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Formula for Harmonic Mean
Weighted Harmonic Mean
Some Important Remarks
Related Resources
Harmonic Mean is one of the several kinds of average. Mathematically, the harmonic mean between two numbers a and b is defined as
H = 2/ (1/a + 1/b)
This can further be written as:
H = 2ab/(a+b)
Likewise, the harmonic mean of n positive numbers x_{1}, x_{2}, …. , x_{n} is defined to be
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 H is the harmonic mean between two numbers say a and b then 1/a, 1/H and 1/b are in A.P.
Let us consider a simple example to understand the concept of harmonic mean:
We wish to find the harmonic mean of 1, 2 and 4. Since these are three in number, so by applying the formula the H.M. is given by
3/ (1/1 + 1/2 + 1/4) = 12/7.
Let us see how exactly we reach the formula for harmonic mean between two numbers a and b:
Let H be the harmonic mean between two numbers a and b.
So, a, H, b are in H.P.
This means that 1/a, 1/H, 1/b are in A.P.
or, 1/H – 1/a = 1/b – 1/H.
or, 2/H = 1/a + 1/b
= a+b/ab
∴ H = 2ab/a+b
On the same lines, we may also find two harmonic means between two numbers.
Let us assume that H_{a} and H_{b} are two harmonic means between a and b.
Then it follows that a, H_{a}, H_{b }and b are in H.P.
Then 1/a, 1/H_{a}, 1/H_{b}, 1/b are in A.P.
Hence, using the formula we have
t_{n} = a + (n-1)d
So, 1/b – 1/a = 3d, where‘d’ denotes the common difference of the A.P.
This implies that 3d = (a-b)/b
Hence, d = (a-b)/3ab
So, 1/H_{1} = 1/a + d = 1/a + a–b/3ab = a+2b/3ab
and 1/H_{2} = 1/a + 2d = 1/a + 2(a–b)/3ab = 2a+2b/3ab
Hence, these are the two harmonic means between a and b.
Another concept closely related to harmonic mean is that of weighted harmonic mean. If we have a set of weights w_{1}, w_{2}, …. , w_{n} associated with the set of values x_{1}, x_{2}, …. , x_{n}, then the weighted harmonic mean is defined as
Harmonic mean is in fact a special case of weighted harmonic mean where all the weights are equal to 1 and is equal to any weighted harmonic mean having all equal weights.
The following figure shows two crossed ladders A and B with each having feet at the base of one side wall and one leaning against a wall at height A and the other leaning against the opposite wall at height B. in such a case, h is half the harmonic mean of A and B. Let us find the harmonic mean H of 2, 20, 10, 5, 1 using the formula stated above.
We explain the whole procedure in the form of steps:
1. First calculate the total number of items which in this case is 5 and hence, N = 5.
2. Now use the formula of harmonic mean i.e.
H = N / (1/a_{1} + 1/a_{2 }+ ….. 1/a_{N})
= 5/ (1/2 + 1/20 + 1/10 + 1/5 + 1/1)
= 5/ (37/20)
= 5 .20 / 37
= 100/37.
Watch this Video for more reference
• If a and b are two non-zero numbers, then the harmonic mean of a and b is a number H such that the numbers a, H, b are in H.P. We have H = 1/H = 1/2 (1/a + 1/b) ⇒ H = 2ab/a+b.
• If a_{1}, a_{2}, ……, a_{n} are n non-zero numbers. then the harmonic mean H of these number is given by 1/H = 1/n (1/a_{1} + 1/a_{2} +...+ 1/a_{n}).
• The n numbers H_{1}, H_{2}, ……, H_{n} are said to be harmonic means between a and b, if a, H_{1}, H_{2} ……, H_{n}, b are in H.P. i.e. if 1/a, 1/H_{1}, 1/H_{2}, ..., 1/H_{n}, 1/b are in A.P. Let d be the common difference of the A.P., Then 1/b = 1/a + (n+1) d ⇒ d = a–b/(n+1)ab.
Thus 1/H_{1} = 1/a + a–b/(n+1)ab,
1/H_{2} = 1/a + 2(a–n)/(n+1)ab,
……….. ……….
1/H_{n} = 1/a + n(a–b)/(n+1)ab.
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You may also like to refer Arithmetic Mean and Geometric Mean.
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