Department of Physics 1st year Laboratory Introduction to Errors Dr S P Tear

Aims To introduce: The concept of random and systematic errors The distinction between precise and accurate measurements Measurements of error: standard deviation and standard error Combining errors Introduction to Errors

Example Consider the results of your measurement of the resistance of a coil at different temperatures: 200.025 Ω at 10°C 200.034 Ω at 20°C Is the difference between these two values significant? We can’t say, unless we know the error on the measurements Yes, if the error in each measurement is 0.001 Ω No, if the error in each measurement is 0.01 Ω Introduction to Errors

Systematic and random errors Systematic errors: Generally constant throughout a set of readings Constant offset, constant factor, or both Caused by factors which are not accounted for in the measurements or theory Examples: Offset error: voltmeter not zeroed properly Gain error: clock running fast or slow Introduction to Errors

Random errors Vary arbitrarily (randomly) from one measurement to the next. Equally likely to be positive or negative Always present in an experiment True value Random only Random & systematic Introduction to Errors

Precision and accuracy A result is precise if the random error is small And accurate if it is free from systematic and random errors Thus, it is possible to have a inaccurate but precise result Introduction to Errors

Reducing systematic errors Careful experimental design (reduce in expt or taken account of) Calibrate measuring instruments (most digital instruments in the lab. will have negligible systematic errors) Maybe the theory is not quite right! No easy way to detect systematic errors correlate with other experiments? discrepancy with theory? Introduction to Errors

Reducing random errors Repeating the measurements and take the mean (detects random errors too) The more readings taken, the closer the mean will approach the true mean (in limit of  readings) Suppose you have a set of n successive measurements of the same quantity: x1, x2, …xn, then the best estimate of the measured value is the arithmetic mean: Introduction to Errors

Error in the mean? What is the error in the mean? If we only have a finite number of measurements, can we determine how close the mean of our sample of n measurements is to the true mean? Standard Deviation Standard Error Introduction to Errors

Standard deviation & standard error Measures the spread in the random errors in the measurement Tells us the error in a single measurement Doesn’t get smaller the more readings taken Standard Error Error in the mean of n measurements Gets smaller the more readings you take, because the mean becomes more precise as more measurements are included in it. Introduction to Errors

Statistically speaking Standard deviation: there is a ~67% chance that the next single measurement I make will lie within 1 s.d. of the true mean (~95% within 2 s.d.) Standard error: there is a ~67% chance that the measured mean of the readings lies within 1 s.e. of the true mean (~95% within 2 s.e.) Note: increasing the number of readings reduces the standard error but only at the rate of 1/n, so time and resources often dictate the optimum number of readings to take. Introduction to Errors

Combining errors: sum & difference Refer to section 4 of the Laboratory Handbook Combining errors - in quadrature Sum and Difference: Z = A + B or Z = A - B Add the squares of the ‘absolute errors’: The absolute error in Z, Z is: where A and B are the errors in A and B. Introduction to Errors

Combining errors: product & quotient Combining errors - in quadrature Products and Quotients: Z = AB or Z = A/B We add the squares of the ‘fractional errors’: The fractional error in Z, is: where A/A and B/B are the fractional errors in A and B Introduction to Errors

Today’s experiments Refer to section 4 of the Laboratory Handbook Circus of 3 activities, 55 minutes each Measurement of time Measurement of length & angle Measurement of voltage and current 20 minute break at 3:10 Notebooks marked at end of each expt. RHS! Introduction to Errors