How error propagation works
Every measurement carries uncertainty. When you combine measured quantities in a calculation, that uncertainty carries through to the answer. Error propagation is the set of rules that tells you how large the uncertainty in the final result is, given the uncertainties in each input. This calculator implements the standard rules for independent, random errors, where uncertainties combine "in quadrature" — the square root of the sum of squares — because independent errors partly cancel rather than always reinforcing each other.
The propagation rules
Let x and y be measured values with absolute uncertainties δx and δy. For the common operations:
- Sum or difference (q = x + y or q = x − y): the absolute uncertainties add in quadrature. δq = √(δx² + δy²). Note that the uncertainty of a difference is the same as the uncertainty of the sum — subtraction does not cancel error.
- Product or quotient (q = x·y or q = x/y): the relative uncertainties add in quadrature. δq/|q| = √((δx/x)² + (δy/y)²), then δq = |q| · (δq/|q|).
- Power (q = xⁿ): the relative uncertainty scales by the magnitude of the exponent. δq/|q| = |n| · (δx/|x|). For example, squaring a value (n = 2) doubles its fractional uncertainty; taking a square root (n = 0.5) halves it.
These rules come from a first-order Taylor expansion of the function around the measured values, keeping only the linear terms — which is why they are sometimes called the "first-order" or "linearized" propagation formulas (Taylor, An Introduction to Error Analysis).
Why quadrature and not a simple sum
If you simply added the errors (δq = δx + δy for a sum), you would be assuming both measurements are wrong in the same direction at the same time — the worst possible case. In reality, independent random errors are just as likely to push in opposite directions, so on average they partially cancel. The root-sum-of-squares captures this: it always gives a smaller, more realistic uncertainty than the linear sum. A linear sum is only appropriate when errors are perfectly correlated or when you deliberately want a conservative worst-case bound.
A worked example
Suppose you measure a length x = 10.0 ± 0.5 cm and a width y = 4.0 ± 0.2 cm and want the area A = x·y. The result is 10.0 × 4.0 = 40.0 cm². The relative uncertainty is √((0.5/10)² + (0.2/4)²) = √(0.0025 + 0.0025) = √0.005 ≈ 0.0707, or about 7.07%. So δA = 40.0 × 0.0707 ≈ 2.83 cm², and you report A = 40.0 ± 2.8 cm².
Frequently Asked Questions
Why are uncertainties added in quadrature instead of just summed?
What is the difference between absolute and relative uncertainty?
Does this calculator assume the errors are independent?
Why doesn't subtracting two numbers reduce the uncertainty?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
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General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |