root mean square

(noun)

The square root of the arithmetic mean of the squares.

Related Terms

Examples of root mean square in the following topics:

  • Root-Mean-Square Speed

    • The root-mean-square speed measures the average speed of particles in a gas, defined as $v_{rms}=\sqrt{\frac{3RT}{M}}$ .
    • The root-mean-square speed is the measure of the speed of particles in a gas, defined as the square root of the average velocity-squared of the molecules in a gas.
    • The root-mean-square speed takes into account both molecular weight and temperature, two factors that directly affect the kinetic energy of a material.
    • What is the root-mean-square speed for a sample of oxygen gas at 298 K?
    • Recall the mathematical formulation of the root-mean-square velocity for a gas.

  • The Root-Mean-Square

    • The root-mean-square, also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity, or set of numbers.
    • Its name comes from its definition as the square root of the mean of the squares of the values.
    • The root-mean-square is always greater than or equal to the average of the unsigned values.
    • Physical scientists often use the term "root-mean-square" as a synonym for standard deviation when referring to the square root of the mean squared deviation of a signal from a given baseline or fit.
    • $G$ is the geometric mean, $H$ is the harmonic mean, $Q$ is the quadratic mean (also known as root-mean-square).

  • Root Mean Square Values

    • The root mean square (RMS) voltage or current is the time-averaged voltage or current in an AC system.
    • Given the current or voltage as a function of time, we can take the root mean square over time to report the average quantities.
    • The root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity.
    • It is especially useful when the function alternates between positive and negative values, e.g., sinusoids.The RMS value of a set of values (or a continuous-time function such as a sinusoid) is the square root of the arithmetic mean of the squares of the original values (or the square of the function).
    • Relate the root mean square voltage and current in an alternating circut with the peak voltage and current and the average power

  • Computing R.M.S. Error

    • Root-mean-square (RMS) error, also known as RMS deviation, is a frequently used measure of the differences between values predicted by a model or an estimator and the values actually observed.
    • Root-mean-square error serves to aggregate the magnitudes of the errors in predictions for various times into a single measure of predictive power.
    • RMS error is the square root of mean squared error (MSE), which is a risk function corresponding to the expected value of the squared error loss or quadratic loss.
    • Like the variance, MSE has the same units of measurement as the square of the quantity being estimated.
    • RMS error is simply the square root of the resulting MSE quantity.

  • The Discriminant

    • The discriminant $\Delta =b^2-4ac$ is the portion of the quadratic formula under the square root.
    • If ${\Delta}$ is positive, the square root in the quadratic formula is positive, and the solutions do not involve imaginary numbers.
    • If ${\Delta}$ is equal to zero, the square root in the quadratic formula is zero:
    • If ${\Delta}$ is less than zero, the value under the square root in the quadratic formula is negative:
    • This means the square root itself is an imaginary number, so the roots of the quadratic function are distinct and not real.

  • Standard Deviation: Definition and Calculation

    • It is therefore more useful to have a quantity that is the square root of the variance.
    • Next, compute the average of these values, and take the square root:
    • This quantity is the population standard deviation, and is equal to the square root of the variance.
    • Using the uncorrected estimator (using $N$) yields lower mean squared error.
    • We can obtain this by determining the standard deviation of the sampled mean, which is the standard deviation divided by the square root of the total amount of numbers in a data set:

  • Imaginary Numbers

    • There is no such value such that when squared it results in a negative value; we therefore classify roots of negative numbers as "imaginary."
    • What does it mean, then, if the value under the radical is negative, such as in $\displaystyle \sqrt{-1}$?
    • This means that there is no real value of $x$ that would make $x^2 =-1$ a true statement.
    • Specifically, the imaginary number, $i$, is defined as the square root of -1: thus, $i=\sqrt{-1}$.
    • We can write the square root of any negative number in terms of $i$.

  • Radical Functions

    • An expression with roots is called a radical function, there are many kinds of roots, square root and cube root being the most common.
    • If fourth root of 2401 is 7, and the square root of 2401 is 49, then what is the third root of 2401?
    • If the square root of a number is taken, the result is a number which when squared gives the first number.
    • Roots do not have to be square.
    • However, using a calculator can approximate the square root of a non-square number:$\sqrt {3} = 1.73205080757$

  • Factoring a Difference of Squares

    • When a quadratic is a difference of squares, there is a helpful formula for factoring it.
    • Taking the square root of both sides of the equation gives the answer $x = \pm a$.
    • If you recognize the first term as the square of $x$ and the term after the minus sign as the square of $4$, you can then factor the expression as:
    • If we recognize the first term as the square of $4x^2$ and the term after the minus sign as the square of $3$, we can rewrite the equation as:
    • This means that either $4x^2-3=0$ or $4x^2+3=0$.

  • Introduction to Radicals

    • Roots are the inverse operation of exponentiation.
    • For example, the following is a radical expression that reverses the above solution, working backwards from 49 to its square root:
    • For now, it is important simplify to recognize the relationship between roots and exponents: if a root $r$ is defined as the $n \text{th}$ root of $x$, it is represented as
    • If the square root of a number $x$ is calculated, the result is a number that when squared (i.e., when raised to an exponent of 2) gives the original number $x$.
    • This is read as "the square root of 36" or "radical 36."