Harmonious mean

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The harmonic mean is an average of a set of numbers. It was already known to Pythagoras . It is the special case of the Hölder mean with parameter −1.

definition

The harmonic mean of the numbers is as ${\displaystyle x_{1},\dotsc ,x_{n}}$

${\displaystyle {\bar {x}}_{\text{harm}}={\frac {n}{{\frac {1}{x_{1}}}+\dotsb +{\frac {1}{x_{n}}}}}}$

Are defined. The reciprocal of the harmonic mean is

${\displaystyle {\frac {1}{{\bar {x}}_{\text{harm}}}}={\frac {{\frac {1}{x_{1}}}+\dotsb +{\frac {1}{x_{n}}}}{n}}}$

and thus the arithmetic mean of the reciprocal values.

With the formula, the harmonic mean is initially only defined for numbers other than zero . But if one of the values approaches zero, the limit value of the harmonic mean exists and is also equal to zero. Therefore it makes sense to define the harmonic mean as zero if at least one of the quantities to be averaged is equal to zero. ${\displaystyle x_{i}}$ ${\displaystyle x_{i}}$

properties

For two values and results ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle {\bar {x}}_{\text{harm}}={\frac {2ab}{a+b}}={\frac {{\bar {x}}_{\text{geom}}^{2}}{{\bar {x}}_{\text{arithm}}}}}$

with the arithmetic mean and the geometric mean . ${\displaystyle {\bar {x}}_{\text{arithm}}}$ ${\displaystyle {\bar {x}}_{\text{geom}}}$

For nonnegatives applies ${\displaystyle x_{i}}$

${\displaystyle \min(x_{1},\dotsc ,x_{n})\leq {\bar {x}}_{\text{harm}}\leq {\bar {x}}_{\text{geom}}\leq {\bar {x}}_{\text{arithm}}\leq \max(x_{1},\dotsc ,x_{n}).}$

example

For the harmonic mean of and applies ${\displaystyle 5}$ ${\displaystyle 20}$

${\displaystyle {\frac {2}{{\frac {1}{5}}+{\frac {1}{20}}}}={\frac {2}{\frac {1}{4}}}=8}$.

If the formula from the Properties section is used, the following applies

${\displaystyle {\frac {2\cdot 5\cdot 20}{5+20}}=8}$.

Weighted harmonic mean

definition

If the positive weights are assigned, the weighted harmonic mean is defined as follows: ${\displaystyle x_{i}}$ ${\displaystyle w_{i}>0}$

${\displaystyle {\bar {x}}_{\mathrm {harm} }={\frac {w_{1}+\cdots +w_{n}}{{\frac {w_{1}}{x_{1}}}+\cdots +{\frac {w_{n}}{x_{n}}}}}}$

If they are all equal, the usual harmonic mean is obtained. ${\displaystyle w_{i}}$

example

In general: Do I need to hop the time (ie, average speed ) and for the hop time (ie average speed ), then for the average speed over the entire distance ${\displaystyle s_{1}}$ ${\displaystyle t_{1}}$ ${\displaystyle v_{1}=s_{1}/t_{1}}$ ${\displaystyle s_{2}}$ ${\displaystyle t_{2}}$ ${\displaystyle v_{2}=s_{2}/t_{2}}$

${\displaystyle v={\frac {s_{1}+s_{2}}{t_{1}+t_{2}}}={\frac {s_{1}+s_{2}}{{\frac {s_{1}}{v_{1}}}+{\frac {s_{2}}{v_{2}}}}}={\frac {t_{1}v_{1}+t_{2}v_{2}}{t_{1}+t_{2}}}}$

The average speed is therefore the harmonic mean of the partial speeds weighted with the distances traveled or the arithmetic mean of the partial speeds weighted with the required time.

If you drive one hour at 50 km / h and then one hour at 100 km / h, you cover a total of 150 km in 2 hours; the average speed is 75 km / h, i.e. the arithmetic mean of 50 and 100. If, on the other hand, one does not refer to the time required, but to the distance traveled, the average speed is described by the harmonic mean: If one drives 100 km with 50 km / h and then 100 km at 100 km / h, you cover 200 km in 3 hours, the average speed is 66.67 km / h, i.e. the harmonic mean of 50 and 100.

${\displaystyle v={\frac {100\ {\text{km}}+100\ {\text{km}}}{2\ {\text{h}}+1\ {\text{h}}}}={\frac {100\ {\text{km}}+100\ {\text{km}}}{{\frac {100\ {\text{km}}}{50\ {\text{km/h}}}}+{\frac {100\ {\text{km}}}{100\ {\text{km/h}}}}}}={\frac {2\ {\text{h}}\cdot 50\ {\text{km/h}}+1\ {\text{h}}\cdot 100\ {\text{km/h}}}{2\ {\text{h}}+1\ {\text{h}}}}={\frac {200}{3}}\ {\text{km/h}}\approx 66{,}67\ {\text{km/h}}}$

See also

• Arithmetic mean
• Geometric mean
• Inequality of harmonic and geometric mean

Web links

• Eric W. Weisstein : Harmonic Mean . In: MathWorld (English).