doPhysics · Mathematical Tools
Measurement and Error
How finely an instrument reads and how far a reading can be trusted. Least count, precision and accuracy, random and systematic errors, absolute and percentage errors, the propagation of errors, the vernier calliper, the screw gauge and the spherometer, and the best straight line through readings.
Start reading0%done
Why this chapter
Why this chapter exists
Every number in physics was once a reading on an instrument, and every reading is uncertain. Young's modulus, the value of \(g\) and Planck's constant are only as good as the readings behind them. This chapter shows how finely an instrument can read, how an error is measured, and how the errors of several readings pass into a result. It reads the vernier calliper, the screw gauge and the spherometer, which examinations ask about every year. It ends with the best straight line, the tool that turns scattered readings into a constant of nature.
You will learn
By the end of this chapter you will
- Give the least count of a scale, a vernier or a screw, and tell precision from accuracy and random from systematic errors.
- Find the mean, the mean absolute error and the relative and percentage error of a set of readings or of an approximation.
- Find the error in a result such as \(g\), Young's modulus or a meter-bridge resistance from the errors of its readings.
- Read a vernier calliper, a screw gauge and a spherometer, correct their zero errors, and read slope and intercept from the best line through readings.
Before you begin
Before you begin
You should be able to work with decimals and percentages, square a number and take a square root, and plot points on a graph. The lessons borrow physics from later chapters, such as Searle's experiment and the photoelectric effect, but explain all that they use. Significant figures are taught in the chapter Numbers, Units and Estimation.
Lessons
1
Least Count, Precision and Accuracy
How finely an instrument can read, how closely repeated readings agree, and how close they come to the truth. Random and systematic errors, and the zero error of an instrument.
Least count precision and accuracyRandom and systematic errorsZero error and its correction
2
Absolute, Relative and Percentage Error
How large an error is, measured against the quantity itself. The mean of repeated readings, the mean absolute error, and the relative and percentage error of a measurement or of an approximation such as \(mgh\).
Absolute relative and percentage errorThe mean and the mean absolute errorPercentage error of an approximation
3
Propagation of Errors
How the errors of measured readings pass into a result calculated from them. Sums and differences add absolute errors; products, quotients and powers add relative errors. The error in \(g\) from a pendulum, in Young's modulus and in a meter-bridge resistance.
Propagation of errorsA difference of close readingsG from a simple pendulum
4
Vernier Calliper, Screw Gauge and Spherometer
Three instruments that read lengths finer than their own marks. The vernier's two scales out of step, the screw gauge's pitch and circular scale, their zero errors, and the radius of a lens by the spherometer.
Vernier calliper and screw gaugeReading a screw gaugeZero errors of the calliper and the gaugeThe spherometer and the radius of a lens
5
The Best Straight Line through Data
Plotting readings on a graph, choosing scales and a false origin, drawing the best straight line by eye, and reading its slope and intercept with their units. Planck's constant and the work function from a voltmeter.
Plotting readings on a graphThe best straight line through dataReading slope and intercept
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- absolute error
- The magnitude of the difference between a measured value and the true value, in the unit of the quantity.
- accuracy
- The closeness of a reading, or of the mean of readings, to the true value.
- backlash
- The small play in a worn screw thread. When the direction of turning is reversed, the screw turns a little without moving the spindle.
- circular scale
- The divided scale round the edge of the thimble of a screw gauge or spherometer. One division is one part of a full turn.
- false origin
- A starting value other than zero on an axis of a graph, marked by a break in the axis. It is used when the readings start far from zero.
- least count
- The smallest change in a quantity that an instrument can show. For a plain scale it is one division; for a screw, the pitch divided by the circular divisions.
- main scale
- The fixed scale of a vernier calliper, marked on the beam, usually in millimetres.
- main-scale reading
- The last main-scale mark before the zero of the vernier scale.
- mean
- The sum of repeated readings divided by their number, \(\bar a\). It is taken as the best value of the quantity.
- mean absolute error
- The average of the sizes of the deviations of the readings from their mean, \(\Delta a\). The result is written \(\bar a \pm \Delta a\).
- mean point
- The point whose coordinates are the means of the x values and the y values of a set of readings. The best straight line passes through it.
- parallax
- The apparent shift of a mark against a scale when the eye is not straight in front of it. Reading always from one side gives a systematic error.
- percentage error
- The relative error multiplied by 100, \(100\,\Delta a/a\,\%\).
- pitch
- The distance the spindle of a screw moves in one full turn.
- precision
- The closeness of repeated readings to one another. Readings with a small scatter are precise.
- random error
- An error that changes in size and sign from reading to reading. It shrinks in the mean of many readings.
- ratchet
- The knob at the end of a screw gauge that slips once the object is gripped. Every reading is then taken with the same pressure.
- relative error
- The absolute error divided by the value of the quantity, \(\Delta a/a\), a pure number.
- screw gauge
- An instrument that holds an object between a fixed anvil and a spindle moved by a fine screw. Its thickness is read from a sleeve and a circular scale.
- sleeve reading
- The last millimetre or half-millimetre mark of a screw gauge's sleeve left uncovered by the thimble.
- spherometer
- A screw gauge on three legs. It measures the height of a curved surface above the plane of its feet, and so its radius of curvature.
- systematic error
- An error of the same size and sign in every reading, from the instrument, the method or the conditions. Averaging cannot reduce it.
- vernier calliper
- An instrument with two jaws, a main scale and a sliding vernier scale. It reads lengths to a fraction of a main division.
- vernier scale
- The short sliding scale of a vernier calliper, whose \(n\) divisions span \(n - 1\) main divisions.
- zero error
- The reading an instrument shows when the quantity measured is zero. The true value is the observed reading minus the zero error, with its sign.