1.5.2. Common Special Cases

If Q = x + y or Q = x – y, and x has uncertainty δx and y has δy, then

δQ = sqrt [ (δx)² + (δy)² ]

1. Note: One never adds absolute uncertainties linearly; instead, add in quadrature (square root of sum of squares).

In the case of adding or subtracting two measured values (Q), we calculate the uncertainty in that result (δQ). For instance, if you measured two distances: x and y with their respective uncertainties (δx and δy), the uncertainty in the final result is determined not by simply adding the uncertainties together but by using a specific formula. We square the individual uncertainties, sum those squares, and then take the square root of that total. This method accounts for the possibility that uncertainties can vary. We call this approach adding uncertainties 'in quadrature'.

Imagine you’re trying to determine the total length of a ribbon formed by two pieces. If one piece is 10 cm long with an uncertainty of ±1 cm and the other is 15 cm long with an uncertainty of ±0.5 cm, you could think that adding the uncertainties directly could give a total uncertainty of ±1.5 cm. However, you actually calculate the combined uncertainty using the quadrature method: √(1² + 0.5²) = √(1 + 0.25) = √1.25 ≈ 1.12 cm. This approach gives a more accurate reflection of measurement uncertainty.

If Q = x × y or Q = x ÷ y, then relative (percent) uncertainties add in quadrature:

δQ/Q = sqrt [ (δx/x)² + (δy/y)² ]

Or, equivalently:

absolute δQ = Q × sqrt [ (δx/x)² + (δy/y)² ]

When multiplying or dividing measurements, we need to consider how relative uncertainties come into play. The formula here shows that relative uncertainties are expressed as a percentage of the actual measured values. If you are calculating a value Q by multiplying two measured quantities (x and y), the total relative uncertainty for Q is calculated using the square root of the sum of the squared relative uncertainties of x and y. This is important because the uncertainty in the final result grows as you combine quantities via multiplication or division.

Think of making a large batch of cookies where you need to multiply the flour and sugar quantities. If you need 2 cups of flour with a ±10% uncertainty and 1 cup of sugar also with a ±10% uncertainty, you wouldn't just add these percentages linearly. Instead, you'd apply the quadrature method to the percentages. That means calculating the total uncertainty based on the proportion each ingredient contributes, ensuring that you know how varying these amounts will impact the total amount of ingredients overall.

If Q = x^n (x raised to power n), then

δQ/Q = |n| × (δx/x)

3. For example, if Q = x², δQ/Q = 2 × (δx/x). Absolute uncertainty δQ = Q × 2 × (δx/x).

When dealing with a value raised to a power, the uncertainty of that value scales with the exponent. The equation shows that the relative uncertainty (δQ/Q) of the resultant value Q, when x is raised to an exponent n, can be calculated by multiplying the absolute value of that exponent |n| by the relative uncertainty of x (δx/x). If Q is x², the relative uncertainty will be double that of x because you're squaring it.

Consider a scientist measuring the area of a square. If each side of the square is measured as 4 cm with an uncertainty of ±0.1 cm, the area (Q) is calculated as side², leading to an area of 16 cm². The uncertainty for this squared term would be larger because it, effectively, includes the side's uncertainty multiplied twice, which means it has greater impact on the final area than a simple length measurement.

Apply the general formula using partial derivatives. For functions of more than two variables, sum all corresponding terms.

In more complex scenarios that involve functions of multiple variables, the uncertainty can be estimated using partial derivatives. This means that you identify how changes in each variable affect the outcome of the function and incorporate that into the uncertainty calculation. You would sum all the contributions from each variable's uncertainty into a total combined uncertainty.

Think of a recipe where the taste of a cake depends on the amounts of sugar, flour, and butter. If you change the amount of one ingredient, you not only affect the outcome, but also the interaction with other ingredients. If you want to know how much the entire cake's taste varies with each ingredient's uncertainty, you would need to account for the effect of changing each ingredient (like measuring how an increase in sugar affects both sweetness and texture) when deciding on the overall uncertainty in the final cake's flavor.

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