Calculators
Weighted Root Mean Square Calculator
Calculate weighted RMS from signed values and non-negative weights privately in your browser.
Calculate the quadratic mean with relative weights.
Weighted root mean square is √(Σ(w × x²) / Σw). For values 3 and 4 with equal weights, RMS is √12.5, approximately 3.53553390593. With weights 1 and 3, it is √14.25, approximately 3.77491721764. Negative values contribute the same as their positive magnitudes. Zero values are accepted; zero-weight pairs are ignored. At least one weight must be positive. Equal weights give the ordinary RMS; multiplying all weights by the same positive constant leaves the result unchanged. Enter up to 10000 pairs, one per line, using commas, semicolons or spaces within each line. Scientific notation is supported; omit thousands separators and use consistent units. RMS retains the values’ units. Log-sum-exp arithmetic avoids direct squares and large weight totals. Results display up to 12 significant digits; extreme contributions may round away and unrepresentably small results require rescaling. All calculations run in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
How is weighted RMS calculated?
Square each value, multiply by its weight, divide the sum by the total weight, and take the square root. Values 3 and 4 with equal weights give approximately 3.53553390593.
Can values be negative or zero?
Yes. Signed finite values and zero are accepted. Squaring removes the sign, so the result is non-negative.
Do weights have to sum to one?
No. Use non-negative relative weights with at least one positive weight. Zero-weight rows are ignored. Scaling all weights by the same positive factor preserves RMS.
How does weighted RMS differ from standard deviation?
RMS measures magnitude relative to zero. Standard deviation measures spread around the mean. RMS is useful when larger magnitudes should contribute more strongly.
Is my data uploaded?
No. Calculation takes place entirely in your browser. Results use floating-point arithmetic and display up to 12 significant digits.