root mean square

Examples of root mean square in the following topics:

• 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).

• 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.

• 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:

• Variance and standard deviation

  • If we square these deviations and then take an average, the result is about equal to the sample variance, denoted by s2:
  • Notice that squaring the deviations does two things.
  • The standard deviation is defined as the square root of the variance:
  • The variance is roughly the average squared distance from the mean.
  • The standard deviation is the square root of the variance.

• Mean, Variance, and Standard Deviation of the Binomial Distribution

  • In this section, we'll examine the mean, variance, and standard deviation of the binomial distribution.
  • If you performed this experiment over and over again, what would the mean number of heads be?
  • Therefore, the mean number of heads would be 6.
  • The mean and standard deviation can therefore be computed as follows:
  • Naturally, the standard deviation ($s$) is the square root of the variance ($s^2$).

• Mean: The Average

  • The three most common averages are the Pythagorean means – the arithmetic mean, the geometric mean, and the harmonic mean.
  • This is because it minimizes the sum of squared deviations from the estimate.
  • The geometric mean is defined as the $n$th root (where $n$ is the count of numbers) of the product of the numbers.
  • For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product; that is $\sqrt{2\cdot8} = 4$.
  • As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2; that is $\sqrt[3]{4\cdot 1 \cdot \frac{1}{32}} = \frac{1}{2}$.

• Mean of All Sample Means (μ x)

  • The mean of the distribution of differences between sample means is equal to the difference between population means.
  • which says that the mean of the distribution of differences between sample means is equal to the difference between population means.
  • Recall that the standard error of a sampling distribution is the standard deviation of the sampling distribution, which is the square root of the above variance.
  • The mean height of Species 1 is 32, while the mean height of Species 2 is 22.
  • Standard error equals the square root of (60 / 10) + (70 / 14) = 3.317.

• Variability in random variable

  • We first computed deviations from the mean (x i − µ), squared those deviations, and took an average to get the variance.
  • This weighted sum of squared deviations equals the variance, and we calculate the standard deviation by taking the square root of the variance, just as we did in Section 1.6.4.
  • The standard deviation of X, labeled σ, is the square root of the variance.
  • The variance of X is σ2 = 3659.3, which means the standard deviation is σ$\sqrt{3659.3}$ = $60.49.
  • The result of part (d) is the square-root of the variance listed on in the total on the last line: σ = $\sqrt{Var(Y)}$= $69.28.

• Chi Square Distribution

  • Define the Chi Square distribution in terms of squared normal deviates
  • The Chi Square distribution is the distribution of the sum of squared standard normal deviates.
  • A Chi Square calculator can be used to find that the probability of a Chi Square (with 2 df) being six or higher is 0.050.
  • The mean of a Chi Square distribution is its degrees of freedom.
  • The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square.

• Variability in linear combinations of random variables

  • The standard deviation is computed as the square root of the variance:
  • The standard deviation of the linear combination may be found by taking the square root of the variance.
  • The negative coefficient for Y in the linear combination was eliminated when we squared the coefficients.
  • The variance of the linear combination is 689, and the standard deviation is the square root of 689: about $26.25.
  • The mean, standard deviation, and variance of the GOOG and XOM stocks.