The measurement of a physical quantity can never be made with perfect accuracy, there will always be some error or uncertainty present
- 2. The uncertainty of measurements The measurement of a physical quantity can never be made with perfect accuracy, there will always be some error or uncertainty present. measurement = (best estimate ± uncertainty) units
- 3. Suppose you want to find the mass of a gold ring using electronic balance. first attempt reading 17.43 g Since the digital display of the balance is limited to 2 decimal places. m = 17.43 ± 0.01 g. Suppose you use the same electronic balance and obtain several more readings: 17.46 g, 17.42 g, 17.44 g the average mass = 17.44 g You decide to use another balance that gives a reading of 17.22 g
- 4. Accuracy is the closeness of agreement between a measured value and a true or accepted value. Measurement error is the amount of inaccuracy. Precision is a measure of how well a result can be determined (without reference to a theoretical or true value). It is the degree of consistency and agreement among independent measurements of the same quantity; also the reliability or reproducibility of the result. The uncertainty estimate associated with a measurement should account for both the accuracy and precision of the measurement.
- 5. Types of Errors In order to determine the uncertainty for a measurement, the nature of the errors affecting the experiment must be examined. There are many different types of errors that can occur in an experiment, but they will generally fall into one of two categories: Random errors Systematic errors
- 6. Random errors Random errors usually result from human or/and accidental errors. vibrations in the equipment, changes in the humidity, fluctuating temperatures, etc miscalculations in analyzing data, the incorrect reading of an instrument, or a personal bias in assuming that particular readings are more reliable than others. Statistical methods are usually used to obtain an estimate of the random
- 7. Systematic errors A systematic error is an error that will occur consistently in only one direction each time the experiment is performed. Systematic errors most commonly arise from defects in the instrumentation or from using improper measuring techniques. For example, measuring a distance using the worn end of a meter stick, using an instrument that is not calibrated, or incorrectly neglecting the effects of viscosity, air resistance and friction. Proper calibration and adjustment of the equipment will help reduce the systematic errors leaving only the accidental and human errors to cause any spread in the data.
- 8. Statistical Methods When several independent measurements of a quantity are made, an expected result to report for that quantity is represented by the average of the measurements. For a set of experimental data containing N elements, or measurements, given by {S1 , S2 , S3 , . . . , SN}, the average S¯, is calculated using the formula
- 9. The data {S1 , S2 , S3 , . . . , SN} are dispersed around the mean, or average. A measure of this dispersion is called the standard deviation and is given by
- 10. Propagation of Errors In many experiments, the quantities measured are not the quantities of final interest. Since all measurements have uncertainties associated with them, clearly any calculated quantity will have an uncertainty that is related to the uncertainties of the direct measurements. The procedure used to estimate the error for the calculated quantities is called the propagation of errors. Suppose we want to determine a quantity R, which depends on A and maybe several other variables B, C, etc. We want to know the error in R if we measure A, B, C... with errors ẟA, ẟB, ẟC …
- 11. Since R is a function of A, B, C, . . . , it can be written as If the errors for A, B, C, . . . are independent, random, and sufficiently small, it can be shown that the uncertainty for R is given by The partial derivative means differentiating R with respect to A holding the other variables fixed.
- 12. Exercise 1. R=A+B 2. R=A-B 3. R=AB 4. R=A/B
- 13. Sketching Graphs