Harmonic mean
The harmonic mean (usually designated by H) of a finite number of numbers is equal to the reciprocal, or inverse, of the arithmetic mean of the reciprocals of those values and is recommended for averaging speeds.
Thus, given n numbers x1, x2,... xn the harmonic mean will be equal to:
H=n␡ ␡ i=1n1xi=n1x1+ +1xn{displaystyle {H}={frac {n}{sum _{i=1}{n}{cfrac {1}{x_{x}}}}}}}}}}{frac {n}{{{cfrac {1}{x_{1}}{x}{x}}{x}}}}}}}
The harmonic mean is little influenced by the existence of certain values much larger than the set of others, being instead sensitive to values much smaller than the set.
The harmonic mean is undefined in the case of any null value.
Features
- The reverse of the harmonic mean is the arithmetic mean of the inverses of the values of the variable.
- You can always move from a harmonic media to an arithmetic media by properly transforming the data.
- The harmonic mean is always less or equal to the arithmetic mean, as for any positive real number 0}" xmlns="http://www.w3.org/1998/Math/MathML">xi▪0{displaystyle x_{i} 20050}
0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ca010976c6eb10a9e66a4ec495766452bfe828" style="vertical-align: -0.671ex; width:6.39ex; height:2.509ex;"/>:
n1x1+ +1xn≤ ≤ x1+ +xnn{displaystyle {frac {n}{{cfrac {1}{x_{1}}}{xx}}{x}}}{x}}}}{x}}}}}}{frac {xx_{1}} +dots +x_{n}}{n}}}{n}}
Advantage
- It considers all the distribution values and in certain cases is more representative than the arithmetic average.
Disadvantages
- The influence of small values and the fact that it cannot be determined in distributions with values equal to zero; therefore its use is not advisable in distributions where there are very small values.
It is usually used to average speeds, times, yields, etc.
Curiosities
The harmonic mean arises naturally when calculating the Paasche index, one of the most common index numbers. Consider a temporary series p1,tq1,t+ +pn,tqn,t{displaystyle p_{1,t}q_{1,t}+cdots +p_{n,t}q_{n,t}} which results from adding the nominal value of production or expenditure pi,tqi,t{displaystyle p_{i,t}q_{i,t} in n{displaystyle n} goods. To isolate changes in price changes, the Laspeyres index sets the prices of the previous period and compares the spending today with yesterday's prices to yesterday's expense
- Lt=␡ ␡ i=1npi,t− − 1qi,t␡ ␡ i=1npi,t− − 1qi,t− − 1=␡ ␡ i=1nqi,tqi,t− − 1pi,t− − 1qi,t− − 1␡ ␡ i=1npi,t− − 1qi,t− − 1{displaystyle L_{t}={frac {sum _{i=1}^{np_{i,t-1}{i,t}{i}{i}{i=1}{n}{i}{i}{i,t-1}{i}}}}{i,t1}}}{i, }{i, }{i, }{i, }{i }{i }, },,,,,,,,,,,, },,,,,,,,, },,,,,,,,,, },,,,,,, },,,,,,, },,,,,,,,,,,,,, },,,,,, }
By leaving fixed prices, it is interpreted that Lt{displaystyle L_{t} only reflects changes in quantities or real. It can also be observed that this is a mean where the change in the quantity of the goods i{displaystyle i} It appears weighted by the weight of spending on this commodity on total expenditure.
The Paasche index, in reverse, proceeds to fix today's prices: it compares spending today with spending yesterday if today's prices had prevailed.
- Pt=␡ ␡ i=1npi,tqi,t␡ ␡ i=1npi,tqi,t− − 1.{displaystyle P_{t}={frac {sum _{i=1}^{np_{i,tq_{i,t}}{sum _{i=1}^{n}p_{i,t}q_{i,t-1}}}}}}}} !
From this definition we cannot obtain a weighted average as before. However, if the inverted formula is considered, it happens that
- 1Pt=␡ ␡ i=1npi,tqi,t− − 1␡ ␡ i=1npi,tqi,t=␡ ␡ i=1nqi,t− − 1qi,tpi,tqi,t␡ ␡ i=1npi,tqi,t{displaystyle {frac {1}{P_{t}}}{frac {sum _{i=1}{n}{n}p_{i,t}q_{i,t-1}}{sum}{i=1}{i}{n}{i, }{i, }{i, }{i, }{n}{i }{i }{i, }{i }{i }{i }{i,,, }{i },, }, } } },,,, }{, }{ },,,,,,, },,,,,,,,,,,,,,,,,,,,,,,,,,,,,, }{ }{ }{ } } }
but then
- Pt=(␡ ␡ i=1n1qi,tqi,t− − 1pi,tqi,t␡ ␡ i=1npi,tqi,t)− − 1.{displaystyle P_{t}=left(sum _{i=1}^{n}{frac {1}{frac {q_{i,t}}{q_{i,t-1}}}}{frac {p_{i,t}q_{i,t}}{sum _{i=1^{n}{i,tq}}{i. !
That is, the Paasche index turns out to be the harmonic mean of the changes in quantities in each of the commodities.