Do the values add? Use arithmetic mean
Scores, counts, distances, and temperatures combine by addition. The arithmetic mean preserves the total: AM × n = sum of all values.
Pythagorean means & dispersion
Statistics Calculators: Arithmetic, Weighted, Geometric, and Harmonic Mean
The three Pythagorean means — arithmetic, geometric, and harmonic — each answer a different averaging question. Arithmetic mean sums values; geometric mean multiplies them; harmonic mean uses reciprocals for rates. This hub links to each calculator, explains which one fits your data, and shows Excel equivalents for each formula.
Calculators in the Statistics section
Each tool below applies a distinct formula. Read the eyebrow label to confirm the formula matches your data before opening the calculator.
Additive average Arithmetic Mean Calculator — Formula, Examples, and When to Use It
Arithmetic mean (AM) is the sum of all values divided by the count: AM = (x₁ + x₂ + ⋯ + xₙ) / n. It is the correct average when every value contributes equally to a total — such as test scores, temperatures, and counts. This calculator computes AM alongside geometric and harmonic mean so you can confirm which average best fits your data.
Unequal-importance average Weighted Mean Calculator - Values, Weights, Formula, and ExamplesWeighted mean is the correct average when each value does not count equally. Formula: weighted mean = sum(value x weight) / sum(weight). Use it for course grades with different credit weights, portfolio returns with different capital allocations, survey results with sample weights, and prices with different quantities. This calculator returns weighted mean, arithmetic mean, total weight, and weighted sum so you can see how much the weights changed the result.
Rate & reciprocal average Harmonic Mean Calculator — Formula, Speed Examples, and Rate AveragingHarmonic mean (HM) is n divided by the sum of reciprocals: HM = n / (1/x₁ + 1/x₂ + ⋯ + 1/xₙ). It is the correct average when each value is a rate applied over an equal distance, budget, or unit — such as speeds over equal distances or prices per equal spending. Classic example: traveling 100 km at 30 mph and 100 km at 60 mph — the true average speed is HM = 40 mph, not the arithmetic mean of 45 mph.
Spread & dispersion Variance and Standard Deviation Calculator — Population vs Sample FormulaVariance measures the average squared distance of each value from the mean. Standard deviation is its square root — bringing the result back to the original unit. Population variance divides by n; sample variance divides by n − 1 (Bessel's correction). For dataset {4, 7, 13, 16, 20}: mean = 12, population variance = 34.8, population standard deviation = 5.90. This calculator returns both population and sample versions side by side.
Three decision rules for choosing the right average
Each mean is correct for a specific data structure. Using the wrong one produces a number that looks reasonable but answers the wrong question.
Do the values multiply? Use geometric mean
Growth factors, return indexes, and ratios combine by multiplication. Geometric mean gives the single factor that, applied n times, reproduces the same product.
Are values rates over equal units? Use harmonic mean
Speeds over equal distances, prices per equal budget, and output rates per equal workload require harmonic mean. It weights slower rates more because they consume more time or resource.
Match the formula to the situation
Each row below shows a real situation, the correct metric, and why a different metric would give a wrong answer.
| Situation | Input | Best next move | Why |
|---|---|---|---|
| Average exam score across a class | 72, 80, 88, 96 | Use arithmetic mean → 84 | Scores are additive; AM preserves the class total and each student's equal weight. |
| Average annual investment return over 4 years | Return factors: 1.08, 0.92, 1.15, 1.06 | Use geometric mean → 1.0494 (4.94%/yr) | Returns compound; AM would give 5.25%/yr but overstates the actual ending value. |
| Average speed on a round trip | 30 mph going, 60 mph returning, same distance | Use harmonic mean → 40 mph | Equal distance at unequal speeds; AM gives 45 mph but would predict the wrong travel time. |
Step-by-step: how to use this calculator correctly
Identify whether your values add, multiply, or are rates: this determines the mean type.
Use arithmetic mean (AM = sum / n) for additive values such as test scores, counts, and temperatures.
Use weighted mean when each value has a different importance, sample size, quantity, credit, or capital allocation.
Use geometric mean (GM = nth root of product) for positive multiplicative data: growth factors, return indexes, and ratios.
Use harmonic mean (HM = n / sum of reciprocals) when averaging speeds, prices per unit, or rates over equal distances.
Add variance or standard deviation to quantify how spread out the values are around the chosen mean.
Same four values, three different averages
Given
Dataset: 4, 9, 16, 25
Work
AM = (4+9+16+25)/4 = 13.50 | GM = (4×9×16×25)^(1/4) = 10.95 | HM = 4/(1/4+1/9+1/16+1/25) = 8.63
Result
The three means produce 13.50, 10.95, and 8.63 from the same four numbers.
Takeaway
The gap between AM and GM grows when values are uneven — a strong signal to check whether the data is truly additive.
Which average fits your data: a decision guide
The same dataset produces different values for arithmetic, geometric, and harmonic mean. Choosing the wrong one gives a numerically plausible but meaningless result.
| Method | Best for | Watch for | Example |
|---|---|---|---|
| Arithmetic mean | Additive values: test scores, temperatures, counts where each observation contributes equally. | One outlier shifts the result sharply. For 4, 9, 16, 100: AM = 32.25, far above most values. | Class scores 72, 80, 88, 96 → AM = 84. |
| Geometric mean | Multiplicative data: growth factors, index ratios, and return factors that compound across periods. | All values must be positive (or percentage returns above −100%). Zero collapses the product. | Annual return factors 1.06, 0.92, 1.15 → GM = 1.038, meaning 3.8% per year. |
| Harmonic mean | Rate data where the denominator is fixed: speeds over equal distances, prices per equal budget. | Wrong for equal-time rates. For 30 mph and 60 mph over equal distance: HM = 40, AM = 45. | 30 mph out, 60 mph back over equal distance → average speed = 40 mph. |
HM ≤ GM ≤ AM: the same four numbers, three averages
For dataset 4, 9, 16, 25: harmonic mean = 8.63, geometric mean = 10.95, arithmetic mean = 13.50. The gap signals whether data is additive, multiplicative, or rate-based.
Correct calculation flow for any mean
The same four steps apply regardless of which mean you use. The formula in step 3 is the only thing that changes.
Determine whether values add, multiply, or are rates over equal units.
Check for zeros, negatives, mixed units, or missing values that would break the formula.
Apply AM = sum/n, GM = (product)^(1/n), or HM = n / Σ(1/xᵢ).
State what the result means in plain terms, including its unit and what it is compared against.
Key facts before you calculate
Choose the mean that matches how the values combine
Arithmetic mean is correct when values add: 72 + 80 + 88 + 96 = 336, average = 84. Geometric mean is correct when values multiply: growth factors 1.06 × 1.08 × 0.95 compound to a single annual rate. Harmonic mean is correct when values are rates: 30 mph and 60 mph over equal distances give a true average of 40 mph, not 45 mph.
Excel formulas for each mean
Use =AVERAGE() for arithmetic mean, =GEOMEAN() for geometric mean, =HARMEAN() for harmonic mean. For spread, use =STDEV.P() when the dataset is the full population, or =STDEV.S() when it is a sample from a larger group.
Common mistakes that produce wrong averages
Most errors come from applying an additive formula to multiplicative data, or vice versa. The result looks plausible but is numerically incorrect.
Averaging investment returns with arithmetic mean
Returns of 50% and −50% give AM = 0%, implying no change — but the actual result is a 25% loss. Use geometric mean: GM = √(1.5 × 0.5) − 1 = −13.4%.
Averaging round-trip speed with arithmetic mean
30 mph and 60 mph gives AM = 45 mph, but the actual average speed is 40 mph (harmonic mean), because more time is spent at the slower pace.
Including a zero in geometric or harmonic mean
A zero makes GM = 0 (product collapses) and HM undefined (division by zero). Treat zero returns as a factor of 1.00, not as the number 0.
Frequently asked questions
What is the fastest way to choose between arithmetic and geometric mean?
Ask: do the values add or multiply? Test scores add — use AM. Annual return factors compound — use GM. If AM is noticeably higher than GM for your data, the values are likely multiplicative or skewed.
Why does the same dataset give three different means?
Each mean answers a different question. AM preserves the sum. GM preserves the product. HM preserves the rate. For 4, 9, 16, 25: AM = 13.50, GM = 10.95, HM = 8.63. None is wrong — each is correct for a different situation.
Why is HM ≤ GM ≤ AM for positive values?
This is the AM-GM-HM inequality, proven from the Cauchy-Schwarz inequality. For dataset 4, 9, 16, 25: HM = 8.63, GM = 10.95, AM = 13.50. The three are equal only when all values are identical.
Are the calculators on this site usable offline?
Yes. Once the page has loaded, all calculations run in your browser using JavaScript. No network connection is required to compute results.