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Independent uncertainties are added in quadrature because they represent random effects that can push a measurement up or down. If the effects are random and independent, their combined influence is neither always cancelling nor always additive.

Conditions: The two uncertainty effects are random.; The two effects are independent from one another.; The combination is modeled by a perpendicular geometric construction.

Independent uncertainties are added in quadrature because they represent random effects that can push a measurement up or down. If the effects are random and independent, their combined influence is neither always cancelling nor always additive.

Conditions: The two uncertainty effects are random.; The two effects are independent from one another.; The combination is modeled by a perpendicular geometric construction.

For addition and subtraction, the absolute uncertainties of the inputs combine in quadrature. The plus or minus signs in the formula do not matter because they are squared in the propagation equation, resulting in the same rule for both operations.

Conditions: The derived quantity is a sum or difference of two measured variables ($z = \pm x \pm y$).; The variables $x$ and $y$ are independent measured quantities.

For addition and subtraction, the absolute uncertainties of the inputs combine in quadrature. The plus or minus signs in the formula do not matter because they are squared in the propagation equation, resulting in the same rule for both operations.

Conditions: The derived quantity is a sum or difference of two measured variables ($z = \pm x \pm y$).; The variables $x$ and $y$ are independent measured quantities.

You can sometimes ignore the smaller relative uncertainty when one measurement is much more accurate, relatively speaking, than the other. In this case, the combined relative uncertainty is approximately equal to the larger one, and the more accurate measurement behaves as if it were an exact constant.

Conditions: The comparison is between relative uncertainties.; One measurement is much more accurate relative to the other.; The approximation is acceptable for the intended level of precision.

You can sometimes ignore the smaller relative uncertainty when one measurement is much more accurate, relatively speaking, than the other. In this case, the combined relative uncertainty is approximately equal to the larger one, and the more accurate measurement behaves as if it were an exact constant.

Conditions: The comparison is between relative uncertainties.; One measurement is much more accurate relative to the other.; The approximation is acceptable for the intended level of precision.

When both mass and velocity are measured, the uncertainty in kinetic energy is found by adding the two uncertainty sources in quadrature. You compute the partial derivative of kinetic energy with respect to each variable, multiply each derivative by the corresponding absolute uncertainty, square those terms, sum them, and take the square root.

Conditions: The kinetic energy formula is $K = \frac{1}{2}mv^2$.; Both mass $m$ and velocity $v$ carry independent absolute uncertainties $\delta m$ and $\delta v$.; The uncertainties are small enough for a first-order linear approximation.

When both mass and velocity are measured, the uncertainty in kinetic energy is found by adding the two uncertainty sources in quadrature. You compute the partial derivative of kinetic energy with respect to each variable, multiply each derivative by the corresponding absolute uncertainty, square those terms, sum them, and take the square root.

Conditions: The kinetic energy formula is $K = \frac{1}{2}mv^2$.; Both mass $m$ and velocity $v$ carry independent absolute uncertainties $\delta m$ and $\delta v$.; The uncertainties are small enough for a first-order linear approximation.

On a two-variable surface, partial derivatives represent the slopes of the tangent plane in the coordinate directions. The partial derivative with respect to $x$ is the slope along the $x$-direction, and the partial derivative with respect to $y$ is the slope along the $y$-direction.

Conditions: The surface is locally smooth enough to have a tangent plane.; The variables are distinguished as $x$ and $y$.

On a two-variable surface, partial derivatives represent the slopes of the tangent plane in the coordinate directions. The partial derivative with respect to $x$ is the slope along the $x$-direction, and the partial derivative with respect to $y$ is the slope along the $y$-direction.

Conditions: The surface is locally smooth enough to have a tangent plane.; The variables are distinguished as $x$ and $y$.