In the Searle's experiment of the elasticity chapter, a load of \(1\,\text{kg}\) stretches a steel wire by only \(0.5\,\text{mm}\). To know Young's modulus within a few per cent, that half millimetre must be read to a hundredth of a millimetre. A metre rule marked in millimetres cannot do this. A micrometer screw can. The same laboratory also hangs a second wire beside the first, only to cancel errors that no screw can see.
The finest step of a scale
Lay a rod along a scale marked in millimetres. Its end usually falls between two marks. You can say which mark is nearer, but you cannot honestly say more. The smallest step that a scale can show is called its least count. For a scale marked in millimetres, the least count is \(1\,\text{mm}\), or \(0.1\,\text{cm}\).
A reading is therefore not one number but a small range. We write the rod of Fig 1.1 as \(L = 3.5 \pm 0.1\,\text{cm}\). The true length lies somewhere between \(3.4\) and \(3.6\,\text{cm}\). Writing \(3.47\,\text{cm}\) from this scale would claim a step the scale does not have.
The least count of an instrument is the smallest change in the measured quantity that the instrument can show. For a plain scale it is one division. For a screw it is the pitch divided by the number of divisions on its circular scale. One reading is uncertain by about one least count.
A micrometer screw moves \(1\,\text{mm}\) in one full turn, and its head carries \(100\) equal divisions. What is its least count, in mm?
Precision and accuracy
Now take many readings of one quantity, such as \(g\) from a pendulum. Two separate questions arise. Do the readings agree with one another? Do they agree with the true value? The first is precision and the second is accuracy. A set of readings can have one without the other.
Watch the chip "accurate, not precise". The readings scatter widely, so the mean wanders at first. But the scatter falls on both sides of the true value. As readings accumulate, the high ones and the low ones cancel, and the mean settles on \(9.80\,\text{m/s}^2\). Now watch "precise, not accurate". The readings agree closely, yet every one is too low. The mean settles on \(9.60\,\text{m/s}^2\) and stays there.
A set of measurements is precise when the readings lie close to one another. It is accurate when the readings, or their mean, lie close to the true value. Precision depends on the size of the scatter; accuracy depends on the bias.
Five students time the same pendulum. One student's readings agree with one another to \(0.01\,\text{s}\), but all of them are \(0.2\,\text{s}\) longer than the true period. How are these readings best described?
Two kinds of error
The difference between a reading and the true value is called its error. A random error changes from reading to reading, in size and in sign. It comes from the judgement of the observer, from the moment a stopwatch is started, from a draught or a small vibration. Because it is as often high as low, it shrinks in the mean of many readings.
A systematic error has the same size and the same sign in every reading. It comes from the instrument, from the method or from the conditions. A stopwatch that runs fast makes every period too long. A scale read always from one side, with the eye not straight above the mark, gives parallax in one direction. Averaging cannot touch such an error, because every reading carries it equally.
A third kind is simply a mistake, a gross error: a 3 read as an 8, or a reading written against the wrong load. Such a reading stands far from the others. It is found by looking at the readings, and it is repeated or discarded, never averaged in.
Which of these are systematic errors?
The zero error
A weighing machine in a sabzi mandi shows \(0.2\,\text{kg}\) with an empty pan. Every weight it shows is then \(0.2\,\text{kg}\) too large. This offset is its zero error. The zero error is a systematic error of a simple kind. It is found once, with nothing to measure, and then subtracted from every reading.
The sign matters. If the pan reads \(0.2\,\text{kg}\) when empty, the zero error is \(+0.2\,\text{kg}\). A bag that reads \(5.6\,\text{kg}\) weighs \(5.6 - 0.2 = 5.4\,\text{kg}\). If the pointer rests at \(-0.1\,\text{kg}\), the zero error is \(-0.1\,\text{kg}\). The same bag would read \(5.3\,\text{kg}\), and its true mass is \(5.3 - (-0.1) = 5.4\,\text{kg}\).
A spring balance reads \(-0.3\,\text{N}\) with nothing hanging from it. A bag of potatoes then reads \(24.7\,\text{N}\). What is the true weight of the bag, in N?
The least count is the floor of the uncertainty, not a promise. Parallax, a pointer that shakes or a coincidence that is hard to judge can make a reading worse than one least count. Averaging many readings shrinks the random part of the error, but no amount of averaging removes a systematic error. A mean of a hundred readings from a fast stopwatch is still wrong.
Back to physics
The Searle's experiment of the elasticity chapter uses every idea of this lesson. Its micrometer screw has a pitch of \(0.5\,\text{mm}\) and \(50\) divisions on its head. The student must read an extension of half a millimetre for each kilogram. The experiment is also designed to remove both kinds of error.
This is the micrometer of Searle's experiment, from Elastic Energy, Thermal Stress and Real Materials, lesson 1, read with the tool. Its screw has a pitch of \(0.5\,\text{mm}\) and \(50\) divisions. Each \(1\,\text{kg}\) stretches the wire by \(0.5\,\text{mm}\). Find the least count, and the turn needed for each kilogram. Say which errors the reference wire and the graph remove.
The pitch is \(p = 0.5\,\text{mm}\) and the head has \(N = 50\) divisions.
\(\text{LC} = p/N = 0.5/50 = 0.01\,\text{mm}\).
An extension of \(0.5\,\text{mm}\) is \(0.5/0.01 = 50\) divisions, exactly one full turn for each kilogram.
The warming of the room and the sag of the beam act on both wires alike, so the reference wire cancels these systematic errors. The line through many readings averages their random errors.
In Searle's experiment the room warms slowly during the readings, and every reading of the extension grows a little. What kind of error is this, and what removes it?
The least count decides what an experiment can hope to measure. In Searle's experiment a micrometer of least count \(0.01\,\text{mm}\) reads an extension of a few millimetres. A reference wire there cancels the systematic errors of warming and sagging. The choice of a long, thin wire in the same lesson makes the extension large compared with the least count. Every practical of the book, from the meter bridge to the calorimeter, rests on the same three words: least count, precision and accuracy.
Return to the half millimetre of the opening. A metre rule has a least count of \(1\,\text{mm}\), twice the stretch to be measured. The micrometer reads it to \(0.01\,\text{mm}\), one part in fifty. The random errors of single readings are averaged by the graph. The systematic errors of warming and sagging are cancelled by the second wire. That is why the laboratory hangs two wires.
- Quoting a reading to more digits than the least count allows.A millimetre scale gives \(3.5\,\text{cm}\), not \(3.47\,\text{cm}\). Write the reading with its least count, \(3.5 \pm 0.1\,\text{cm}\).
- Calling a set of closely agreeing readings accurate.Agreement shows precision only. Accuracy needs the readings to lie near the true value.
- Hoping that more readings will remove a systematic error.Averaging shrinks only random errors. Find the cause of a systematic error, or cancel it by design.
- Subtracting a negative zero error as if it were positive.True value = observed reading − zero error, with the sign. Subtracting \(-0.1\) adds \(0.1\).