Statistics - Root Mean Square


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Root Mean Square, RMS is defined as the square root of mean square where mean square is the arithmetic mean of the squares of numbers. RMS is also termed as the quadratic mean.

Formula

${ x_{rms} = \sqrt{ \frac{1}{n} ( {x_1}^2 + {x_2}^2 + ... + {x_n}^2 } }$

Where −

  • ${x_i}$ = items under observation.

  • ${n}$ = total number of items.

Example

Problem Statement:

Compute the RMS of following data.

5 6 7 8 9

Solution:

Step 1: Compute squares of each no.

${ {x_1}^2 + {x_2}^2 + ... + {x_n}^2 \\[7pt] = 6^2 + 7^2 + 8^2 + 9^2 \\[7pt] = 36 + 49 + 64 + 81 \\[7pt] = 230 }$

Step 2: Compute mean of squares of each no.

${ \frac{1}{n} ({x_1}^2 + {x_2}^2 + ... + {x_n}^2 ) \\[7pt] = \frac{1}{4} (230) \\[7pt] = \frac{230}{4} \\[7pt] = 57.5 }$

Step 3: Compute RMS by taking sqrt of means of squares.

${ x_{rms} = \sqrt{ \frac{1}{n} ( {x_1}^2 + {x_2}^2 + ... + {x_n}^2 } \\[7pt] = \sqrt {57.5} \\[7pt] = \frac{230}{4} \\[7pt] = 7.58 }$

As a result, RMS is ${7.58}$.

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