Mean Calculator
The mean with its proof attached: the deviations table showing Σ(x−x̄) = 0 live, and the frequency-weighted mean as a first-class mode — because weighted data deserves weighted arithmetic.
Mean Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- 2 to 500 values; weights must match one-to-one
- Arithmetic mean only — geometric/harmonic means are different tools
In short: The mean of 2, 4, 4, 4, 5, 5, 7, 9 is 40/8 = 5 — and the page’s deviations card proves the mean is the balance point on your own numbers: subtract 5 from every value and the results are −3, −1, −1, −1, 0, 0, 2, 4, summing to exactly 0. Weighted data is the other half of the story: values 3 and 5 taken 2 and 3 times average to (3×2 + 5×3)/(2 + 3) = 4.2, not 4 — weights change the centre, and pretending they do not is how averaging averages goes wrong.
Formula
x̄ = Σx / n · x̄w = Σwᵢxᵢ / Σwᵢ
identity: Σ(xᵢ − x̄) = 0, always
the mean is the value that balances the deviations — the identity is not a coincidence, it is the definition holding.
Worked Example
- Sum and divide. the arithmetic mean — but the page does not stop at the division.
- Prove the balance. subtract the mean from every value and sum: exactly zero, printed. That identity is the mean’s defining property, checked live.
- Weight if needed. give a weight per value and the mean becomes Σwx/Σw — repeated values, credit hours and reliability weights all live here.
2, 4, 4, 4, 5, 5, 7, 9: Σx = 40, n = 8, mean 5. Deviations sum to 0. Weighted 3 and 5 with weights 2 and 3: 21/5 = 4.2. The plain average of 3 and 5 would have said 4 — the weights moved the centre.
Strengths & Limits Of This Model
Where this engine is strong
- The zero-sum identity is verified live on every run
- Weighted mean is a mode of the page, not an afterthought
Where it stops
- No geometric or harmonic mean
- No outlier diagnostics
Practical Use Cases
Gradebooks
scores weighted by credit or importance
Repeated measurements
weights = rep counts, honestly
Teaching the identity
the zero-sum card is the lesson
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Mean Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
What is the Σ(x − x̄) = 0 card actually proving?
That the mean is the balance point. Move the mean to the left side of Σx = n·x̄ and the deviations must cancel — what sits below the mean exactly offsets what sits above. It is not an extra feature; it is the defining property of the mean, verified on your own numbers every run.
When do I need weights?
Whenever values repeat, matter differently, or arrive in batches: a score earned twice, a course worth 4 credits, a poll weighting by reliability. The weighted mean Σwx/Σw counts each value as many times as it deserves. The unweighted mean of {3, 5} says 4; weighted 2-vs-3 it says 4.2 — weights are information, and dropping them is a silent lie.
How is this different from the Average Calculator?
The average page answers WHICH average — mean, median or mode — and when each one lies. This page owns the mean itself: the identity, the deviations, the weights. If you are arguing about outliers and skew, that is the average page’s question; if you want the centre computed and proven, it is this one.
Can weights be zero or negative?
Zero is legal and means “ignore this value” — it drops out of both sums. Negative weights are refused: a negative count of observations is not a thing, and admitting one would let the mean land outside every value it averages. All-zero weights are refused too — the denominator would be zero.
Does a big outlier drag the mean?
Yes, by exactly its deviation divided by n — and the deviations card shows that contribution in plain sight. That sensitivity is the mean’s defining trait and the median’s entire sales pitch; when one wild value should not move the centre, the median page is the right door.
Is the weighted mean always between the smallest and largest values?
Yes — it is a convex combination: every weight positive, everything inside the hull. The page will not state it as a check unless the weights are legal, which is precisely why negative weights are refused.
Why does the page cap the deviation table at twelve entries?
Readability. The identity Σ(x−x̄) = 0 is printed over ALL values regardless; the table shows the first dozen so the pattern is visible without scrolling forever. The math does not change with display.
Is the mean of several group means the overall mean?
Only when the groups are the same size. Ten values averaging 4 and two averaging 8 do not average to 6 — they average to (10×4 + 2×8)/12 ≈ 4.67, because the bigger group gets more votes. Weight the group means by their sizes, or pool the raw lists; the weighted card on this page is exactly that computation.