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Fitting Models by Judgment ©2005 Dr. B. C. Paul Least Squares Formulations Statistics can fit a variety of mathematical models to data by minimizing the sum of the squared departures of individual data This a mathematical technique The distribution of other mathematical model is being fit to an actual data or situation – sometimes your knowledge of the subject is as powerful a tool as least squares. Helps Available Histograms and basic statistics give us some great initial ideas Cumulative Probability Plots But histograms can have some wild interval jumps even for a perfectly normal distribution. Computer can show them – but you can also take control and plot them on paper. Special Types of Paper linearize data so that you can read key model parameters off of the slope and intercept. The Normal Probability Plot We have seen those Cumulative probability plots Come out as a straight line Put an appropriate linear scale for your data down here Plot the percentage of your Sample data here. Example – with 10 samples Plot the first sample by its X axis value with its Y Axis value at 10. The second sample would Plot at 20%. What Can Happen to Results It might plot as a straight line – (if it is Perfectly normal) You can read off your mean Woopy – I got it! More Reading Read Off Your Standard Deviation Difference is your standard deviation Why Graph for Statistics Computers Spit Out Instantly? Because a lot of almost normal distributions Are not normal A common defect is that much engineering Data has a definite range over which it can Physically occur – The normal model goes From plus to minus infinity. This constriction of outer limit data causes Droopy tails on the graph. The slope determines standard deviation. Simple mathematics fitting can over-emphasize tail Heavy Tails Common to have deviations of Physical situations in the tail areas. One of the reasons I cautioned Against doing 99.99% confidence Intervals – those extreme values Are likely to give you garbage. I have found normal models useful If only my outer 5% or less tails Bent off in unwanted directions. (But this also gives you a clue how Far you can go with a confidence Interval). By setting your own model slope you can preferentially fit your model to the central Data you will be more likely to use. The Lognormal Distribution Lognormal Distributions can be normalized By taking the Log of the data (but the transform can have issues in Dealing with real world data). Lognormal Probability Paper Can Allow Read off of mean and Logvariance – (not equal to Traditional variance). Web Site Resources Web site has ability to download normal, lognormal, exponential, and wiebul distribution paper.