MEASUREMENT AND DATA PROCESSING

11.1 UNCERTAINTY AND ERROR IN MEASUREMENT Measurement involves comparing to a standard Base units MeasurementUnitSymbol LengthMeterm MassKilogramkg TimeSecondS AmountMolemol Electric currentAmpereA TemperatureKelvinK

ERRORS Errors and uncertainties caused by: Imperfections in apparatus Imperfections in experimental methods Judgments made by experimenter Two types of errors: Random uncertainties Systematic errors

RANDOM UNCERTAINTIES Make measured value larger or smaller than the true value Cannot be avoided Ex: Reading a scale Recording digital readout Reading scale from wrong position Taking a reading which changes with time

SYSTEMATIC ERROR Cause bias in experimental measurement always in the same direction (always larger or always smaller) Accuracy is a measure of systematic error A value experimentally determined is different than the true value by a large margin Ex: Non-zero reading on a meter Incorrectly calibrated scale Reaction time of experimenter

PRECISION VS. ACCURACY Precision  A series of measurements are close together If precise results have been obtained, the experiment is repeatable Accuracy  results are close to the true value

REDUCING THE EFFECT OF RANDOM UNCERTAINTY Random errors are beyond the control of the experimenter Effects can be reduced by repeating measurements The average should give a better estimate of the true value

RANDOM UNCERTAINTY AS AN UNCERTAIN RANGE Reported as uncertainty range (± next smallest measured unit) 23.5s ±1s The last digit is an estimate Usually expressed as a single significant digit

SIGNIFICANT FIGURES A method for expressing uncertainty in measurement Assumed that all but the last digit are a known value Total number of digits in a number

RULES OF SIGNIFICANT FIGURES All non-zero digits are significant 1.67 and 245  3 sig.fig. Zeros left of first non-zero not significant  2 sig. fig. Zeros between non-zeros are significant 4503  4 sig. fig. Zeros to right of decimal point significant 2.50  3 sig. fig. Exact and irrational numbers have infinite sig. figs. 4 or π If a number ends in zeros that are not right of decimal, they may or may not be counted  2-5 sig. fig. Write in scientific notation

SIG. FIG. IN CALCULATIONS Rounding after calculaing Less than 5 decimal places  rounded to 2 sig. fig. Addition/ subtraction Final result takes the same # of decimal places as the number with the least decimal places Multiplication/ division Final result has the same number of sig. figs. as the number with the least Combinations of calculations Go back to previous calculations and the final result should be rounded to the number with the least sig. figs.

11.2 UNCERTAINTIES IN CALCULATED RESULTS Uncertainty is in the same units as the measurement Uncertainty represented by the symbol δx;x is the measurement Absolute uncertainty converted to percent uncertainty (δx/x)

UNCERTAINTIES IN RESULTS Adding/ subtracting Add uncertainties of both numbers If both have uncertainty of ±0.05, final uncertainty is ±0.1 Multiplying/ dividing, add percentage uncertainties Multiplying/ dividing by a whole number Multiply or divide the uncertainty by that whole number also Powers Multiply the percentage uncertainty by the power being raised to Averaging Final answer will have the same uncertainty as the original measurements

11.3 GRAPHICAL TECHNIQUES Dependent variable measured after the independent variable is changed y-axis  dependent x-axis  independent

CONSTRUCTING GRAPHS Choose scale to suit measurements and make use of graph space Label x and y axes Plot points with noticeable markers Use appropriate type of line (i.e. best fit or curved) Title the graph Have a key if multiple lines are present Add trend line (if applicable) Identify and anomalous data points

INTERCEPT, AREA, GRADIENT, DOT-TO-DOT Intercept  where the straight-line intercepts the y-axis Gradient (slope)  Δy/Δx Units are y units/ x units Interpolation  used to determine data points between what was measured Extrapolation  used to find values outside of what was measure (i.e. make predictions) Area  determining the area under a graph, use area of triangle for straight-line graphs, use integration for curved lines Dot-to-dot  useful for showing patters. Dots are simple connected