Class 11 Physics Units and Measurement – Error Analysis Part 1

Error Analysis Part 1 – Measurement and Errors

[Watch Video for Detailed Explanation with Numerical Problem]

Every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error in measurement.

Accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity.

Precision tells us to what resolution or limit the quantity is measured or it is how close the measured values are.

Error in measurement can be divided mainly as 1) systematic errors and 2) random errors.

1) Systematic Errors

Systematic errors are those errors whose cause are known. They tend to be in one direction, can be minimized. Few causes of systematic errors are:

  • Error in measurement due to error in instruments
  • Imperfection in experimental techniques or procedure
  • Error due to individual bias or carelessness in taking observations without observing proper precautions

2) Random Errors

Random error are those errors, which occur irregularly. Causes of random errors are unknown. Cannot be avoided completely. Can be minimized by repeating the measurement a large number of times and taking the arithmetic mean of all the observations.

Least Count Error

The smallest value that can be measured by the measuring instrument is called its least count. All the readings or measured values are good only up to this value.

Absolute Error

The magnitude of the difference between the true value of the quantity and the individual measurement value is called the absolute error of the measurement. Usually the mean value of the measurement  is taken as the true value.

| \Delta x |=| x - x_{mean} |

Mean Absolute Error

The arithmetic mean of all the absolute errors is taken as the final or mean absolute error of the value of the physical quantity.

\Delta x_{mean}= \frac {| \Delta x_1 | + | \Delta x_2 | + | \Delta x_3 | + ... + | \Delta x_n |}{n} = \sum\limits_{i=1}^n \frac {\Delta x_n}{n}

Relative Error

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

Relative Error = \frac{\Delta x_{mean}}{x_{mean}}

Percentage Error

When relative error is expressed in percentage it is called percentage error.

Percentage Error= \frac{\Delta x_{mean}}{x_{mean}} \times 100\%

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