In mathematics, the harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average, and 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. As a simple example, the harmonic mean of 1, 2, and 4 is
The harmonic mean H of the positive real numbers
is defined to be
From the third formula in the above equation, it is more apparent that the harmonic mean is related to the arithmetic and geometric means. It is the reciprocal dual of the arithmetic mean for positive inputs:
The harmonic mean is a Schur-concave function, and dominated by the minimum of its arguments, in the sense that for any positive set of arguments,
. Thus, the harmonic mean cannot be made arbitrarily large by changing some values to bigger ones (while having at least one value unchanged).
Relationship with other means
The harmonic mean is one of the three Pythagorean means. For all positive data sets containing at least one pair of nonequal values, the harmonic mean is always the least of the three means, 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.g. the harmonic, geometric, and arithmetic means of {2,2,2} are all2.)
See all my videos at http://www.zstatistics.com/
0:00 Introduction
1:21 Arithmetic mean
3:25 Geometric mean
8:59 Harmonic mean
14:29 Challenge Question
published: 07 Jan 2019
Geometry: Arithmetic, Geometric, Harmonic Means
published: 10 Sep 2015
Harmonic Mean|How To Calculate Harmonic Mean?|Use Of harmonic Mean
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.
Examples of how to calculate Harmonic Mean in group and ungroup data
or in frequency distribution.
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published: 08 May 2020
Harmonic mean 😎| Simple Calculation✍ |
easy calculation of HARMONIC MEAN..
published: 06 Jun 2020
Harmonic mean, Geometric mean,Airthmatic mean and root mean square
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published: 13 Dec 2019
Expected Math MCQ Harmonic Mean Questions for BCOM First Semester Examination | Mathematics videos
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0:00 Introduction
1:21 Arithmetic mean
3:25 Geometric mean
8:59 Harmonic mean
14:29 Challenge Question
See all my videos at http://www.zstatistics.com/
0:00 Introduction
1:21 Arithmetic mean
3:25 Geometric mean
8:59 Harmonic mean
14:29 Challenge Question
See all my videos at http://www.zstatistics.com/
0:00 Introduction
1:21 Arithmetic mean
3:25 Geometric mean
8:59 Harmonic mean
14:29 Challenge Question
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 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.
Examples of how to calculate Harmonic Mean in group and ungroup data
or in frequency distribution.
Kindly do like and subscribe this channel.
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.
Examples of how to calculate Harmonic Mean in group and ungroup data
or in frequency distribution.
Kindly do like and subscribe this channel.
In this video, we are going to find the visual proof HM GM AM RM with graphs.
////here is a free gift for you
-➕➗✖️➖-A FREE COURSE ON HUMAN CALCULATOR-➕➗✖️➖-
h...
In this video, we are going to find the visual proof HM GM AM RM with graphs.
////here is a free gift for you
-➕➗✖️➖-A FREE COURSE ON HUMAN CALCULATOR-➕➗✖️➖-
https://www.youtube.com/playlist?list=PLMTt7lFOKaqpRfEMWHHTNzYeH9n1tm-70
if you have any questions you can ask me in comments.
I hope you will watch this video till the end and subscribe to my channel for more upcoming videos.
thanks for watching
follow us on Facebook
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also, follow us on Instagram
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In this video, we are going to find the visual proof HM GM AM RM with graphs.
////here is a free gift for you
-➕➗✖️➖-A FREE COURSE ON HUMAN CALCULATOR-➕➗✖️➖-
https://www.youtube.com/playlist?list=PLMTt7lFOKaqpRfEMWHHTNzYeH9n1tm-70
if you have any questions you can ask me in comments.
I hope you will watch this video till the end and subscribe to my channel for more upcoming videos.
thanks for watching
follow us on Facebook
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also, follow us on Instagram
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Download my App from Google Play Store:
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Subscribe for Government Exams preparation [...
Download my App from Google Play Store:
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See all my videos at http://www.zstatistics.com/
0:00 Introduction
1:21 Arithmetic mean
3:25 Geometric mean
8:59 Harmonic mean
14:29 Challenge Question
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.
Examples of how to calculate Harmonic Mean in group and ungroup data
or in frequency distribution.
Kindly do like and subscribe this channel.
In this video, we are going to find the visual proof HM GM AM RM with graphs.
////here is a free gift for you
-➕➗✖️➖-A FREE COURSE ON HUMAN CALCULATOR-➕➗✖️➖-
https://www.youtube.com/playlist?list=PLMTt7lFOKaqpRfEMWHHTNzYeH9n1tm-70
if you have any questions you can ask me in comments.
I hope you will watch this video till the end and subscribe to my channel for more upcoming videos.
thanks for watching
follow us on Facebook
https://www.facebook.com/mathocube
also, follow us on Instagram
https://www.Instagram.com/mathocube
Download my App from Google Play Store:
https://play.google.com/store/apps/details?id=co.iron.peumr&hl=en_IN&gl=US
Subscribe for Government Exams preparation [SSC, CGL, CHSL, Bank, Railway and many more]: https://www.youtube.com/channel/UCZtcHe6hiqkDF4vHMUN8ytg
Purchase/Enquire about *Pendrive Course*: https://bit.ly/msc-course-enquiry
Set-up for recording:
1. Mic - https://amzn.to/2Q8PJWj
2. Tripod - https://amzn.to/2SA6tqk
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4. Voice Recorder - https://amzn.to/2RIGGvJ
Follow me on:
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WhatsApp - 9830489610
In mathematics, the harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average, and 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. As a simple example, the harmonic mean of 1, 2, and 4 is
The harmonic mean H of the positive real numbers
is defined to be
From the third formula in the above equation, it is more apparent that the harmonic mean is related to the arithmetic and geometric means. It is the reciprocal dual of the arithmetic mean for positive inputs:
The harmonic mean is a Schur-concave function, and dominated by the minimum of its arguments, in the sense that for any positive set of arguments,
. Thus, the harmonic mean cannot be made arbitrarily large by changing some values to bigger ones (while having at least one value unchanged).
Relationship with other means
The harmonic mean is one of the three Pythagorean means. For all positive data sets containing at least one pair of nonequal values, the harmonic mean is always the least of the three means, 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.g. the harmonic, geometric, and arithmetic means of {2,2,2} are all2.)