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If n is the number of numbers, it is found by dividing the number of numbers by the reciprocal of each number. Example 1: Find the harmonic mean of 3 and 4.

A car travels with a speed of 40 miles per hour for the first half of the way. Then, the car travels with a speed of 60 miles per hour for the second half of the way. What is the average speed? However, with some manipulation, we can still tackle the problem. See an example of harmonic mean related to the stock market. Top-notch introduction to physics.

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Harmonic mean The harmonic mean H of n numbers x 1x 2x 3Homepage Pre-algebra lessons Harmonic mean. Recent Articles. Check out some of our top basic mathematics lessons.The harmonic mean is a type of numerical average. It is calculated by dividing the number of observations by the reciprocal of each number in the series. Thus, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. The harmonic mean of 1, 4, and 4 is:. The harmonic mean helps to find multiplicative or divisor relationships between fractions without worrying about common denominators.

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Harmonic means are often used in averaging things like rates e. The weighted harmonic mean is used in finance to average multiples like the price-earnings ratio because it gives equal weight to each data point. Using a weighted arithmetic mean to average these ratios would give greater weight to high data points than low data points because price-earnings ratios aren't price-normalized while the earnings are equalized.

The harmonic mean is the weighted harmonic mean, where the weights are equal to 1. Other ways to calculate averages include the simple arithmetic mean and the geometric mean. An arithmetic average is the sum of a series of numbers divided by the count of that series of numbers.

If you were asked to find the class arithmetic average of test scores, you would simply add up all the test scores of the students, and then divide that sum by the number of students. The geometric mean is the average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio.

Harmonic Mean of 2 numbers

The harmonic mean is best used for fractions such as rates or multiples. As an example, take two firms. As can be seen, the weighted arithmetic mean significantly overestimates the mean price-earnings ratio. Financial Ratios. Advanced Technical Analysis Concepts. Real Estate Investing.

harmonic mean of two numbers

Investing Essentials. Your Money. Personal Finance.In mathematicsthe harmonic mean sometimes called the subcontrary mean is one of several kinds of averageand in particular, one of the Pythagorean means. Typically, it is appropriate for situations when the average of rates is desired.

The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations.

As a simple example, the harmonic mean of 1, 4, and 4 is. The third formula in the above equation expresses the harmonic mean as the reciprocal of the arithmetic mean of the reciprocals. It is the reciprocal dual of the arithmetic mean for positive inputs:.

Thus, the harmonic mean cannot be made arbitrarily large by changing some values to bigger ones while having at least one value unchanged. The harmonic mean is also concavewhich is an even stronger property than Schur-concavity. One has to take care to only use positive numbers though, since the mean fails to be concave if negative values are used.

The harmonic mean is one of the three Pythagorean means. For all positive data sets containing at least one pair of nonequal valuesthe harmonic mean is always the least of the three means, [2] while the arithmetic mean is always the greatest of the three and the geometric mean is always in between. If all values in a nonempty dataset are equal, the three means are always equal to one another; e.

Harmonic Mean

Since the harmonic mean of a list of numbers tends strongly toward the least elements of the list, it tends compared to the arithmetic mean to mitigate the impact of large outliers and aggravate the impact of small ones. The arithmetic mean is often mistakenly used in places calling for the harmonic mean. The harmonic mean is related to the other Pythagorean means, as seen in the equation below. This can be seen by interpreting the denominator to be the arithmetic mean of the product of numbers n times but each time omitting the j -th term.

That is, for the first term, we multiply all n numbers except the first; for the second, we multiply all n numbers except the second; and so on.

Thus the n -th harmonic mean is related to the n -th geometric and arithmetic means. The general formula is. If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are "spread apart" from each other while leaving the arithmetic mean unchanged — then the harmonic mean always decreases.

Three positive numbers HGand A are respectively the harmonic, geometric, and arithmetic means of three positive numbers if and only if [5] : p. The unweighted harmonic mean can be regarded as the special case where all of the weights are equal.

In many situations involving rates and ratiosthe harmonic mean provides the correct average. For instance, if a vehicle travels a certain distance d outbound at a speed x e.

The total travel time is the same as if it had traveled the whole distance at that average speed. This can be proven as follows: [6]. The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers the same distancethen the average speed is the harmonic mean of all the sub-trip speeds; and if each sub-trip takes the same amount of timethen the average speed is the arithmetic mean of all the sub-trip speeds.

If neither is the case, then a weighted harmonic mean or weighted arithmetic mean is needed. For the arithmetic mean, the speed of each portion of the trip is weighted by the duration of that portion, while for the harmonic mean, the corresponding weight is the distance.

In both cases, the resulting formula reduces to dividing the total distance by the total time. However one may avoid the use of the harmonic mean for the case of "weighting by distance".

Pose the problem as finding "slowness" of the trip where "slowness" in hours per kilometre is the inverse of speed. When trip slowness is found, invert it so as to find the "true" average trip speed. Then take the weighted arithmetic mean of the s i 's weighted by their respective distances optionally with the weights normalized so they sum to 1 by dividing them by trip length. This gives the true average slowness in time per kilometre. It turns out that this procedure, which can be done with no knowledge of the harmonic mean, amounts to the same mathematical operations as one would use in solving this problem by using the harmonic mean.

Thus it illustrates why the harmonic mean works in this case. Similarly, if one wishes to estimate the density of an alloy given the densities of its constituent elements and their mass fractions or, equivalently, percentages by massthen the predicted density of the alloy exclusive of typically minor volume changes due to atom packing effects is the weighted harmonic mean of the individual densities, weighted by mass, rather than the weighted arithmetic mean as one might at first expect.

To use the weighted arithmetic mean, the densities would have to be weighted by volume. Applying dimensional analysis to the problem while labeling the mass units by element and making sure that only like element-masses cancel makes this clear.Last Updated: April 30, References.

This article was co-authored by our trained team of editors and researchers who validated it for accuracy and comprehensiveness. This article has been viewedtimes.

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Learn more The harmonic mean is a way to calculate the mean, or average, of a set of numbers. Using the harmonic mean is most appropriate when the set of numbers contains outliers that might skew the result. Most people are familiar with calculating the arithmetic mean, in which the sum of values is divided by the number of values. Calculating the harmonic mean is a little more complicated. If working with a small set of numbers you may be able to solve by hand using the formula.

Otherwise, you can easily use Microsoft Excel to find the harmonic mean. Every day at wikiHow, we work hard to give you access to instructions and information that will help you live a better life, whether it's keeping you safer, healthier, or improving your well-being. Amid the current public health and economic crises, when the world is shifting dramatically and we are all learning and adapting to changes in daily life, people need wikiHow more than ever.

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To calculate the harmonic meaning, start by determining the number of values in your set of numbers. For example, if you're working with 10, 12, 16, and 8, you have 4 numbers, so the value is 4. Then, rewrite the numbers you're working with as denominators over the number 1. Then, divide 4 by the sum of the fractions to find the harmonic mean.

To calculate the harmonic meaning using a calculator, keep reading! Did this summary help you?

Harmonic Mean Calculator

Yes No. Please help us continue to provide you with our trusted how-to guides and videos for free by whitelisting wikiHow on your ad blocker.Use this harmonic mean calculator to easily calculate the Harmonic mean of a set of numbers. The harmonic mean archaic: subcontrary mean is a specialized average of a set of numbers.

It is one of the three Pythagorean means that provides the most accurate average.

harmonic mean of two numbers

The harmonic mean is more complex to solve than the arithmetic, although they might seem similar at first. The difference between the two is that the harmonic mean calculates the reciprocal of the arithmetic mean of reciprocals.

The harmonic mean is largely used in situations dealing with quantitative data, such as finding the average of rates or ratiosdue to the fact that it is not seriously affected by fluctuations.

The formula for calculating the harmonic mean of a set of non-zero positive numbers is: where n is number of items and X 1 …X 2 are the numbers from 1 to n. To put it simply, all you need to do is divide the number of items in the set by the sum of their reciprocals. The above formula is what we use in this harmonic mean calculator. If you're given the set of numbers: 3, 12, 20, 24, and have to find their harmonic mean, the first thing you should do is find a common denominator.

It'll come in handy when you add up the reciprocals. In this case, all of the numbers are divisible by Next up, find the sum of the reciprocals:.

An important note to be made is that you can't find the harmonic mean in a set of numbers, if one of them is 0. Our harmonic mean calculator does the math for you! You just need to write down the number dataset, separating the items with a comma and a blank space. The weighted harmonic mean is the reciprocal of the weighted average of the reciprocals of a weights dataset corresponding to a set of numbers.

It is calculated with the formula:.

harmonic mean of two numbers

In specific cases where the set consists of two numbers only, you can calculate the harmonic mean using this formula:. Even in this simple case it might be useful to turn to a calculator if the numbers are big or fractions. Three real, non-negative numbers A, G and H are respectively the average, geometric and harmonic means of three real, non-negative numbers if and only if this inequality is true:.Sign your name to a piece of paper, then hold that paper up to the camera on your Mac.

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