V. Rouillard  Introduction to measurement and statistical analysis ASSESSING EXPERIMENTAL DATA : ERRORS Remember: no measurement is perfect – errors always exist. Measurement error is defined as the difference between the true value and the measured value. We can only estimate the size of the error or its likelihood that it exceeds a certain value. Errors can be estimated statistically when large number of measurements are taken. However, must ensure that measurement systems are calibrated.

V. Rouillard  Introduction to measurement and statistical analysis TYPES OF ERROR Most errors can be put into two classes: Bias errors and Precision errors. Bias errors are also referred to as systematic errors and remain the same for every measurement made. Precision errors are also called random errors and are different for each measurement made. However, the average value of the random error is zero. If enough measurements are repeated, the distribution of precision errors will be revealed and the likely size of the error can be estimated statistically. Because bias errors are fixed and do not produce a statistical distribution, they cannot be estimated using statistical techniques. They can only be estimated by comparison with a standard or another instrument or even by experience and common sense.

V. Rouillard  Introduction to measurement and statistical analysis TYPES OF ERROR Large bias error & small random error Small bias error & large random error Frequency of occurrence

V. Rouillard  Introduction to measurement and statistical analysis COMMON SOURCES OF ERROR Bias errors: Calibration (eg: zero-offset and scale adjustments) Certain consistently recurring human errors (eg: parallax, poor synchronisation) Certain errors caused by defective equipment (eg: poor design, fabrication and maintenance) Loading errors (eg: microphone, vehicle speed gun) Resolution limitations (eg: lack of significant figures in digital displays) Random errors: Certain human errors (eg: lack of concentration) Disturbances to equipment (eg: ground vibrations, atmospheric conditions) Fluctuating experimental conditions (eg: poor experimental design, must account for inherent oscillations/variations of the measurand) System sensitivity imitations (eg: use bathroom scale to measure mass of small animal)

V. Rouillard  Introduction to measurement and statistical analysis COMMON SOURCES OF ERROR Combined errors: Backlash, friction and hysteresis (eg: in mechanical indicators such as pressure gauges) Calibration drift or reaction to changing environmental conditions. Variations in procedure (eg: when short cuts are taken or personnel changes) Illegitimate errors (mistakes): Blunders and mistakes (eg: forgot to switch on amplifier, write phone number instead of reading) Computational errors (eg: use wrong calibration constant)

V. Rouillard  Introduction to measurement and statistical analysis INSTRUMENT PERFORMANCE : TERMINOLOGY Accuracy: (expected) closeness with which a measurement approaches the true value. Precision: indication of the reproducibility of measurements. If a variable is fixed, precision is the measure of the degree to which successive measurements differ from one another. Resolution: The smallest change in the measurand that the instrument will detect. Sensitivity: The ratio of the instrument response to an change in the measured quantity. Eg: and accelerometer with a sensitivity of 100 mV/g is more sensitive than one with a sensitivity of 10 mV/g. Error: Difference between the true value and the measured value.

V. Rouillard  Introduction to measurement and statistical analysis UNCERTAINTY : ESTIMATING THE LEVEL OF MEASUREMENT ERROR. Total uncertainty, U, combines the bias and random uncertainties as follows: This method is based on the assumption that the sources of bias and random errors are independent and they are therefore unlikely to coincide. Remember: The bias uncertainty is estimated from calibration checks while the random uncertainty is estimated by statistical analysis of repeat measurements.

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis A measurement sample is drawn from the population to make an estimate of the measurand. In may be that no two samples (4 blades) will have precisely the same value. But each sample (and specimen) should approximate the average value for the population Population

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis A measurement sample is drawn from the population to make an estimate of the measurand. In may be that no two samples (4 blades) will have precisely the same value. But each sample (and specimen) should approximate the average value for the population Population Sample (random selection from population)

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis Manufacturing (production) uncertainty: Analyse repeat measurements from the sample (each specimen is measured only once). Experimental uncertainty: Analyse repeat measurements from one individual specimen only.

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis Statistical analysis and interpretation meaningful only of a (relatively) large number of measurements are made. Systematic errors should be kept small. Statistical treatment cannot remove systematic (bias) errors. Arithmetic mean: Deviation from the mean: Difference between an individual reading and the mean of the group of readings: (note: the algebraic sum of all deviations = zero)

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis Average deviation: an indication of the precision of the measurements: Standard deviation: the root-mean-square (RMS) deviation of the measurements. For a finite number of readings: Variance: mean-square deviation =  2

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis Case Study: Measurement of the mass of turbine blades for use in jet propulsion systems. Blades are supplied by different manufacturers. Mass must be established based on random sample.

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis Probability Distribution of errors: The frequency distribution of observations can be calculated and displayed graphically using a histogram or frequency distribution plot: Mass [g] Number of observations

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis If more observations were made it is expected that the frequency distribution of the observations will become more defined: Mass [g] Number of observations

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis This bell-shaped curve has been shown to approach the distribution function called the Normal or Gaussian distribution. Mass [g] Number of observations

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis This bell-shaped curve has been shown to approach the distribution function called the Normal or Gaussian distribution.

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis The normal distribution function characteristically has few observations at the high and low ends and many in the middle. It has been shown to be very useful in for evaluating random errors. Mass [g] Number of observations

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis Comments on the normal distribution of random errors: All observations include small, disturbing effects called random errors. Random errors can be positive or negative with equal probability. Small errors are more likely to occur that large errors. Very large errors (> 3  ) are very improbable The probability of a given error will be symmetrical about zero.

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis Interpretation of the normal distribution of random errors:

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis Turbine blade mass case study: Mass [g] Number of observations Standard deviation: 0.75 g Mean: g

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE STATISTICAL DISTRIBUTION OF A RANDOM PROCESS Case Study: Analysis of maximum daily wave height for the design of an offshore structure. Measurements made continuously by a wave rider buoy which stores the daily maximum wave height and transmits the data to a base station.

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE STATISTICAL DISTRIBUTION OF A RANDOM PROCESS Day of year Daily max. wave height [m] Sample: record for one year (random?)

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE STATISTICAL DISTRIBUTION OF A RANDOM PROCESS The frequency distribution of observations can be calculated and displayed graphically using a histogram or frequency distribution plot: Wave height [m] Number of observations

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE STATISTICAL DISTRIBUTION OF A RANDOM PROCESS With more observations, it is expected that the frequency distribution will approach the Normal or Gaussian distribution Wave height [m] Number of observations

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE STATISTICAL DISTRIBUTION OF A RANDOM PROCESS The normal distribution has been shown to be very useful in for describing many random variables such as test scores, people height, weight etc., Wave height [m] Number of observations

V. Rouillard  Introduction to measurement and statistical analysis ESTIMATING THE RANDOM (PRECISION) UNCERTAINTY Statistical Analysis The normal (Gaussian) distribution is a function of the mean and standard deviation of the sample: Where  is the mean And  the standard deviation. In this example: The mean daily max. height = 7.5 m The standard deviation is = 2.0 m 3  (99.7%  332 days per 333 days) the expected ann. max. wave height is: (2.0) = 13.7 m 4  (99.994%  all but 1 day per 45 yrs) the expected max. wave height over 45 yrs is: (2.0) = 15.7 m Wave height [m] Number of observations m