Harmonic Mean Calculator NZ

This calculator finds the harmonic mean of a set of numbers, the correct average to use for rates, speeds and ratios where the quantity is measured per unit of something. Of the three classical averages, the harmonic mean is the least familiar but the most appropriate in a surprising number of situations. It is defined as the number of values divided by the sum of their reciprocals, which gives more weight to the smaller values and is exactly what you need when averaging things like speeds over equal distances, prices in cost-per-unit terms, or rates of work. The classic example is average speed: if you drive somewhere at 40 kilometres an hour and back at 60, your average speed for the round trip is not 50 but the harmonic mean of 40 and 60, which is 48, because you spend more time at the slower speed. Getting this wrong is a common mistake, and the harmonic mean fixes it. This tool computes it for you. You paste or type your numbers, which should be positive, and the calculator returns the harmonic mean, along with the arithmetic and geometric means for comparison and the count of values. For any set of positive numbers, the harmonic mean is the smallest of the three, the arithmetic the largest, and the geometric in between. The results update as you type. Use it to average speeds or rates correctly, to combine ratios, in finance for averaging price multiples, or for statistics study. The key is to reach for the harmonic mean whenever you are averaging quantities expressed as something per unit, especially over equal amounts of the denominator, where the ordinary average would mislead.

Advertise on this page

Your numbers (positive, space/comma separated)

1.714

harmonic mean

Arithmetic mean2.333

Geometric mean2

Count3

Harmonic mean = n / sum of reciprocals. Best for averaging rates and speeds over equal distances. Always the smallest of the three classical means.

How it works

The harmonic mean is the number of values divided by the sum of their reciprocals. In other words, take one over each value, add those up, divide that into the count of values. This weights smaller values more heavily, which is why it is the right average for rates measured over equal amounts.

Worked example

For 1, 2 and 4, the reciprocals are 1, 0.5 and 0.25, which sum to 1.75. The harmonic mean is 3 divided by 1.75, about 1.714. For comparison the arithmetic mean is 2.333 and the geometric mean is 2, so the harmonic mean is the smallest, as it always is for positive numbers.

Related calculators

Geometric Mean: average of ratios.
Mean Median Mode: common averages.
Weighted Average: weighted mean.
Average Calculator: simple average.

Frequently asked questions

When should I use the harmonic mean?

Use it for averaging rates, speeds and ratios, especially over equal amounts of the denominator. The classic case is average speed over equal distances, where the harmonic mean, not the arithmetic mean, gives the correct round-trip average because more time is spent at the slower speed.

How do I calculate the harmonic mean?

Divide the number of values by the sum of their reciprocals. So for two speeds, it is two divided by one over the first plus one over the second. This calculator does it for any list of positive numbers and shows the other means for comparison.

Why is the harmonic mean smaller than the arithmetic mean?

Because it weights smaller values more heavily through their reciprocals. For any set of positive numbers that are not all equal, the harmonic mean is the smallest of the three classical means, the arithmetic mean is the largest, and the geometric mean sits in between.