Simplify Your Workflow: Search MiniWebtool.
Add Extension
Home Page > Math > Statistics And Data Analysis > Harmonic Mean Calculator

Harmonic Mean Calculator

Calculate the harmonic mean of a data set with step-by-step formulas, comparison with arithmetic and geometric means, interactive visualization, and practical examples for rates, speeds, and financial analysis.

Harmonic Mean Calculator

Try example datasets:

Speed (40, 60) Powers of 2 Arithmetic Seq F1 Scores
Note: Harmonic mean requires all positive, non-zero values. It's ideal for averaging rates, speeds, and financial ratios. For mixed data, use our Number Extractor.

Embed Harmonic Mean Calculator Widget

About Harmonic Mean Calculator

Welcome to the Harmonic Mean Calculator, a comprehensive tool for calculating the harmonic mean with step-by-step solutions, interactive visualizations, and comparisons with arithmetic and geometric means. The harmonic mean is essential for averaging rates, ratios, and speeds, and is widely used in physics, finance, and data science.

What is the Harmonic Mean?

The harmonic mean is a type of average calculated as the reciprocal of the arithmetic mean of reciprocals. For a dataset of n positive numbers x₁, x₂, ..., xₙ, the harmonic mean H is defined as:

Harmonic Mean Formula
$$H = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$$

The harmonic mean gives greater weight to smaller values in the dataset, making it particularly useful when dealing with rates, ratios, and situations where reciprocals are meaningful.

The AM-GM-HM Inequality

A fundamental relationship in mathematics connects the three Pythagorean means:

For any set of positive numbers:
H ≤ G ≤ A
Harmonic Mean ≤ Geometric Mean ≤ Arithmetic Mean
Equality holds if and only if all values are identical.

When to Use Harmonic Mean

The harmonic mean is the appropriate average when:

🚗
Average Speed
When traveling equal distances at different speeds, the average speed is the harmonic mean of the individual speeds.
📈
Financial Ratios
P/E ratios, price-to-book ratios, and other financial metrics are properly averaged using harmonic mean.
🤖
F1 Score
The F1 score in machine learning is the harmonic mean of precision and recall.
⚡
Parallel Circuits
The equivalent resistance of parallel resistors uses harmonic mean principles.

Average Speed Example

If you drive 100 km at 40 km/h and return 100 km at 60 km/h, your average speed is:

Average Speed Calculation
$$H = \frac{2}{\frac{1}{40} + \frac{1}{60}} = \frac{2}{\frac{3+2}{120}} = \frac{2 \times 120}{5} = 48 \text{ km/h}$$

Note: This is less than the arithmetic mean of 50 km/h because you spend more time at the slower speed.

How to Use This Calculator

  1. Enter your data: Input positive numbers separated by commas, spaces, or line breaks. Try the example buttons for quick testing.
  2. Set precision: Choose decimal places (2-15) based on your accuracy requirements.
  3. Calculate: Click the button to see the harmonic mean with step-by-step calculations.
  4. Compare means: Review the comparison of harmonic, geometric, and arithmetic means.
  5. Visualize: Examine the interactive charts to understand your data distribution.

Comparing the Three Means

Mean Type Formula Best Used For
Harmonic Mean n / Σ(1/xᵢ) Rates, ratios, speeds (equal distances)
Geometric Mean (∏xᵢ)^(1/n) Growth rates, percentages, ratios
Arithmetic Mean Σxᵢ / n Additive quantities (heights, weights)

Practical Applications

Finance and Investment

In financial analysis, the harmonic mean is used to average price ratios. When calculating the average P/E ratio of a portfolio or index, the harmonic mean provides a more accurate representation because it accounts for the relative sizes of investments at different P/E levels.

Machine Learning - F1 Score

The F1 score, a crucial metric for evaluating classification models, is defined as:

F1 Score Formula
$$F_1 = 2 \times \frac{\text{Precision} \times \text{Recall}}{\text{Precision} + \text{Recall}} = \frac{2}{\frac{1}{\text{Precision}} + \frac{1}{\text{Recall}}}$$

Using harmonic mean ensures that both precision and recall must be reasonably high for a good F1 score.

Physics - Parallel Resistors

For n identical resistors R in parallel, the equivalent resistance is R/n. For different resistors, the formula uses harmonic relationships.

Limitations and Considerations

Frequently Asked Questions

What is the harmonic mean?

The harmonic mean is a type of average calculated as the reciprocal of the arithmetic mean of reciprocals. For a dataset of n positive numbers, the harmonic mean H = n / (1/x₁ + 1/x₂ + ... + 1/xₙ). It is particularly useful for averaging rates, ratios, and speeds, and always gives a value less than or equal to the geometric and arithmetic means.

When should I use harmonic mean instead of arithmetic mean?

Use harmonic mean when: (1) Averaging rates or ratios like speed, efficiency, or price-to-earnings ratios; (2) Equal amounts of time or resources are spent at different rates; (3) Calculating average speed for equal distances; (4) Finding the effective resistance of parallel resistors; (5) Working with F-scores in machine learning. Arithmetic mean is better for additive quantities like heights, weights, or scores.

What is the relationship between harmonic, geometric, and arithmetic means?

For any set of positive numbers, the three means satisfy the inequality: Harmonic Mean ≤ Geometric Mean ≤ Arithmetic Mean (H ≤ G ≤ A). Equality holds only when all values in the dataset are identical. This relationship is known as the AM-GM-HM inequality and is fundamental in mathematics and statistics.

Why can't harmonic mean be calculated with zero or negative numbers?

The harmonic mean requires calculating reciprocals (1/x) of each value. Division by zero is undefined, so zeros cannot be included. Negative numbers would make the sum of reciprocals potentially zero or negative, making the result undefined or meaningless. The harmonic mean is designed for positive ratio-scale data.

How do I calculate average speed using harmonic mean?

When traveling equal distances at different speeds, the average speed is the harmonic mean of the speeds. For example, if you drive 100 km at 40 km/h and return 100 km at 60 km/h, the average speed is H = 2 / (1/40 + 1/60) = 48 km/h, not the arithmetic mean of 50 km/h. This is because you spend more time traveling at the slower speed.

What is the F1 score and how does it relate to harmonic mean?

The F1 score in machine learning is the harmonic mean of precision and recall: F1 = 2 × (precision × recall) / (precision + recall). Using harmonic mean ensures that both metrics must be reasonably high for a good F1 score - having high precision but low recall (or vice versa) results in a low F1 score, making it a balanced measure of classifier performance.

Additional Resources

Reference this content, page, or tool as:

"Harmonic Mean Calculator" at https://MiniWebtool.com/harmonic-mean-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 29, 2026

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

Related MiniWebtools:

Arithmetic Mean CalculatorFeatured
Geometric Mean Calculator

Statistics And Data Analysis:

Top & Updated:

Random PickerRandom Name PickerRelative Standard Deviation CalculatorLine CounterSort NumbersFPS ConverterRemove SpacesMAC Address GeneratorFeet and Inches to Cm ConverterMAC Address LookupWord to Phone Number ConverterBitwise CalculatorBatting Average CalculatorFacebook User ID LookupRandom Quote GeneratorPercent Off CalculatorLog Base 10 CalculatorSum CalculatorNumber of Digits CalculatorSHA256 Hash GeneratorSaturn Return CalculatorERA CalculatorDecimal to BCD ConverterRoman Numerals ConverterRandom Truth or Dare GeneratorRandom Credit Card GeneratorAI ParaphraserSun, Moon & Rising Sign Calculator 🌞🌙✨Binary to Gray Code ConverterCm to Feet and Inches ConverterOctal CalculatorVideo to Image ExtractorAI Punctuation AdderMP3 LooperSquare Root (√) CalculatorFirst n Digits of PiPhone Number ExtractorBCD to Decimal ConverterOutlier CalculatorNumber to Word ConverterQuotient and Remainder CalculatorCompound Growth CalculatorText FormatterSalary Conversion CalculatorGray Code to Binary ConverterDay of Year CalendarRandom Birthday GeneratorMerge VideosRemove Audio from VideoTime Duration CalculatorStandard Error CalculatorScientific Notation to Decimal ConverterOn Base Percentage CalculatorConvolution CalculatorAverage Deviation CalculatorSlugging Percentage CalculatorAudio SplitterRandom Group GeneratorBinary to BCD ConverterCM to Inches ConverterRemove AccentPercent Growth Rate CalculatorCompare Two StringsLeap Years ListIP Address to Hex ConverterOne Rep Max (1RM) CalculatorImage SplitterRandom Meal GeneratorMAC Address AnalyzerModulo CalculatorAntilog CalculatorList of Prime NumbersCoin FlipperMedian CalculatorList RandomizerRemove Leading Trailing SpacesMaster Number CalculatorDay of the Year Calculator - What Day of the Year Is It Today?Random User-Agent GeneratorFirst n Digits of eLog Base 2 CalculatorPER CalculatorSample Standard Deviation CalculatorPercentage Increase CalculatorName Number CalculatorBonus CalculatorSHA512 Hash GeneratorEmail ExtractorMaze GeneratorRandom Emoji GeneratorAdd or Replace Audio in VideoExponential Decay CalculatorNatural Log CalculatorRandom Object GeneratorHypotenuse CalculatorRandom Time GeneratorSquare Numbers ListFraction CalculatorReverse VideoWAR CalculatorURL ExtractorAdd Prefix and Suffix to TextOPS CalculatorWHIP CalculatorLove Compatibility CalculatorVideo CompressorSort Lines AlphabeticallyHex to BCD ConverterBCD to Binary ConverterLottery Number GeneratorBCD to Hex ConverterBreak Line by CharactersAverage CalculatorPVIFA CalculatorActual Cash Value CalculatorNumber ExtractorAngel Number CalculatorRoot Mean Square CalculatorSum of Positive Integers CalculatorSHA3-256 Hash GeneratorAI Sentence Expander📅 Date CalculatorLbs to Kg ConverterHex to Decimal ConverterRandom String GeneratorAmortization CalculatorMarkup CalculatorPVIF CalculatorDecimal to Hex ConverterList of Fibonacci NumbersRandom Number PickerBlood Donation Time CalculatorField Goal Percentage CalculatorRandom Line PickerRatio to Percentage CalculatorAPY to APR CalculatorSort Text By LengthName RandomizerRandom Animal GeneratorFixed Deposit CalculatorPPM to Percent ConverterCube Numbers ListCaesar Cipher ToolMorse Code GeneratorPercent to PPM ConverterSRT to TXT Converterkg to lbs ConverterXML ValidatorArithmetic Mean CalculatorBoard Foot CalculatorSquareness Checker (3-4-5 Rule) 📐Miter Angle Calculator 📐Slope and Grade Calculator 📐Sine CalculatorCosine CalculatorHigh-Precision Tangent CalculatorArcsin CalculatorArccos (Inverse Cosine) CalculatorArctan CalculatorArctan2 CalculatorCosecant/Secant/Cotangent CalculatorHyperbolic Functions CalculatorRight Triangle CalculatorGeneral Triangle SolverLaw of Sines CalculatorLaw of Cosines CalculatorDMS to Decimal Degrees ConverterDecimal Degrees to DMS ConverterPolar to Cartesian ConverterHigh-Precision Cartesian to Polar Coordinates ConverterInteractive Unit Circle VisualizerTrigonometric Identities CalculatorTrigonometric Function GrapherAlgebraic Expression SimplifierRadical SimplifierRational Expression CalculatorPolynomial Factoring CalculatorExpand Polynomials CalculatorPolynomial Long Division CalculatorSynthetic Division CalculatorLinear Equation SolverAbsolute Value Equation SolverRadical Equation SolverSystem of Linear Equations SolverInequality SolverAbsolute Value Inequality SolverSystem of Inequalities GrapherFunction GrapherDomain and Range CalculatorInverse Function CalculatorFunction Composition CalculatorVertex and Axis of Symmetry CalculatorX and Y Intercept CalculatorHTML Entity Encoder/DecoderURL Slug GeneratorJSON String Escape/UnescapeMarkdown Table GeneratorSmart Quotes RemoverDaily Time Savings CalculatorLorem Ipsum GeneratorReadability Score CalculatorKeyword Density CheckerBionic Reading ConverterText to Speech ReaderFancy Text GeneratorZalgo Text GeneratorUpside Down Text GeneratorInvisible Text GeneratorASCII Art GeneratorList Difference CalculatorText Column ExtractorText to SQL List ConverterInvisible Character RemoverStock Average CalculatorPosition Size CalculatorMargin Call CalculatorShort Selling Profit CalculatorRisk of Ruin CalculatorBlack-Scholes Option Pricing CalculatorOption Greeks CalculatorImplied Volatility CalculatorOptions Profit CalculatorImpermanent Loss CalculatorCrypto Arbitrage CalculatorSatoshi to USD ConverterCrypto Leverage CalculatorPivot Point CalculatorFibonacci Extension CalculatorStop Loss & Take Profit CalculatorKelly Criterion CalculatorMartingale Strategy CalculatorCompound Daily Interest CalculatorPip Value CalculatorGaussian Distribution GeneratorRandom Tournament Bracket GeneratorRandom Color Palette GeneratorRandom Country GeneratorRandom JSON GeneratorRandom Coordinate GeneratorRandom Date GeneratorRandom IP Address GeneratorRandom Playing Card GeneratorMagic 8-BallRock Paper Scissors GeneratorDice RollerSpin the WheelAquarium Volume & Stocking CalculatorAquarium Substrate CalculatorScale Model Conversion CalculatorPlant Spacing CalculatorDice Roll Probability CalculatorDepth of Field (DoF) CalculatorGolden Hour / Blue Hour CalculatorPrint Size & Resolution Calculator (DPI/PPI)Soap Making Lye Calculator (SAP)Candle Wax & Fragrance Oil CalculatorHomebrew ABV Calculator (Alcohol by Volume)Coffee Brew Ratio CalculatorBrine and Salinity CalculatorYarn CalculatorFabric CalculatorCross-Stitch Size CalculatorBaker's Percentage CalculatorDough Hydration CalculatorBaking Pan Size ConverterYeast Conversion CalculatorJSON to YAML ConverterJWT DecodercURL to JSON ConverterSQL FormatterCrontab Expression GeneratorFavicon GeneratorSVG OptimizerHtaccess Redirect GeneratorResistor Color Code CalculatorOhm's Law CalculatorVoltage Drop CalculatorPCB Trace Width CalculatorBattery Life CalculatorRandom PIN GeneratorRandom NanoID GeneratorRandom Port Number GeneratorRandom Fake Address GeneratorRandom User Persona GeneratorRandom IMEI GeneratorRandom Writing Prompt GeneratorRandom Haiku GeneratorRandom Superpower GeneratorRandom Domain Name GeneratorRandom Movie PickerRandom Cocktail Recipe GeneratorRandom Activity GeneratorRandom Excuse GeneratorRandom US State GeneratorLemonade Stand CalculatorPizza Value CalculatorVampire Apocalypse CalculatorChristmas Tree CalculatorParking Ratio CalculatorGooglebot Crawl Size Checker