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.
• A value given without an uncertainty: the absolute uncertainty is one unit in its last significant figure.

Finding the uncertainty

Repeated readings

• Remove an anomalous reading before taking the range.
• Calculate the mean and the half range: percentage uncertainty = half range / mean × 100 %.

One reading and two readings

• One reading on a scale: the uncertainty is half the smallest division.
• A length found from two readings, one at each end: subtraction of data leads to addition of uncertainties, so the absolute uncertainty is 2 × the uncertainty in each reading.

Reducing the percentage uncertainty

• Measure a larger quantity: the absolute uncertainties are unchanged and the value is larger, so the percentage uncertainties reduced.
• Measuring multiple fringes to reduce the percentage uncertainty in fringe spacing; time many oscillations for the same reason.
• A time too short to measure with a stopwatch has a large percentage uncertainty.

Combining uncertainties

• A reciprocal, f = 1/T: the percentage uncertainty in f is the same as that in T.
• A difference of two quantities: add the absolute uncertainties, then find the percentage uncertainty for the difference.

Error bars

• An error bar extends the absolute uncertainty above and below the point: error bar consistent with the plotted point and its uncertainty, drawn for all error bars.
• The line of best fit should pass through all error bars.

Uncertainty in a gradient and an intercept

• Maximum gradient line: from the bottom of first error bar to the top of the last. Minimum gradient line: top of first to bottom of last.
• Horizontal error bars, positive gradient: the minimum gradient line runs from the left-hand end of the first error bar to the right-hand end of the last.
• Each line should go through all the error bars, and the lines must extend to cover the range of all the points.
• Best gradient: the mean of G_max and G_min. Percentage uncertainty of gradient = ΔG / best gradient × 100 %.
• Intercept: use the two intercepts of the maximum and minimum gradient lines, find the mean of the two values, and state the absolute uncertainty as the difference between the mean and either of the two values.

Significant figures of an uncertainty

• An absolute or percentage uncertainty: 1 or 2 sf only. Never write an uncertainty to 3 or more significant figures.
• Give the value to the same decimal place as its absolute uncertainty, written as value ± uncertainty with its unit.

Quantities and units