Quadratic Mean CalculatorFree online math calculator — instant results, no sign-up
Calculate the root mean square of a list of values.
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About the Quadratic Mean Calculator
The Quadratic Mean Calculator finds the root mean square (RMS) of a list of values by squaring each value, averaging the squares, and taking the square root. Also called the quadratic mean, this measure describes magnitude without allowing positive and negative values to cancel. For the default values 2, 4, 6, 8, 10, the root mean square is 6.6332495807.
- Values
- Enter the numerical observations whose root mean square you want to calculate; all values must share the same units, or all be dimensionless.
Formula and method
R = sqrt((Σ_{i=1}^{n} x_i²) / n); for the default values, R = sqrt((2² + 4² + 6² + 8² + 10²) / 5) = sqrt(44) ≈ 6.6332495807.- RThe root mean square, or quadratic mean, expressed in the same units as the input values.
- x_iThe value of the observation at position i in the list.
- iThe observation index, running from 1 to n.
- nThe total number of observations in the list.
- ΣSummation over all observations.
- sqrtThe nonnegative square root.
Each observation receives equal weight. For 2, 4, 6, 8, 10, the squared values are 4, 16, 36, 64, and 100. Their sum is 220; dividing by the five observations gives 44. Taking the square root gives an RMS of approximately 6.6332495807.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Values2, 4, 6, 8, 10
Results
- Root mean square6.6332495807
How to use the Quadratic Mean Calculator
- 1Enter your list in Values, or keep the default list: 2, 4, 6, 8, 10.
- 2Check that every entry is numerical and that all entries use the same units.
- 3Run the calculation to square the values, average their squares, and take the square root.
- 4Read the Root mean square output; the default list returns 6.6332495807.
- 5Interpret the result in the same units as your original values.
Reading your result
The Root mean square row reports the nonnegative quadratic mean of the entered list. For 2, 4, 6, 8, 10, it displays 6.6332495807, rounded to ten decimal places. This exceeds the arithmetic mean of 6 because squaring gives larger magnitudes more influence. If the inputs have units, the RMS retains those units.
Limitations
The calculation requires at least one finite numerical value and assumes equal weighting. Values measured in different units must be converted to a common unit before entry. Squaring removes sign information and makes the result sensitive to large magnitudes and outliers. RMS measures magnitude relative to zero, not spread around the mean, so it should not be interpreted as standard deviation. Extremely large or small inputs may be affected by numerical precision limits, and displayed decimals are rounded.
Common questions
Are quadratic mean and root mean square the same thing?+
Yes. Both names refer to the square root of the arithmetic mean of the squared values, which is the quantity this calculator returns.
Why is the RMS of 2, 4, 6, 8, 10 greater than their arithmetic mean?+
Their arithmetic mean is 6, while their RMS is 6.6332495807. Squaring emphasizes larger magnitudes. For nonnegative values, RMS is at least as large as the arithmetic mean, with equality only when every value is identical.
Can I include negative values or zeros in the RMS calculation?+
Yes. Negative values contribute their positive squares, so changing a value's sign does not change its RMS contribution. Zeros contribute nothing to the sum of squares but still count in the number of observations.
Is this calculator's RMS output the same as standard deviation?+
No. RMS squares the original values, whereas standard deviation uses deviations from the mean. For the same equally weighted list, RMS squared equals the population variance plus the square of the arithmetic mean.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.