. This is a strong check and a strong shortcut in MCQs.

Both GM and HM need all values to be positive (GM and HM are not defined in the usual way if a value is zero or negative). Both are affected by every value. GM and HM are never larger than AM. HM is the most influenced by small values and the least influenced by large ones.

Key formulas to remember

GM (ungrouped): GM = (x₁ × x₂ × … × xₙ)^(1/n); For positive values. With logs: GM = antilog[Σ log x ÷ n].
GM (grouped data): GM = antilog[Σ f log x ÷ N], where N = Σf; x is the class mid-point for continuous classes.
Weighted GM: GM = antilog[Σ w log x ÷ Σ w]; Use when values carry different weights.
HM (ungrouped): HM = n ÷ Σ(1/x); For positive values. Reciprocal of the AM of reciprocals.
HM (grouped data): HM = N ÷ Σ(f/x), where N = Σf; x is the mid-point for continuous classes.
Weighted HM: HM = Σw ÷ Σ(w/x); For average speed with unequal distances, weights are the distances.
Two numbers a and b: HM = 2ab ÷ (a + b); GM = √(ab); GM² = AM × HM; The GM² rule holds for exactly two positive numbers only.
Order of means: AM ≥ GM ≥ HM; For positive values. Equality holds only when all values are equal.
Average growth rate: Average growth factor = (Final ÷ Initial)^(1/n); rate = factor − 1; This is the GM of the yearly growth factors.
Average speed, equal distances: Average speed = HM of the speeds; For two speeds u and v: 2uv ÷ (u + v). For equal times, use AM.

How to solve Geometric Mean and Harmonic Mean questions

Use this method for any GM or HM question. Decide the mean first, then calculate.

1Read what the values represent. Growth factors, ratios or index numbers point to GM. Rates over equal distances or equal cost or quantity point to HM.
2Check for equal times in a speed problem. If times are equal, use AM, not HM.
3Convert percentage growth into factors. A 20% rise is 1.20 and a 10% fall is 0.90. Never average the raw percentages.
4Write the right formula. For grouped data, find mid-points x first and then f log x or f/x.
5Calculate. For GM use roots or logs. For HM sum the reciprocals and divide.
6Check the answer against AM ≥ GM ≥ HM. GM must lie between HM and AM.
7If asked for a growth rate, subtract 1 from the GM factor and convert to a percentage.

Quickest way: Option elimination with AM ≥ GM ≥ HM

When to use it: Use in MCQs when options include AM, GM and HM-like values, or when the calculation is long.

Compute the AM quickly. It is the upper limit for GM and HM.
Remove any option larger than the AM for a GM or HM question.
For two numbers, use GM² = AM × HM to find a missing mean in seconds.
For equal-distance speed with two speeds, use 2uv ÷ (u + v). For two unequal speeds, the HM is below the AM and closer to the smaller speed.
If all values are equal, all three means equal that value. Pick it at once.
If a grouped-data log calculation will take over two minutes, mark it and move on. Wrong answers cost 0.25 marks.

Common mistakes in Geometric Mean and Harmonic Mean

Using AM for average speed over equal distances
AM is the default average and feels natural.
Fix: Ask whether distances or times are equal. Equal distances need HM. Equal times need AM.
Averaging growth percentages directly
Students treat 100% and 700% as plain numbers to add.
Fix: Convert to factors (2 and 8), take the GM (4), then subtract 1 to get 300%.
Applying GM² = AM × HM to three or more values
The two-number rule is memorised without its condition.
Fix: Use it only for exactly two positive numbers. For more values, only AM ≥ GM ≥ HM holds.
Using class limits instead of mid-points in grouped data
Students rush to the formula and skip the x column.
Fix: Compute mid-point x = (lower + upper) ÷ 2 first, then log x or 1/x.
Writing HM = Σ(1/x) ÷ n
Students confuse HM with the AM of reciprocals.
Fix: That expression is the AM of reciprocals. HM is its reciprocal: n ÷ Σ(1/x).
Including zero or negative values in GM or HM
The formula is applied without checking the data.
Fix: GM and HM are used for positive values. A zero makes the product zero and 1/x undefined.

Worked examples

Example 1

A car covers a journey at 40 km/h and returns the same distance at 60 km/h. The average speed for the whole trip is: (A) 48 km/h (B) 50 km/h (C) 52 km/h (D) 45 km/h

Show the solution

Distances are equal, so average speed is the HM of 40 and 60.
HM = 2 × 40 × 60 ÷ (40 + 60) = 4800 ÷ 100 = 48.
Check: the AM is 50, and 48 is below it, as HM ≤ AM requires.

Answer: (A) 48 km/h

Example 2

The AM of two positive numbers is 10 and their HM is 6.4. Their GM is: (A) 8 (B) 7.2 (C) 9 (D) 6.4

Show the solution

For two positive numbers, GM² = AM × HM.
GM² = 10 × 6.4 = 64.
GM = √64 = 8.
Check: 6.4 ≤ 8 ≤ 10, so the order HM ≤ GM ≤ AM holds.

Answer: (A) 8

Example 3

The sales of a firm grew by 100% in the first year and by 700% in the second year. The average growth rate per year, using GM, is: (A) 300% (B) 400% (C) 350% (D) 200%

Show the solution

Convert rates to growth factors: 100% growth gives 2 and 700% growth gives 8.
GM of the factors = √(2 × 8) = √16 = 4.
Average growth rate = 4 − 1 = 3, which is 300%.
Check: a steady factor of 4 each year gives 4 × 4 = 16, the same as 2 × 8. The AM of the rates (400%) is a trap and does not reproduce 16.

Answer: (A) 300%

Exam tips

Read the wording for clues. Words like average speed, average rate of growth, index numbers and ratios signal HM or GM.
Always test the answer with AM ≥ GM ≥ HM. It removes at least one option in many questions.
Remember that growth questions need factors, and the final answer needs the factor minus 1.
For grouped data, build the table with x, f and log x or f/x in a neat order to avoid slips. Skip it if it threatens your time.
With 0.25 negative marking, leave a long log-based question for the end and attempt the shortcut-friendly ones first.

Practice questions from Measures of Central Tendency and Dispersion

Geometric Mean and Harmonic Mean: frequently asked questions

When should I use HM instead of GM?

Use HM for averages of rates where the fixed quantity is the same, such as equal distances at different speeds. Use GM for growth factors, ratios and index numbers, where values multiply across periods.

What is the relation between AM, GM and HM?

For positive values, AM ≥ GM ≥ HM. They are equal only when all values are the same. For two positive numbers, GM² = AM × HM as well.

How do I find GM and HM for grouped data?

For GM, find mid-points x and use antilog[Σ f log x ÷ N]. For HM, use N ÷ Σ(f/x). In both, N is the total frequency.

Can GM or HM be found when a value is zero or negative?

Not in the usual way. These means are meant for positive values. A zero makes the GM zero and makes 1/x undefined for HM.