Arithmetic mean is used when the data values have the same units. The geometric mean is used when the data set values have differing units. When the values are expressed in rates we use harmonic mean. Harmonic mean is a mathematical mean that is usually used to find the average of variables when they are expressed as a ratio of different measuring units. Given below are the merits and demerits of harmonic mean:. The harmonic mean is completely based on observations and is very useful in averaging certain types of rates.
Other merits of the harmonic mean are given below. To calculate the harmonic mean, all elements of the series must be known. In case of unknown elements, we cannot determine the harmonic mean. Given below are other demerits of harmonic mean.
An important property of harmonic mean is that without taking a common denominator it can be used to find multiplicative and divisor relationships between fractions. This can be a very helpful tool in industries such as finance. Given below are some other real-life applications of harmonic mean.
Weighted harmonic mean is used when we want to find the average of a set of observations such that equal weight is given to each data point. Then the formula for weighted harmonic mean is given as follows:.
If we have normalized weights then all weights sum up to 1. When we take the reciprocal of the arithmetic mean of the reciprocal terms in a data set we get the harmonic mean.
Furthermore, if there are certain weights associated with each observation then we can calculate the weighted harmonic mean. We first take the sum of the reciprocals of each term in the given data set.
Then we divide the total number of terms n in the data set by this value to get the harmonic mean. When we have a data set, the geometric mean can be determined by taking the n th root of the product of all the n terms. To find the harmonic mean we divide n by the sum of the reciprocals of the terms.
The harmonic mean will always have a lower value than the geometric mean. Harmonic mean is not the reciprocal of the arithmetic mean. Harmonic mean is the reciprocal of the arithmetic mean of the reciprocal terms in a given set of observations. The harmonic mean has a rigid value and does not get affected by fluctuations in the sample however if the sample contains a zero term we cannot calculate the harmonic mean.
Also, the formula to determine the harmonic mean can result in complex computations. Harmonic mean sees widespread use in geometry and music. Apply market research to generate audience insights. Measure content performance. Develop and improve products.
List of Partners vendors. Your Money. Personal Finance. Your Practice. Popular Courses. What Is a Harmonic Mean? Harmonic means are used in finance to average data like price multiples. Harmonic means can also be used by market technicians to identify patterns such as Fibonacci sequences. Compare Accounts. The offers that appear in this table are from partnerships from which Investopedia receives compensation.
This compensation may impact how and where listings appear. Investopedia does not include all offers available in the marketplace. Related Terms Arithmetic Mean Definition The arithmetic mean is the sum of all the numbers in the series divided by the count of all numbers in the series. How to Use the Winsorized Mean Winsorized mean is an averaging method that involves replacing the smallest and largest values of a data set with the observations closest to them. T-Test Definition A t-test is a type of inferential statistic used to determine if there is a significant difference between the means of two groups, which may be related in certain features.
What Is Annualized Total Return? Annualized total return gives the yearly return of a fund calculated to demonstrate the rate of return necessary to achieve a cumulative return. It's also a choice of what set of summary statistics to describe the population or process of interest.
One shouldn't think that all that's necessary is a single number to describe something of maybe great complexity. Add a comment. Active Oldest Votes. The distance—velocity—time example appears to be popular, so let's use it. Varying distances, constant time Now, let's change the situation. New means from old As an exercise, think about what the "natural" mean is in the situation where you let both the distances and times vary in the first example.
Exercise : What is the "natural" mean in this situation? Improve this answer. However, I think it's incomplete in an important way: in many cases the right mean to use is determined by the question we are trying to answer rather than by any mathematical structure in the data. A good example of this occurs in environmental risk assessment: regulatory authorities want to estimate a population's total exposure to contaminants over time.
This requires an appropriately weighted arithmetic mean, even though environmental concentration data usually have a multiplicative structure. The geometric mean would be the wrong estimator or estimand. On my path to constructing an answer, I took a decidedly nonstatistical fork, so I'm glad you mentioned this.
It's a topic worthy of a complete answer hint. The issue I've run into there in the past is that they sometimes want to also dictate the way that statistical estimation is carried out! Honestly, your explanations are one of the best I've seen on Stats.
Sometimes perhaps often! Show 4 more comments. Peter Flom Peter Flom And what if say extracurricular-activities tend to cluster in a much narrower band than its theoretical range? It seems like it would make more sense to take an arithmetic mean of percentiles or other adjusted values than a geometric mean of raw values. I think an arithmetic mean of percentiles will be close to the geometric mean in its conclusions even though not in the actual numbers.
Here's a non-automotive example courtesy of David Giles : For instance, consider data on "hours worked per week" a rate. However, they work for different numbers of hours per week, as follows: Person Total Hours Hours per Week Weeks Taken 1 2, 40 50 2 2, 45 Ira Nirenberg Ira Nirenberg 1 1 1 bronze badge. See our editing help for more information. In that case does it ever make sense to use geometric or harmonic mean? Featured on Meta.
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