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The harmonic mean between two numbers a and b is
A.$\dfrac{2}{\dfrac{1}{a}+\dfrac{1}{b}}$
B.$\dfrac{a b}{\dfrac{1}{a}+\dfrac{1}{b}}$
C.$\dfrac{2 n}{\dfrac{1}{a}+\dfrac{1}{b}}$
D.$\dfrac{1}{\dfrac{1}{a}-\dfrac{1}{b}}$

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Last updated date: 20th Jun 2024
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Answer
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Hint: The harmonic mean of two numbers is in fact the reciprocal of arithmetic mean of the reciprocal of the numbers. This simply means that if $\mathrm{H}$ is the harmonic mean between two numbers say a and $\mathrm{b}$ then $1 / \mathrm{a}, 1 / \mathrm{H}$ and $1 / \mathrm{b}$ are in A.P. Harmonic Mean is the reciprocal of the arithmetic mean of the reciprocals. In other words, it is the number of observations, divided by the sum of reciprocals of the observations. This also is one of several kinds of average and it is appropriate for situations when the average of rates is desired. The mean, also referred to by statisticians as the average, is the most common statistic used to measure the center of a numerical data set. The mean is the sum of all the values in the data set divided by the number of values in the data set.

Complete step-by-step answer:
The harmonic mean helps to find multiplicative or divisor relationships between fractions without worrying about common denominators. 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.
The arithmetic mean is appropriate if the values have the same units, whereas the geometric mean is appropriate if the values have differing units. The harmonic mean is appropriate if the data values are ratios of two variables with different measures,
called rates.
The harmonic mean has the following merits. It is rigidly defined. It is based on all the observations of a series i.e. it cannot be calculated ignoring any item of a series. It is capable of further algebraic treatment. It gives better results when the ends to be achieved are the same for the different means adopted.
Harmonic mean $=\dfrac{\mathrm{n}}{\sum_{\mathrm{i}=1}^{\mathrm{n}} \dfrac{1}{\mathrm{x}_{1}}}=\dfrac{2}{\left(\dfrac{1}{\mathrm{a}}+\dfrac{1}{\mathrm{b}}\right)}$
Hence, the correct answer is Option A.

Note: Harmonic mean has the least value among all the three means. The relationship between arithmetic mean, geometric mean and harmonic mean is: "The product of arithmetic mean and harmonic mean of any two numbers a and $b$ in such a way that $a>b>0$ is equal to the square of their geometric mean." $A M \times H M=G M^{2}$. For two numbers $x$ and y, let x, a, y be a sequence of three numbers. If x, a, y is an arithmetic progression then 'a' is called arithmetic mean. If $\mathrm{x}, \mathrm{a}, \mathrm{y}$ is a geometric progression then 'a' is called geometric mean. If $\mathrm{x}, \mathrm{a}, \mathrm{y}$ form a harmonic progression then 'a' is called harmonic mean.
It is the value that is most common. An important property of the mean is that it includes every value in your data set as part of the calculation. In addition, the mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero.