Error Bars and Worst-Fit Line Calculator

What is a worst-fit line?

A worst-fit line is the steepest or the shallowest straight line that still passes through every error bar on a graph. The two of them bracket every gradient the measurements allow, and the uncertainty in the gradient is half the difference between them: Δm = (m_max − m_min) ÷ 2. Dividing by the best gradient gives the percentage uncertainty, Δm ÷ m × 100.

The same two lines give the uncertainty in the y-intercept, as half the difference between the points where they cross the y-axis. Both matter in a write-up, because a gradient or an intercept usually stands for something, such as a resistance, a spring constant or a stopping voltage, and a value without its uncertainty cannot be compared with anything.

Using the calculator

Type each reading into the table: x, y and ±y, which is half the length of the y error bar. Switch on error bars on x for a ±x column as well. A block of columns copied from a spreadsheet can be pasted into any box, and Enter moves down a row. Name the axes with the unit in brackets, as in Voltage (V), and the gradient and intercept are given in those units.

The graph shows the readings, their error bars, the least-squares line of best fit and the two worst-fit lines. The readouts give the gradient and the intercept with their uncertainties, rounded so that the uncertainty keeps one significant figure, or two when it starts with a 1, and the value stops at the same decimal place. Below them are the percentage uncertainty, the maximum and minimum gradients, and whether a line through the origin fits. The working names the two corners of the error bars that each worst-fit line rests on, so the same lines can be drawn on paper.

With error bars on both axes, each point’s two bars outline a box, and a line counts as passing through them when it crosses that box. The calculator tries the line through every pair of box corners, which finds the true steepest and shallowest lines even when they do not run between the end bars.

Choosing the size of each error bar

An error bar shows the uncertainty in one plotted value. For a single reading it usually comes from the instrument: half the smallest division on an analogue scale, or one unit in the last digit of a digital meter. For repeated readings, half the range is the usual choice: repeats of 4.0, 4.3 and 4.2 V have a mean of 4.17 V and an uncertainty of (4.3 − 4.0) ÷ 2 = 0.15 V.

When the plotted quantity is calculated, its error bar follows from the reading’s. Squaring doubles the percentage uncertainty, so a period of 2.00 ± 0.02 s, which is 1 percent, gives T² = 4.00 ± 0.08 s², which is 2 percent.

Worked example: a resistor’s voltage against current

The calculator opens on five readings of the voltage across a resistor at currents from 1 to 5 mA, each voltage read to ±0.5 V.

• Best fit. The means are 3 mA and 6.1 V, and least squares gives m = Sxy ÷ Sxx = 19.9 ÷ 10 = 1.99 V/mA and c = 6.1 − 1.99 × 3 = 0.13 V.
• The obvious steepest line fails. From the bottom of the first bar, (1, 1.7), to the top of the last, (5, 10.8), the gradient is 9.1 ÷ 4 = 2.275, but at 4 mA that line is at 1.7 + 2.275 × 3 = 8.525 V, above the top of point 4’s bar at 8.3 V.
• The steepest line pivots on point 4 instead: (8.3 − 1.7) ÷ (4 − 1) = 2.2 V/mA, crossing the y-axis at 1.7 − 2.2 × 1 = −0.5 V.
• The shallowest line runs from the top of the first bar to the bottom of the last: (9.8 − 2.7) ÷ (5 − 1) = 1.775 V/mA, with intercept 2.7 − 1.775 × 1 = 0.925 V.
• The uncertainties. Δm = (2.2 − 1.775) ÷ 2 = 0.2125 V/mA, which is 0.2125 ÷ 1.99 × 100 = 10.7 percent, and Δc = (0.925 − (−0.5)) ÷ 2 = 0.7125 V.
• The result. The gradient is 2.0 ± 0.2 V/mA, a resistance of 2.0 ± 0.2 kΩ since a volt per milliamp is a kilohm, and the intercept is 0.1 ± 0.7 V.

The intercept’s range, −0.6 V to 0.8 V, includes zero, and lines through the origin with gradients from 1.96 to 2.075 V/mA pass through every bar, so the readings are consistent with the voltage being proportional to the current, as Ohm’s law expects.

Half the spread, or one worst line?

Two conventions are in use. Half the spread, which the calculator leads with, treats the steepest and shallowest lines alike and is the version usually taught for IB Physics, whose syllabus asks for the uncertainty in gradients and intercepts. Some A level practical handbooks instead use one worst acceptable line, either the steepest or the shallowest, and quote the difference between its gradient and the best one. For the example that gives 0.21 V/mA with the steepest line and 0.215 V/mA with the shallowest, and the Single worst line readout shows the larger. All three round to 0.2 V/mA here, but they need not agree, so use the one your course asks for.

None of them is the standard error a statistics package reports. Pasted into the linear regression calculator, the same five points give a standard error in the slope of 0.068 V/mA, about a third of the worst-fit figure. The standard error comes from how far the points scatter about the line, while the worst-fit lines come from the error bars, so they still give an uncertainty when the points happen to lie on a perfect line.

What this model leaves out

• Weighting. The best-fit line is unweighted least squares, so a point with a short error bar has no more say than one with a long bar. A weighted fit, with each point weighted by 1/Δy², is the usual next step at university.
• Curves. Only straight lines are tried. A curved relationship has to be plotted in a form that should be straight first, such as T² against L for a pendulum, with the uncertainties worked out for those values.
• Systematic error. A zero error shifts every reading by the same amount, which moves the intercept, and a miscalibrated scale stretches them all, which changes the gradient. Neither shows in the error bars or the worst-fit lines.
• Anomalies. No reading is left out automatically. When no straight line passes through every error bar, the calculator names the reading to check, if a single one is to blame, and leaves the decision to you.
• Confidence. The worst-fit range is not a confidence interval. It shows what the error bars allow, and it is only as honest as the uncertainties typed in.
• How repeats are spread. An error bar uses only the ends of a range. With many repeats of one reading, a histogram of them, which the frequency density and histogram calculator draws, shows whether half the range is a fair uncertainty.

Common mistakes

• Assuming the worst-fit lines join the end bars. They often do, but a reading off the trend can block them, as point 4 does here. Check each line against every bar.
• Reading the intercept off the edge of the grid. Where the line meets the left edge is the y-intercept only if that edge is x = 0. Otherwise work it out as c = y − mx from a point on the line.
• Taking the gradient from two readings. A gradient is read from two points on the line, far apart. For a worst-fit line, the corners of the bars it rests on are those points.
• Quoting too many figures. 1.99 ± 0.2125 claims digits the uncertainty rules out. Round the uncertainty to one significant figure, or two if it starts with a 1, and the value to the same decimal place, which the significant figures calculator can help with.
• Calling it a percentage error. The percentage uncertainty is the spread the error bars allow. A percentage error compares a result with an accepted value, and the percent error calculator works that out.
• Losing the sign. On a falling graph the steepest line has the most negative gradient, which is the minimum. The uncertainty is still half the spread and is quoted as a positive number.