Units, Dimensions, and Errors in Measurements

Vocabulary flashcards detailing key error analysis definitions and rules for combination of errors from JEE Main Physics Chapter 2.

Mean Value ($a_m$)

The arithmetic mean of observed values $a_1, a_2, \dots, a_n$, which is taken as the true value of a measured quantity, given by $a_m = \frac{a_1 + a_2 + \dots + a_n}{n} = \frac{1}{n} \sum_{i=1}^{n} a_i$.

Absolute Error

The difference between the true value and the observed value, expressed as $|\Delta a_i| = |a_m - a_i|$, having the same unit as the quantity itself.

Mean Absolute Error ($\Delta a_m$)

The arithmetic mean of the magnitudes of absolute errors across all measurements, given by $\Delta a_m = \frac{|\Delta a_1| + |\Delta a_2| + \dots + |\Delta a_n|}{n} = \frac{1}{n} \sum_{i=1}^{n} |\Delta a_i|$.

Relative Error

The ratio of the mean absolute error to the mean value of the quantity, calculated as $\frac{\Delta a_m}{a_m}$, which has no unit.

Percentage Error

The relative error of a quantity expressed in percentage form, calculated as $\frac{\Delta a_m}{a_m} \times 100$.

Error in Sum ($Z = X + Y$)

The maximum possible absolute error in $Z$ when $Z = X + Y$, given by $\Delta Z = \pm(\Delta X + \Delta Y)$.

Error in Difference ($Z = X - Y$)

The maximum possible absolute error in $Z$ when $Z = X - Y$, given by $\Delta Z = \pm(\Delta X + \Delta Y)$.

Fractional Error in Product ($Z = XY$)

The fractional change in $Z$ when $Z = XY$, given by $\frac{\Delta Z}{Z} = \pm \left( \frac{\Delta X}{X} + \frac{\Delta Y}{Y} \right)$.

Fractional Error in Quotient ($Z = \frac{X}{Y}$)

The fractional change in $Z$ when $Z = \frac{X}{Y}$, given by $\frac{\Delta Z}{Z} = \pm \left( \frac{\Delta X}{X} + \frac{\Delta Y}{Y} \right)$.

Fractional Error in Exponential ($Z = X^n$)

The fractional change in $Z$ when $Z = X^n$, given by $\frac{\Delta Z}{Z} = \pm n \frac{\Delta X}{X}$.

Fractional Error in Power Combination ($Z = \frac{X^a Y^b}{W^c}$)

The fractional change in $Z$ when $Z = \frac{X^a Y^b}{W^c}$, given by $\frac{\Delta Z}{Z} = \pm \left( a \frac{\Delta X}{X} + b \frac{\Delta Y}{Y} + c \frac{\Delta W}{W} \right)$.

Percentage Error in Power Combination ($Z = \frac{X^a Y^b}{W^c}$)

The percentage error in $Z$ when $Z = \frac{X^a Y^b}{W^c}$, calculated as $\frac{\Delta Z}{Z} \times 100 = \pm \left( a \frac{\Delta X}{X} + b \frac{\Delta Y}{Y} + c \frac{\Delta W}{W} \right) \times 100$.

Fractional Error in Combination ($Z = \frac{XY}{X+Y}$)

The fractional change in $Z$ when $Z = \frac{XY}{X+Y}$, given by $\frac{\Delta Z}{Z} = \frac{\Delta X}{X} + \frac{\Delta Y}{Y} + \frac{\Delta X + \Delta Y}{X + Y}$.