Geometric and harmonic means

Some numbers combine by multiplying. A ratio of 1.5 followed by a ratio of 0.8 gives 1.5 × 0.8 = 1.2. The average of such numbers should be the single ratio that, used every time, gives the same product. That is the geometric mean: the n-th root of the product.

The GM of 2, 4 and 8 is (2 × 4 × 8)^1/3 = 64^1/3 = 4 (CUET 2025). The GM of 4 and 9 is √36 = 6. The log version is the one DSE 2013 asks: the arithmetic mean of the logarithm of wages is the log of the geometric mean of wages, which is why wage regressions in logs describe a typical wage rather than an average one.

Two conditions follow from the definition. The values must be positive. If any value is zero the product, and so the GM, is zero (JNU 2015): the GM of 4, 12, 20, 7, 0 and 5 is 0, which says nothing about the data. And the GM is never larger than the arithmetic mean, a fact the third picture of this lesson shows.

Rates such as speed (kilometres per hour) and price (rupees per litre) have a trap in them. A car drives 40 km/h out and 60 km/h back. The simple average is 50, but the car spends longer at the slow speed, so the true average over the round trip is less. For distance d each way, the time is d ÷ 40 + d ÷ 60 = d ÷ 24 hours for 2d kilometres, so the average speed is 48 km/h.

The question to ask is what is held fixed? The same prices can be averaged two ways, depending on how the family shops. JAM 2025: milk cost ₹28 a litre in September and ₹36 in October, and the family spends the same amount of money in each month. The average price it pays is the harmonic mean, ₹31.50. Had it bought the same number of litres, the average would be the plain mean, ₹32. Try other prices.

JAM 2026 gives twelve observations, 15, 20, 20, 5, 20, 20, 15, 20, 15, 20, 20, 20, and asks for their harmonic mean. Count them: three 15s, one 5 and eight 20s, so the sum of reciprocals is 3 ÷ 15 + 1 ÷ 5 + 8 ÷ 20 = 0.2 + 0.2 + 0.4 = 0.8, and HM = 12 ÷ 0.8 = 15.0.

Hyderabad 2012 lists the properties: the HM uses every observation, it gives the largest weight to the smallest items (a small value has a big reciprocal), and it cannot be computed if any value is zero.

For any positive numbers that are not all equal, the three means come in a fixed order: arithmetic mean at least the geometric, geometric at least the harmonic. Equality holds only when every number is the same (Hyderabad 2012 and 2017). JNU 2010 and 2015 ask the first half, GM ≤ AM.

For two numbers a and b there is a picture that shows it. Put a and b end to end on a line and draw the semicircle whose diameter is the whole line. Stand at the join and go up to the arc, to a point D. Then three lengths run out of D: the radius to the centre is the arithmetic mean; the straight drop to the join is the geometric mean; and the short segment to the foot of the perpendicular dropped from the join onto the radius is the harmonic mean. Three lengths from one point, and plainly in order.

Two facts to take away. First, for two numbers GM² = AM × HM: for 4 and 9 the means are 6.5, 6 and 5.54, and 6.5 × 5.538 = 36 = 6². Second, the picture gives a proof: since AM × HM = GM² and AM ≥ GM, we get HM = GM² ÷ AM ≤ GM. The other half is the algebra (√a − √b)² ≥ 0, which rearranges to a + b ≥ 2√(ab).

In words: the arithmetic mean treats differences additively, the geometric mean multiplicatively and the harmonic mean through reciprocals. The further the numbers are spread, the further the three drift apart. The gap between AM and HM for two numbers is (a − b)² ÷ (2(a + b)).