Editing Harmonic mean (section)

===In physics===
====Average speed====
In many situations involving [[rate (mathematics)|rate]]s and [[ratio]]s, the harmonic mean provides the correct [[average]]. For instance, if a vehicle travels a certain distance ''s'' outbound at a speed ''v''<sub>1</sub> (e.g. 60 km/h) and returns the same distance at a speed ''v''<sub>2</sub> (e.g. 20 km/h), then its average speed is the harmonic mean of ''v''<sub>1</sub> and ''v''<sub>2</sub> (30 km/h), not the arithmetic mean (40 km/h). The total travel time is the same as if it had traveled the whole distance at that average speed. This can be proven as follows:<ref>{{cite web|url=https://learningpundits.com/module-view/48-averages/1-tips-on-averages/|title=Average: How to calculate Average, Formula, Weighted average|website=learningpundits.com|access-date=8 May 2018|url-status=live|archive-url=https://web.archive.org/web/20171229231610/https://learningpundits.com/module-view/48-averages/1-tips-on-averages/|archive-date=29 December 2017}}</ref>

Average speed for the entire journey
{{=}} <big>{{sfrac|Total distance traveled|Sum of time for each segment}}</big>
{{=}} {{sfrac|2''s''|''t''<sub>1</sub> + ''t''<sub>2</sub>}} = {{sfrac|2''s''|<big>{{sfrac|''s''|''v''<sub>1</sub>}} + {{sfrac|''s''|''v''<sub>2</sub>}}</big>}} = {{sfrac|2''v''<sub>1</sub>''v''<sub>2</sub>|''v''<sub>1</sub> + ''v''<sub>2</sub>}}

However, if the vehicle travels for a certain amount of ''time'' at a speed ''v''<sub>1</sub> and then the same amount of time at a speed ''v''<sub>2</sub>, then its average speed is the [[arithmetic mean]] of ''v''<sub>1</sub> and ''v''<sub>2</sub>, which in the above example is 40 km/h.{{cn|date=March 2025}}

Average speed for the entire journey
{{=}} <big>{{sfrac|Total distance traveled|Sum of time for each segment}}</big>
{{=}} {{sfrac|''s''<sub>1</sub> + ''s''<sub>2</sub>|2''t''}} = {{sfrac|''v''<sub>1</sub>''t'' + ''v''<sub>2</sub>''t''|2''t''}}
{{=}} {{sfrac|''v''<sub>1</sub> + ''v''<sub>2</sub>|2}}

The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers the same ''distance'', then the average speed is the ''harmonic'' mean of all the sub-trip speeds; and if each sub-trip takes the same amount of ''time'', then the average speed is the ''arithmetic'' mean of all the sub-trip speeds. (If neither is the case, then a [[#Weighted harmonic mean|weighted harmonic mean]] or [[weighted arithmetic mean]] is needed. For the arithmetic mean, the speed of each portion of the trip is weighted by the duration of that portion, while for the harmonic mean, the corresponding weight is the distance.  In both cases, the resulting formula reduces to dividing the total distance by the total time.)

However, one may avoid the use of the harmonic mean for the case of "weighting by distance". Pose the problem as finding "slowness" of the trip where "slowness" (in hours per kilometre) is the inverse of speed. When trip slowness is found, invert it so as to find the "true" average trip speed. For each trip segment i, the slowness s<sub>i</sub> = 1/speed<sub>i</sub>. Then take the weighted [[arithmetic mean]] of the s<sub>i</sub>'s weighted by their respective distances (optionally with the weights normalized so they sum to 1 by dividing them by trip length). This gives the true average slowness (in time per kilometre). It turns out that this procedure, which can be done with no knowledge of the harmonic mean, amounts to the same mathematical operations as one would use in solving this problem by using the harmonic mean. Thus it illustrates why the harmonic mean works in this case.{{cn|date=April 2025}}

====Density====
{{Unreferenced section|date=December 2019}}
Similarly, if one wishes to estimate the density of an [[alloy]] given the densities of its constituent elements and their mass fractions (or, equivalently, percentages by mass), then the predicted density of the alloy (exclusive of typically minor volume changes due to atom packing effects) is the weighted harmonic mean of the individual densities, weighted by mass, rather than the weighted arithmetic mean as one might at first expect. To use the weighted arithmetic mean, the densities would have to be weighted by volume. Applying [[dimensional analysis]] to the problem while labeling the mass units by element and making sure that only like element-masses cancel makes this clear.

====Electricity====
{{also|Parallel (operator)}}
If one connects two electrical [[resistor]]s in parallel, one having resistance ''R''<sub>1</sub> (e.g., 60&nbsp;[[Ohm|Ω]]) and one having resistance ''R''<sub>2</sub> (e.g., 40&nbsp;Ω), then the effect is the same as if one had used two resistors with the same resistance, both equal to the harmonic mean of ''R''<sub>1</sub> and ''R''<sub>2</sub> (48&nbsp;Ω): the equivalent resistance, in either case, is 24&nbsp;Ω (one-half of the harmonic mean). This same principle applies to [[capacitor]]s in series or to [[inductor]]s in parallel.

Average resistance for both resistors in parallel
{{=}} <big>{{sfrac|Total voltage|Sum of current for each resistor}}</big>
{{=}} {{sfrac|2''V''|''I''<sub>1</sub> + ''I''<sub>2</sub>}} = {{sfrac|2''V''|<big>{{sfrac|''V''|''R''<sub>1</sub>}} + {{sfrac|''V''|''R''<sub>2</sub>}}</big>}} = {{sfrac|2''R''<sub>1</sub>''R''<sub>2</sub>|''R''<sub>1</sub> + ''R''<sub>2</sub>}}

However, if one connects the resistors in series, then the average resistance is the arithmetic mean of ''R''<sub>1</sub> and ''R''<sub>2</sub> (50&nbsp;Ω), with total resistance equal to twice this, the sum of ''R''<sub>1</sub> and ''R''<sub>2</sub> (100&nbsp;Ω). This principle applies to [[capacitor]]s in parallel or to [[inductor]]s in series.

Average resistance for both resistors in series
{{=}} <big>{{sfrac|Total voltage|Sum of current for each resistor}}</big>
{{=}} {{sfrac|''V''<sub>1</sub> + ''V''<sub>2</sub>|2''I''}} = {{sfrac|''R''<sub>1</sub>''I'' + ''R''<sub>2</sub>''I''|2''I''}}
{{=}} {{sfrac|''R''<sub>1</sub> + ''R''<sub>2</sub>|2}}

As with the previous example, the same principle applies when more than two resistors, capacitors or inductors are connected, provided that all are in parallel or all are in series.{{cn|date=May 2025}}

The "conductivity effective mass" of a semiconductor is also defined as the harmonic mean of the effective masses along the three crystallographic directions.<ref>{{cite web|url=http://ecee.colorado.edu/~bart/book/effmass.htm|title=Effective mass in semiconductors|website=ecee.colorado.edu|access-date=8 May 2018|url-status=dead|archive-url=https://web.archive.org/web/20171020040802/http://ecee.colorado.edu/~bart/book/effmass.htm|archive-date=20 October 2017}}</ref>

====Optics====
As for other [[optic equation]]s, the [[focal length#Thin lens approximation|thin lens equation]] {{sfrac|''f''}} = {{sfrac|''u''}} + {{sfrac|''v''}} can be rewritten such that the focal length ''f'' is one-half of the harmonic mean of the distances of the subject ''u'' and object ''v'' from the lens.<ref name="Hecht1">{{cite book |first=Eugene |last=Hecht |year=2002 |title=Optics |edition=4th |publisher=[[Addison Wesley]] |isbn=978-0805385663 |page=168}}</ref>

Two thin lenses of focal length ''f''<sub>1</sub> and ''f''<sub>2</sub> in series is equivalent to two thin lenses of focal length ''f''<sub>hm</sub>, their harmonic mean, in series. Expressed as [[optical power]], two thin lenses of optical powers ''P''<sub>1</sub> and ''P''<sub>2</sub> in series is equivalent to two thin lenses of optical power ''P''<sub>am</sub>, their arithmetic mean, in series.