Units, Dimensions, and Errors in Measurements

Core Concepts of Errors in Measurements

• Mean Value (True Value):
  • If $a_1, a_2, \dots, a_n$ are the observed values of a physical quantity from $n$ measurements, the arithmetic mean of these values is taken as the true value $a$ (or $a_{\text{mean}}$).
  • Formula:

$a = \frac{a_1 + a_2 + \dots + a_n}{n} = \frac{1}{n} \sum_{i=1}^{n} a_i$

• Absolute Error:
  • The magnitude of the difference between the true value of the quantity and the individual observed value is called the absolute error of the measurement.
  • Definition formula:

$\text{Absolute error} = \text{true value} - \text{observed value}$

• For the $i^{\text{th}}$ measurement, the absolute error is given by:

$\Delta a_i = a - a_i$

• The absolute error magnitude is expressed as:

$|\Delta a_i| = |a_i - a|$

• Mean Absolute Error:
  • The arithmetic mean of the numerical values (magnitudes) of the absolute errors in all measurements is called the mean absolute error $\Delta a_m$ (or $\Delta a_{\text{mean}}$).
  • Formula:

$\Delta a_m = \frac{|\Delta a_1| + |\Delta a_2| + \dots + |\Delta a_n|}{n} = \frac{1}{n} \sum_{i=1}^{n} |\Delta a_i|$

• Relative Error (Fractional Error):
  • The ratio of the mean absolute error to the mean value (true value) of the measured quantity is called relative error or fractional error.
  • Formula:

$\text{Relative error} = \frac{\Delta a_m}{a}$

• Percentage Error:
  • When the relative error is expressed in percentage, it is called percentage error.
  • Formula:

$\text{Percentage error} = \frac{\Delta a_m}{a} \times 100$

Rules for Combination of Errors

• Let $\pm \Delta X$ and $\pm \Delta Y$ be the absolute errors in physical quantities $X$ and $Y$, respectively.

• Sum of Quantities ($Z = X + Y$):

  • The maximum possible absolute error in $Z$ is the sum of the absolute errors of individual quantities.
  • Formula:

$\Delta Z = \pm (\Delta X + \Delta Y)$

• Difference of Quantities ($Z = X - Y$):
  • The maximum possible absolute error in $Z$ is the sum of the absolute errors of individual quantities.
  • Formula:

$\Delta Z = \pm (\Delta X + \Delta Y)$

• Product of Quantities ($Z = X Y$):
  • The maximum fractional error in $Z$ is the sum of the fractional errors of individual quantities.
  • Fractional change formula:

$\frac{\Delta Z}{Z} = \pm \left( \frac{\Delta X}{X} + \frac{\Delta Y}{Y} \right)$

• Quotient/Division of Quantities ($Z = \frac{X}{Y}$):
  • The maximum fractional error in $Z$ is the sum of the fractional errors of individual quantities.
  • Fractional change formula:

$\frac{\Delta Z}{Z} = \pm \left( \frac{\Delta X}{X} + \frac{\Delta Y}{Y} \right)$

• Power of a Quantity ($Z = X^n$):
  • The fractional error in a quantity raised to the power $n$ is $n$ times the fractional error in the quantity itself.
  • Fractional change formula:

$\frac{\Delta Z}{Z} = \pm n \left( \frac{\Delta X}{X} \right)$

• General Power Combination Formula ($Z = \frac{X^a Y^b}{W^c}$):
  • For a general algebraic expression involving multiple powers and terms in numerator and denominator, absolute fractional errors always add up.
  • Fractional change formula:

$\frac{\Delta Z}{Z} = \pm \left( a \frac{\Delta X}{X} + b \frac{\Delta Y}{Y} + c \frac{\Delta W}{W} \right)$

• Percentage error formula:

$\frac{\Delta Z}{Z} \times 100 = \pm \left( a \frac{\Delta X}{X} \times 100 + b \frac{\Delta Y}{Y} \times 100 + c \frac{\Delta W}{W} \times 100 \right)$

Special Error Combination Cases

• Error in Equivalent Parallel-like Expression ($Z = \frac{X Y}{X + Y}$):
  • When $Z$ is defined as the product over the sum of two variables $X$ and $Y$, the fractional change in $Z$ is given by:

$\frac{\Delta Z}{Z} = \frac{\Delta X}{X} + \frac{\Delta Y}{Y} + \frac{\Delta X + \Delta Y}{X + Y}$

Properties and Units of Errors

• Unit of Absolute Error:

  • The absolute error always possesses the exact same unit as the physical quantity itself.

• Unit of Fractional and Percentage Errors:

  • Fractional change (relative error) and percentage error are dimensionless quantities and have no units.