Revision: 11th Std >> Error Analysis MAH-MHT CET (PCM/PCB) Error Analysis

The deviation of a measured value from the true value of a quantity arising due to human error, instrument limitations, or environmental conditions is called error.

Mathematically: Error = Measured value − True value

The error arising due to improper design or calibration, least count of the instrument, or zero error of the instrument is called instrumental error.

1. For a given set of measurements of a quantity, the magnitude of the difference between mean value (Most probable value) and each individual value is called absolute error (Δa) in the measurement of that quantity.
2. absolute error = |mean value - measured value|
   Δa₁ = |a_mean - a₁|
   Similarly,
   Δa₂ = |a_mean - a₂|,
             `\vdots           \vdots             \vdots`
   Δa_n = |a_mean - a_n|

When relative error is represented as percentage it is called the percentage error.

Percentage error = `(triangle"a"_"mean")/("a"_"mean") xx 100`

When a physical quantity is measured incorrectly, it can result in an error.

Systematic errors are consistent deviations from the true value caused by flaws in the measurement system.

OR

The type of error that consistently occurs in the same direction (either positive or negative), arising from imperfect design or calibration of measuring instruments, imperfection in experimental technique, or carelessness of an individual is called systematic error.

The ratio of the mean absolute error in the measurement of a physical quantity to its arithmetic mean value is called relative error.

Relative error = `(triangle "a"_"mean")/"a"_"mean"`

For a given set of measurements of the same quantity, the arithmetic mean of all the absolute errors is called mean absolute error in the measurement of that physical quantity.

`triangle "a"_"mean" = (triangle"a"_1 + triangle"a"_2 + ......+ triangle"a"_"n")/"n" = 1/"n"` \[\sum_{i=1}^n\triangle a_i\]

Random errors are unpredictable fluctuations in measurements that vary in both magnitude and direction.

OR

The error that occurs irregularly with respect to sign and size, being unpredictable and varying in magnitude and direction — which can be minimised by taking a large number of observations — is called random error.

The arithmetic mean of the magnitudes of absolute errors in all the measurements of a quantity is called the mean absolute error.

The ratio of the mean absolute error to the mean value of the quantity measured is called relative error or fractional error.

When the relative/fractional error is expressed in percentage, it is called percentage error.

The magnitude of the difference between the true value and the measured value of a quantity is called absolute error.

The measured value of a physical quantity denoting the number of digits in which we have confidence — where a larger number indicates greater accuracy of measurement — is called significant figures.

How far each reading is from the mean:

\[\Delta a_i=
\begin{vmatrix}
a-a_i
\end{vmatrix}\]

Average error over all readings:

\[\Delta a_{\mathrm{mean}}=\frac{\sum_{i=1}^n\Delta a_i}{n}\]

The relative error as a percent: 

Percemtage Error: \[\frac{\Delta a_{\mathrm{mean}}}{a}\times100\%\]

How big the error is, compared to the mean value (no units):

Relative Error: \[\frac {Δa_{mean}}{a}\]