Descriptive Statistics
Arithmetic Mean Calculator
The arithmetic mean is the headline, and when every value is positive the geometric and harmonic means appear only as comparison figures.
Worksheet
Press Calculate to see the result.
The arithmetic mean is the headline, and when every value is positive the geometric and harmonic means appear only as comparison figures.
What this calculator answers
The numbers 6, 10, 15, and 20 can be added, multiplied, or inverted, and each operation answers a different question. The arithmetic mean splits their sum into an equal share, which is the headline. Because every value is positive, the geometric mean of the logs and the harmonic mean of the reciprocals can be computed too. They are printed as comparison figures, not as rival headlines. If a zero or a negative value appears, those two comparisons are withheld and the arithmetic mean still stands, because addition does not need a logarithm.
The headline is always the arithmetic mean. Geometric and harmonic means are extra figures, shown only when every typed value is strictly positive. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Arithmetic mean, with positive-only comparisons uses this relationship.
arithmetic = sum / n. If every x > 0, also show exp(mean of ln x) and n / sum of 1/x.
- sum. The values added once each.
- n. How many values were typed.
- geometric. Comparison only, from the logs, when every value is positive.
- harmonic. Comparison only, from the reciprocals, when every value is positive.
The arithmetic mean is the additive equal share. The geometric and harmonic means answer multiplicative and reciprocal questions. Showing them beside a positive list makes the gap visible without changing which number is the result. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
The values are 6, 10, 15, 20. Their sum is 51 and the count is 4. The arithmetic mean is 51 / 4 = 12.75.
That arithmetic mean is the headline. Every value is positive, so the geometric mean is 11.5829 and the harmonic mean is 10.4348. Those two figures are comparisons. They do not replace the arithmetic mean of 12.75.
Nothing was trimmed and nothing was weighted. The unit of the arithmetic mean is the unit of the typed values.
| Arithmetic mean | 12.75 |
|---|---|
| Sum | 51 |
| Count | 4 |
| Geometric mean (comparison) | 11.5829 |
| Harmonic mean (comparison) | 10.4348 |
A second situation
20 is replaced by 0, so the geometric and harmonic means are withheld
The values are 6, 10, 15, 0. Their sum is 31 and the count is 4. The arithmetic mean is 31 / 4 = 7.75.
That arithmetic mean is the headline. A zero or a negative value is present, so the geometric and harmonic means are withheld. The arithmetic mean does not need a logarithm, so 7.75 is still defined.
Nothing was trimmed and nothing was weighted. The unit of the arithmetic mean is the unit of the typed values.
| Arithmetic mean | 7.75 |
|---|---|
| Sum | 31 |
| Count | 4 |
Replacing 20 with 0 drops the arithmetic mean from 12.75 to 7.75. The geometric and harmonic comparisons disappear, because a zero has no real logarithm and cannot be a reciprocal. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
20 was replaced by 0. The headline result went from 12.75 to 7.75.
The nudged inputs
The values are 6, 10, 15, 0. Their sum is 31 and the count is 4. The arithmetic mean is 31 / 4 = 7.75.
That arithmetic mean is the headline. A zero or a negative value is present, so the geometric and harmonic means are withheld. The arithmetic mean does not need a logarithm, so 7.75 is still defined.
Nothing was trimmed and nothing was weighted. The unit of the arithmetic mean is the unit of the typed values.
| Arithmetic mean | 7.75 |
|---|---|
| Sum | 31 |
| Count | 4 |
How to read the result
Lead with 12.75, the arithmetic mean. The geometric mean 11.5829 and the harmonic mean 10.4348 are smaller here, which is the usual order for a varied positive list, and they are not substitutes for 12.75. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- Adding the values is meaningful.
- Each value should pull once.
- Geometric and harmonic figures are comparisons, not the answer, and only when every value is positive.
- A missing value stays blank rather than becoming zero.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Reporting the geometric mean as the arithmetic headline.
- Forcing a geometric mean through a zero.
- Averaging percentages that came from groups of different sizes.
- Using this page's comparisons as a trimmed mean.
Where a different calculator is the honest choice
Use the geometric mean calculator when the product is the real question. Use the harmonic mean calculator for equal-distance rates. Use the mean calculator when you want the arithmetic mean without the other two figures. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Geometric Mean Calculator, Harmonic Mean Calculator, Mean Calculator.
Check yourself
Values are 6, 10, 15, and 20. What is the arithmetic mean, and what are the two comparison means?
The arithmetic mean is 12.75. The geometric mean is 11.5829 and the harmonic mean is 10.4348.
Using the result in a sentence
A first look at a positive batch, before you decide whether addition, multiplication, or reciprocals match the story, is what these three numbers are for. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
The comparison means are withheld as soon as one value is not positive, even if the rest of the list is a clean growth series. Split that value out before you ask for a geometric mean. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Arithmetic Mean Calculator commit to?
This page uses Arithmetic mean, with positive-only comparisons: arithmetic = sum / n. If every x > 0, also show exp(mean of ln x) and n / sum of 1/x.. The arithmetic mean is the additive equal share. The geometric and harmonic means answer multiplicative and reciprocal questions. Showing them beside a positive list makes the gap visible without changing which number is the result. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Numbers, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the arithmetic mean is 12.75. Lead with 12.75, the arithmetic mean. The geometric mean 11.5829 and the harmonic mean 10.4348 are smaller here, which is the usual order for a varied positive list, and they are not substitutes for 12.75. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Reporting the geometric mean as the arithmetic headline. The comparison means are withheld as soon as one value is not positive, even if the rest of the list is a clean growth series. Split that value out before you ask for a geometric mean. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the geometric mean calculator when the product is the real question. Use the harmonic mean calculator for equal-distance rates. Use the mean calculator when you want the arithmetic mean without the other two figures. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The arithmetic mean is 12.75. The geometric mean is 11.5829 and the harmonic mean is 10.4348. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.