Calculation Methodology
Arithmetic mean and stable addition
The arithmetic mean is the sum of the observations divided by their count. The engine uses compensated summation to reduce rounding loss when adding numbers of different magnitudes. This improves ordinary floating-point addition but is not arbitrary-precision or symbolic mathematics.
Weighted and combined means
A weighted mean is Σ(value × weight) ÷ Σweights. Weights must be nonnegative and their total must be positive. They are normalized by the actual total; percentage weights need not total 100. Combined group averages use positive whole-number group counts as weights. Overlapping groups should not be combined as if independent.
Median, range, and endpoints
Median sorts a copy of the dataset. With an odd count it selects the middle value; with an even count it averages the middle two. Range is maximum minus minimum. The midpoint of a minimum and maximum is only the mean of those endpoints, not evidence of the full dataset mean.
New averages and one missing number
An updated mean is (old mean × old count + sum of new observations) ÷ (old count + new count). Old count must be a positive whole number no greater than 1,000,000. The missing-number solver requires exactly total count minus one known values; the answer is target mean × total count minus their sum.
Percentages and educational scores
Equal percentage averaging gives each row equal weight. Pooled percentage uses total earned points divided by total possible points, multiplied by 100. Each possible score must be positive and each earned score between zero and its maximum. Extra-credit points above the maximum are not supported. No letter-grade or GPA conversion is assumed.
Purchase costs, wages, and units
Purchase price is weighted by purchased quantity. Sales, fees, taxes, and accounting adjustments are excluded. Blended hourly wage weights each rate by hours worked. An equal average of rates is displayed separately where relevant. Salary, income, and weight unit selectors label inputs; they never silently convert values.
Circular mean
Angles are reduced modulo 360 degrees and converted to radians. The engine sums their sine and cosine components, then uses atan2 to recover the direction, normalized to [0°, 360°). A resultant magnitude divided by the observation count below 10⁻¹² is treated as undefined or ambiguous. Near-zero wrap results within 10⁻¹⁰ degrees normalize to zero.
Input format and limits
Paste at most 10,000 observations, or use up to 100 individual rows. Complete numeric tokens may contain a sign, dot decimal separator, and scientific notation. Commas, whitespace, and semicolons separate observations; commas never denote thousands. NaN, Infinity, malformed tokens, underflow-to-zero values, and input magnitudes above 1 trillion are rejected. Contextual fields can impose narrower bounds.
Rounding and long result tables
Values are rounded for display, with selectable zero to twelve decimal places. Very small nonzero results use scientific notation. Computation retains ordinary floating-point precision and CSV exports retain the calculated numbers. Long tables and charts show at most 100 evenly sampled rows; all observations remain in the calculation and export.
Verification
Shared numerical tests cover means, weights, group sizes, reverse averages, percentages, purchase quantities, wages, running totals, and opposing angles. Route checks inspect generated page metadata, links, keyword ownership, and sitemap coverage. These checks do not replace independent verification for consequential decisions.