Harmonic Mean Calculator
Built for Rates and Speeds
The harmonic mean is built for rates, speeds, and prices per unit, where plain averages mislead. Drivers comparing trip speeds and traders averaging purchase prices rely on it because small values pull the result down sharply.
The Reciprocal Formula
Take n, divide it by the sum of reciprocals, and you have the harmonic mean. For 4 and 6, the reciprocals are 1/4 and 1/6, which sum to 5/12, and 2 divided by 5/12 equals 4.8.
Speeds show why this matters. Travel out at 60 km/h, back at 40 km/h: the harmonic mean is 48 km/h, not 50, since the distances stayed equal.
Equal spending, not equal holdings, is where the mean shines for prices. Suppose you invest the same amount at 10, 20, and 40 dollars per share; the true average price paid is the harmonic mean of the three. The reciprocals run 0.1 plus 0.05 plus 0.025 = 0.175, and 3 divided by 0.175 gives 17.14 dollars, less than the plain average of 23.33 because cheap prices earn you more shares. Buy at 10 and 90 dollars with equal spends instead, and the result drops to 18, versus an average of 50. Enter 10, 20, 40, press Calculate, and the tool confirms 17.14, exactly the cost basis a purchase log should record.
Running the Calculation
Enter your positive numbers, separated by commas, then press Calculate. Zero and negative entries break the method because their reciprocals are undefined. The result always sits at or below the ordinary mean: 4 and 6 give 4.8 against an average of 5.
Frequently Asked Questions
What is the harmonic mean of 4 and 6?
Add the reciprocals so 1/4 plus 1/6 equals 5/12. Dividing the count, 2, by that sum gives 2 x 12/5, which equals 4.8, the value the calculator reports.
Why is it best for average speed?
Equal distances demand equal time weighting, which the harmonic mean provides. Driving out at 60 km/h and back at 40 km/h takes a total time equivalent to 48 km/h overall, not the misleading 50.
How does it compare with the arithmetic mean?
The harmonic mean never exceeds the arithmetic mean. For 4 and 6 the arithmetic value is 5 while the harmonic value is 4.8, and the gap widens as the numbers spread further apart.
Can I include zero or negative numbers?
Zero fails first because 1/0 is undefined, and negatives can distort the sum of reciprocals. Keep every entry positive; inputting 0, 4, and 6 would trigger an error rather than a mean.
How many values can the calculator accept?
Any reasonable list, since the formula divides the count n by the sum of all reciprocals. Three numbers like 2, 4, and 8 combine as 3 over 0.875, producing a harmonic mean of about 3.43.