## Physical World and Measurements

# Errors in Measurements and Accuracy and precision of measuring instruments

- Uncertainty in measurement of a physical quantity is called the error in measurement. The difference between the measured value and true value as per standard method without mistakes is called error.
- Accuracy refers to how closely a measured value agrees with the true values. Precision refers to what limit or resolution the given physical quantity can be measured.
- The errors arise due to external conditions like charges in environment changes in temperature pressure, humidity etc.
- In the measurement of a physical quantity the arithmetic mean of all readings which is found to be very close to the most accurate reading is to be taken as true value of the quantities.
- Absolute errors, mean absolute error, relative error, percentage error.

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1. The absolute error in measured values is given by

Δa_{n} = a_{m} − a_{n}

2. Mean Absolute Error: The arithmetic mean of the magnitude of absolute errors in all the measurements is called mean absolute error.

\tt \overline{\Delta a}=\frac{\mid\Delta a_{1}\mid+\mid\Delta a_{2}\mid+...+\mid\Delta a_{n}\mid}{n}

3. Relative error = \tt\frac{Mean\ absolute\ error}{True \ value}=\frac{\overline{\Delta a}}{a_{m}}

4. Percentage error = \tt\ \frac{\overline{\Delta a}}{a_{m}}\times100\%

5. Error in Addition or Subtraction: Maximum absolute error in their addition or subtraction

Δ*x* = ± (Δ*a* + Δ*b*)

6. Error in Multiplication or Division: Maximum relative error \tt \frac{\Delta x}{x}=\pm\left(\frac{\Delta a}{a}+\frac{\Delta b}{b} \right)

7. Error in Raised to a Power: Maximum error \tt \frac{\Delta Z}{Z}=p\left(\frac{\Delta A}{A}\right)+q\left(\frac{\Delta B}{B}\right)+r\left(\frac{\Delta C}{C}\right)