Uncertainties and Error Bars

Absolute, fractional and percentage uncertainty

• The absolute uncertainty has the same unit as the quantity. The value in the denominator is the mean of the repeated readings.
• Write a percentage uncertainty with %. Never write a percentage uncertainty as a decimal.
• Same absolute uncertainty, larger value: absolute uncertainty is the same but value is larger, so the percentage uncertainty is smaller.
• For a small measured value write percentage uncertainty is large, naming the quantity. Never write small distances are hard to measure.

Finding the uncertainty

Repeated readings

• Remove an anomalous reading before taking the range.
• Percentage uncertainty = ½ range / mean × 100.

One reading and two readings

• The uncertainty in one reading is half the resolution of the scale when a scale is read. Where a spot or line has to be located, it can be a whole division.
• A length found from two readings (one at each end): the uncertainties in each reading are added, so absolute uncertainty = 2 × uncertainty in each reading. Never write because the smallest division is 1 mm as the reason.
• Locating the exact position of a spot or line adds to the uncertainty: uncertainty in locating exact position.
• Timing many oscillations: the uncertainty of the one timing is spread over all of them, so percentage uncertainty = uncertainty in the timing / total time × 100 %.

Reducing the percentage uncertainty

• Make the measured quantity larger: larger values, so the percentage uncertainty is reduced.
• Time many oscillations, such as 20T: this reduces the percentage uncertainty.

Combining uncertainties

Error bars

• An error bar extends the absolute uncertainty above and below the point.
• Draw a straight line of best fit passing through all error bars.
• A constant absolute uncertainty: the error bars are the same length. A constant percentage uncertainty: the bars grow longer as the value grows.
• Compare the lengths of the error bars with a number: how many times longer one bar is than another.

Uncertainty in a gradient and an intercept

• Steepest line: ruled through bottom of first error bar and through top of last error bar it can pass through. Shallowest line: top of the first, bottom of the last.
• Both are lines that pass through all the error bars, each a thin, single, continuous ruled line.
• Best gradient: the mean of G_max and G_min.
• Percentage uncertainty in the gradient = ΔG / best gradient × 100 %.
• Uncertainty in an intercept: the same method, using the intercepts of the steepest and shallowest lines.

Significant figures

• Give the result with sf consistent with uncertainty: the value to the same decimal place as its absolute uncertainty, written as value ± uncertainty with its unit.
• Give a percentage uncertainty to two significant figures.

Quantities and units