Probability & Statistics: Introduction. Robert Leishman Mark Colton ME 363 Spring 2011

Transcription

1 Probability & Statistics: Introduction Robert Leishman Mark Colton ME 363 Spring 2011

2 Why do we care? Why do we care about probability and statistics in an instrumentation class?

3 Example Measure the strength of 30 titanium hip joints Questions: What do we report the strength to be? How do we quantify the spread in the data? What is the probability of a hip joint breaking at 350 lb? If we measure a hip joint to have a strength of 500 lb, how likely is the measurement to be correct? How confident are we in these measurements? Are they right? F break (lb) i

4 Example Measure temperature in a laboratory experiment Questions: Is this good data? What is the variation due to? Noise? Actual variations in the measured variable? Is there a trend underlying the data? T (deg F) t (s)

5 Why do we care? Why do we care about probability and statistics in an instrumentation class? Instrumentation is the science of measurement We measure a value because we don t know what it is We want to get a good measurement What is good?

6 Measurement Error Definition: Difference between the true value and the measured value e = x true - x m During the design and execution of an experiment the objective is to minimize the measurement error We don t usually know the error We can come up with uncertainty of our measurements Statistics and probability are the science of uncertainty

7 Measurement Error We often want to determine a bound on the measurement error: -u e u (n:1) where u is the uncertainty estimated at odds of n:1 Only 1 measurement in n will have an error whose magnitude is greater than u Other ways to express this idea: -u x true - x m u (n:1) x m - u x true x m + u (n:1) x true - u x m x true + u (n:1)

8 Measurement Error One more way to express the error: x true = x m ± u (P%) where P is the probability of a measured value falling within the bounds described by u Example: In measuring maximum torque of motors, the true value was found to be 45 N m, u was calculated to be 6.5 N m, and P was 96.25% Interpretations: It is 96.25% probable that all motors have a maximum torque between 38.5 and 51.5 N m It is unlikely (only 3.75% probability) that a motor will have a measured maximum torque less than 38.5 or greater than 51.5 N m Questions: How do we estimate the uncertainty u? How do we estimate the probability P?

9 Error Sources Noise Quantization Instrument limitations (precision) Calibration errors Human errors

10 Error Types Bias (systematic) Precision (random) Illegitimate (blunders)

11 Bias Error Also called systematic error Errors that occur the same way each time a measurement is made Example: A scale may have a fixed offset so that every reading of W may be higher than W true by an amount W offset Sources: Calibration Consistently recurring human error Defective equipment Loading errors Limitations of system resolution (sometimes) Cannot be treated using statistical techniques

12 Precision Error Also called random error Errors that are different for each measurement, but have an average value of zero Example: Vibration may cause the reading of a measurement device to fluctuate about the true value sometimes reading high and other times reading low (a distribution of values surrounding the true value) Sources: Certain human errors Fluctuating experimental conditions Insufficient measurement system sensitivity Actual fluctuations in measured variable (e.g., the strength of a certain material) Noise The distribution of precision error can be characterized through statistical analysis

13 Precision vs. Bias Error

14 Precision vs. Bias Error

15 Random Variables Precision error in measurements results in the measured variable being a random variable If we take repeated measurements of a variable under fixed conditions, the measured value will change due to errors in the measurement process Even though the measurements change, they tend to assume values in the neighborhood of a particular value Central tendency

16 Histogram A histogram is a plot that quantifies the tendency and density of measured data Idea: Plot the number of occurrences of a random variable in several ranges (bins)

17 Example: Blood Oxygen Measure the blood oxygen level in N = 100 subjects O% i

18 Example: Blood Oxygen To describe tendency and variation, create a histogram O max = 99.0% O min = 84.2% Divide data into K = 10 bins Count the number of data points falling in each bin

19 Example: Blood Oxygen Data tends to 92% Distributed around 92% Fairly symmetric n O%

20 Histogram Shapes Symmetric Skewed J-shaped Bimodal Uniform

21 Comments on Histograms Can also plot frequency (%) of occurrences, rather than number of occurrences, by dividing each bin count by the total number of data points N Selecting the number of bins (K): For small N: K should be chosen so that at least one bin has at least 5 occurrences For larger N: K = 1.87(N 1) For large N: K = N ½

22 Example: ASTM-A242 Steel n (%) n (%) S (Mpa) N = 100 K = 12 S ave = MPa S (Mpa) N = 10,000 K = 75 S ave = MPa