Fundamentals of Business Mathematics and Statistics · Measures of Central Tendency and Dispersion

Geometric Mean and Harmonic Mean: Formulas and Uses

Q: When should I use the harmonic mean?

Use it when you average rates and the same quantity sits in the numerator for every item. Typical cases are speed over equal distances and price per unit when equal amounts of money are spent. In these cases AM would give a wrong, higher answer.

Q: What is the relation between AM, GM and HM?

For positive values, AM ≥ GM ≥ HM. The three are equal only when all values are the same. For exactly two positive numbers there is also the relation GM² = AM × HM.

Q: How do I find the geometric mean for grouped data?

Find the mid-point x of each class and its log. Multiply each log by the frequency f, add them, and divide by N. That gives log GM. Take the antilog to get GM.

Q: Can GM or HM be used when a value is zero or negative?

No. A zero makes the GM zero and makes the reciprocal in HM undefined. Negative values can make the root meaningless. Both means are defined for positive values.

Updated 10 October 2026 · Fact-checked

The geometric mean (GM) of n positive values is the nth root of their product. The harmonic mean (HM) is n divided by the sum of their 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

An average gives one value that stands for a whole set of numbers. The arithmetic mean adds and divides. It works well for plain quantities like marks or wages. It fails when the numbers are rates, ratios or growth factors, and that is where GM and HM come in.

The geometric mean multiplies the values and then takes the nth root. Think of it as the one constant factor that gives the same total product. If a price is multiplied by 2 and then by 8, the GM is √16 = 4. Multiplying by 4 twice also gives 16. That is why GM suits growth over several periods, index numbers and averages of ratios.

The harmonic mean is the reciprocal of the arithmetic mean of reciprocals. It is the right average when the quantity is a rate (something per unit) and the thing held fixed is the numerator. A classic case is speed over equal distances. The slower speed takes more time, so it should pull the average down more than AM does. HM does exactly that.

GM and HM need all values to be positive. A zero or a negative value breaks the formula. For positive values that are not all equal, AM > GM > HM. If all values are equal, the three means are the same. For exactly two positive numbers, GM² = AM × HM.

Key formulas to remember

Geometric mean (ungrouped): GM = (x₁ × x₂ × … × xₙ)^(1/n); All values must be positive. For large numbers use logs: log GM = Σ log x ÷ n, then take the antilog.
Geometric mean (grouped data): log GM = Σ f log x ÷ N, where N = Σ f; Here x is the mid-point of each class. Take the antilog of the result to get GM.
Weighted geometric mean: log GM = Σ w log x ÷ Σ w; Use when values carry different weights.
Harmonic mean (ungrouped): HM = n ÷ Σ(1/x); All values must be non-zero and positive. Do not forget the n in the numerator.
Harmonic mean (grouped data): HM = N ÷ Σ(f/x); x is the value or class mid-point. N = Σ f.
Weighted harmonic mean: HM = Σ w ÷ Σ(w/x); For speeds over unequal distances, use the distances as weights.
HM of two numbers: HM = 2ab ÷ (a + b); Shortcut for two values only, such as two speeds over equal distances.
Relation among the means: AM ≥ GM ≥ HM; For positive values. Equality holds only when all values are equal.
Relation for two numbers: GM² = AM × HM; True only for two positive numbers. Do not use it for three or more.
Average growth rate: Average growth factor = GM of the yearly factors; rate = GM − 1; A 10% rise is a factor of 1.10. A 20% fall is a factor of 0.80.

How to solve Geometric Mean and Harmonic Mean questions

Pick the right mean first, then compute carefully. This method works for most questions on this topic.

1Read what the data is. Growth factors, ratios or index numbers point to GM. Rates like speed, price per unit or output per hour over equal quantities point to HM.
2Check that every value is positive. If a value is zero or negative, GM and HM cannot be used.
3For GM, write the product of the values and take the nth root. If the numbers are big or the data is grouped, find the logs and use log GM = Σ f log x ÷ N.
4For HM, write each reciprocal, add them, and then compute n ÷ Σ(1/x). For grouped data use N ÷ Σ(f/x), with mid-points as x.
5For growth questions, convert percentages to factors first. Take the GM of the factors and subtract 1 at the end.
6For questions that give AM, GM or HM, use GM² = AM × HM if there are exactly two numbers.
7Check your answer. It should satisfy AM ≥ GM ≥ HM and lie between the smallest and largest value.

Quickest way: Elimination with AM ≥ GM ≥ HM

When to use it: Use this when the options are close and you have little time, or when the question gives one mean and asks about another.

Compute the AM quickly. It is the upper limit for GM and HM.
Discard any option for GM or HM that is larger than the AM.
For two numbers, use HM = 2ab ÷ (a + b) directly. Or use GM² = AM × HM to get the missing mean in one step.
For GM of a few numbers, factorise into primes and look for perfect powers. For example, 2 × 4 × 8 = 64, and the cube root is 4.
For equal-distance speed questions, go straight to the two-number HM formula.

Common mistakes in Geometric Mean and Harmonic Mean

Using AM to average speeds over equal distances
AM is the average students know best, so they use it for everything.
Fix: If the distance is the same and speeds differ, the right average is HM. Compute 2ab ÷ (a + b) for two speeds.
Writing HM as the reciprocal of AM, or leaving out n
Students remember that reciprocals are involved but mix up the steps.
Fix: Take reciprocals, add them, then divide n by that sum. HM = n ÷ Σ(1/x).
Averaging growth percentages with AM or taking GM of the percentages
Growth is given as 10%, 20% and so on, and students work with those numbers directly.
Fix: Convert to factors such as 1.10 and 1.20, take the GM, then subtract 1.
Forgetting the antilog in the log method
Students stop once they get Σ f log x ÷ N, which is only log GM.
Fix: Always write the last line GM = antilog (the value). Check that the answer is in the range of the data.
Using GM² = AM × HM for three or more numbers
The relation is memorised without its condition.
Fix: This holds only for two positive numbers. For more numbers, only AM ≥ GM ≥ HM holds.
Using class limits instead of mid-points in grouped data
Students rush and take the lower limit of each class.
Fix: Find the mid-point of each class first, (lower + upper) ÷ 2, and use it as x.

Worked examples

Example 1

A delivery van travels from Pune to a town 120 km away at 40 km/h and returns the same 120 km at 60 km/h. What is the average speed for the whole trip?

Show the solution

The distances are equal and the speeds differ, so use HM.
HM = 2ab ÷ (a + b) = 2 × 40 × 60 ÷ (40 + 60).
= 4,800 ÷ 100 = 48.
Check: time going = 120 ÷ 40 = 3 hours. Time returning = 120 ÷ 60 = 2 hours. Total distance 240 km in 5 hours = 48 km/h.
The AM would be 50, and 48 < 50, which agrees with AM ≥ HM.

Answer: 48 km/h

Example 2

The arithmetic mean of two positive numbers is 10 and their geometric mean is 8. Find their harmonic mean.

Show the solution

There are exactly two numbers, so GM² = AM × HM applies.
GM² = 8 × 8 = 64.
HM = GM² ÷ AM = 64 ÷ 10 = 6.4.
Check the order: AM = 10 ≥ GM = 8 ≥ HM = 6.4. It holds.

Answer: HM = 6.4

Exam tips

First decide the type of data. Growth or ratio means GM. Rate over equal distance or equal cost means HM. This choice alone often settles the answer.
Use AM ≥ GM ≥ HM to remove options. If a GM or HM option is above the AM, it is wrong.
Questions that give two of AM, GM, HM for two numbers are quick marks. Use GM² = AM × HM and avoid solving for the numbers.
Watch the word 'equal' in speed problems. If distances are not equal, a plain HM is wrong and weighted HM is needed.
There is no negative marking, so always mark an answer. Eliminate options that break the AM ≥ GM ≥ HM order, then pick from the rest.

Practice questions from Measures of Central Tendency and Dispersion

Geometric Mean and Harmonic Mean in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CFA Level IArithmetic Mean vs Geometric Mean Return

Geometric Mean and Harmonic Mean: frequently asked questions

When should I use the harmonic mean?

Use it when you average rates and the same quantity sits in the numerator for every item. Typical cases are speed over equal distances and price per unit when equal amounts of money are spent. In these cases AM would give a wrong, higher answer.

What is the relation between AM, GM and HM?

For positive values, AM ≥ GM ≥ HM. The three are equal only when all values are the same. For exactly two positive numbers there is also the relation GM² = AM × HM.

How do I find the geometric mean for grouped data?

Find the mid-point x of each class and its log. Multiply each log by the frequency f, add them, and divide by N. That gives log GM. Take the antilog to get GM.

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

No. A zero makes the GM zero and makes the reciprocal in HM undefined. Negative values can make the root meaningless. Both means are defined for positive values.