6.2 Uncertainties and logarithmic graphs (A2)

How do you combine uncertainties when quantities are added or subtracted?

At AS you found the uncertainty of one reading at a time. At A2 most results are calculated from several readings, so you need rules for combining uncertainties. These rules give a cautious, worst-case estimate. Start with a result that is a sum or a difference of two measured values. Each reading could be out by its full uncertainty, and the two errors could happen to push the result the same way. That is true even for a subtraction: if the first reading is slightly too high and the second slightly too low, the difference is too large by both amounts together. So the absolute uncertainties add.

Typical cases: a temperature rise, an extension from two scale readings, and a count rate corrected for background. Work out the absolute uncertainty first; turn it into a percentage only if the result goes on into a product or quotient.

The danger is a small difference between two large readings: the absolute uncertainty stays the same size but becomes a much larger fraction of the result. Design the experiment so the difference is as large as practical.

How do you combine uncertainties for products, quotients and powers?

For products and quotients the percentage uncertainties combine. If a is 2% too big and b is 3% too big, ab is about 5% too big. If b is 3% too small and on the bottom of a fraction, a/b is again about 5% too big: uncertainties never cancel.

• Powers. A square doubles the percentage uncertainty, a cube triples it and a square root halves it. For small uncertainties \((1+p)^2 \approx 1 + 2p\), which is where the factor comes from.
• Exact numbers (2, π, ½, a count of 20 oscillations) have no uncertainty and contribute nothing.
• A constant multiplier changes the value but not the percentage uncertainty: T = (time for 20 swings) ÷ 20 has the same percentage uncertainty as the time.
• Mixed calculations such as \(c = E \div m\Delta\theta\): absolute uncertainty of the difference first, then its percentage, then add percentages.
• Rounding. Give the final absolute uncertainty to 1 (or 2) significant figures and the value to the same decimal place.

The quantity with the highest power usually dominates the total, so it deserves the most precise instrument: for a wire, measure the diameter with a micrometer at several places and average.

How do you decide whether a result agrees with an accepted value?

Once a result has an uncertainty, you can ask whether it is any good. Two different percentages answer two different questions.

• Percentage difference < percentage uncertainty: the accepted value lies inside the range of your result, so the result agrees with (is consistent with) it within the experimental uncertainty.
• Percentage difference > percentage uncertainty: the accepted value lies outside your range. The uncertainties you estimated cannot explain the gap, so there is probably a systematic error, or an uncertainty you have not allowed for. Say what it might be and which way it pushes the result.
• Equivalent check: does the accepted value fall inside value ± uncertainty?
• With no accepted value (two methods, or a tolerance such as "470 μF ± 10%"), compare the two ranges: if they overlap, the values are consistent.

How do you find the uncertainty in a gradient and an intercept?

Most Unit 6 results come from the gradient or intercept of a straight-line graph, so the uncertainty is found from the graph itself.

• Error bars. A vertical bar through each point, ± the uncertainty in y (horizontal too if x is uncertain), from the resolution or half-range of repeats.
• Best-fit line. As at AS, with points scattered evenly about it; it should pass through (almost) all the error bars.
• Steepest and shallowest lines. The steepest and least steep straight lines that still pass through all the error bars. In practice they usually join the bottom of the first bar to the top of the last bar, and the top of the first to the bottom of the last.
• Uncertainty in the gradient ≈ (steepest − shallowest) ÷ 2. Another common method uses |best − worst|, with whichever extreme line differs more from the best fit; either way, show the gradients you used.
• Uncertainty in the intercept: half the difference between the intercepts of the steepest and shallowest lines. A long extrapolation makes it far less certain than the gradient.
• Percentage uncertainty in the gradient = uncertainty ÷ best gradient × 100%; it feeds into anything calculated from the gradient.

What do you need to know about logarithms?

Exponential and power-law relationships give curves, which are hard to judge by eye. Taking logs turns both into straight lines whose gradient and intercept give the constants. First, the log skills themselves.

• Two bases. ln uses base e (≈ 2.718): \(\ln e^x = x\). log (or lg) uses base 10: \(\log 10^x = x\). Use the one the question uses, and stay with it.
• The three rules (true for either base): \(\log(ab) = \log a + \log b\), \(\log(a/b) = \log a - \log b\), \(\log(a^n) = n\log a\).
• Units. Only a pure number has a log, so divide by the unit first: ln (V / V), log (T / s). Logs have no unit.
• Significant figures. Give a log to the same number of decimal places as the number has significant figures. 5.83 V (3 s.f.) → ln 5.83 = 1.763; 0.64 V (2 s.f.) → ln 0.64 = −0.45. The digits before the decimal point only say where the decimal point was.
• Error bars on a log graph. \(\Delta(\ln y) \approx \Delta y \div y\), the fractional uncertainty. For 2.38 V ± 0.05 V: ln 2.43 = 0.888 and ln 2.33 = 0.846 either side of ln 2.38 = 0.867, about ± 0.021. The bars grow as y gets smaller.
• Changing base. \(\log y = 0.434\ln y\). So if you plot log y for an exponential, the gradient is 0.434k, not k: another reason to use ln for exponentials.

How do you test an exponential relationship with a log graph?

An exponential change has a constant ratio in equal intervals of x: the same fraction is lost every 15 s, every half-life, every millimetre of absorber. A falling curve could be many things, so the curve alone is never convincing.

Take natural logs of \(y = Ae^{kx}\):

This matches y = mx + c with ln y against x. A straight line means the relationship is exponential; the gradient is k and the intercept ln A, so \(A = e^{\text{intercept}}\), with the unit of y.

• For a decay k is negative: the gradient gives the decay constant (−λ) or −1/RC.
• Half-life (time to halve) = ln 2 ÷ |gradient|.
• Correct the data first if the physics needs it: subtract the background count from a count rate, or the room temperature from a cooling temperature, before taking logs. Uncorrected data give a curve on the log graph.

How do you test a power law with a log–log graph?

For \(y = kx^n\) with n unknown, y against x is a curve you cannot read n from. Take logs of both sides:

Plot log y against log x; a straight line confirms a power law.

• The gradient is the power n and has no unit. Either base works and gives the same n; just use the same base on both axes.
• The intercept is log k, but only where log x = 0. The log x axis rarely starts at 0, so calculate it: log k = log y − n log x using a point on the line, then \(k = 10^{\log k}\).
• If theory predicts the power (0.5 for a pendulum, −2 for an inverse-square law), compare your n with it using its uncertainty from the extreme lines.
• If you already know n, a direct linear graph (T² against l) is simpler; use log–log when n itself is the question.

Exam tips for 6.2

Common misconceptions

How this topic connects

3.1 Measurements, uncertainties and graphs (AS, Unit 3)

Half the resolution, half the range, single percentage uncertainties and linearising to y = mx + c are the starting point for everything on this page.

Capacitors (Unit 4) and nuclear decay, oscillations and gravitation (Unit 5)

Capacitor discharge, Q = Q₀e^−t/RC, is the classic ln graph. ln (count rate) against t gives the decay constant; log–log graphs test T ∝ l^0.5 and T ∝ r^1.5; error bars and extreme lines judge absolute zero.

6.3 The A2 core practicals

Capacitor discharge, gas pressure, gamma absorption by lead and resonance all finish with the analysis practised here. Planning, precision and accuracy are on 6.1.

Key equations

Check your understanding

Specification checklist

• (analysis) I can find the uncertainty of a sum or difference by adding absolute uncertainties.
• (analysis) I can find the percentage uncertainty of a product, quotient or power (adding percentages times powers) and convert it back to an absolute uncertainty.
• (analysis) I can give a final value and its uncertainty to consistent, sensible significant figures, with the correct unit.
• (evaluation) I can compare a percentage difference with the percentage uncertainty to decide whether a result agrees with an accepted value.
• (analysis) I can draw error bars and steepest and shallowest lines and find the uncertainty in a gradient and an intercept.
• (analysis) I can use the rules of logarithms with base e and base 10, and give logs to the correct number of decimal places.
• (analysis) I can plot ln y against x to test an exponential relationship and find its constants.
• (analysis) I can plot log y against log x to test a power law and find n and k.
• (evaluation) I can identify the largest contribution to the uncertainty and suggest how to reduce it.