Practical Basics of Statistical Analysis
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1 Practical Basics of Statistical Aalysis David Keffer Dept. of Materials Sciece & Egieerig The Uiversity of Teessee Koxville, TN Goveror s School Uiversity of Teessee, Koxville Jue 14, 2018
2 Purpose Use a materials example to explore basic statistical tools icludig Mea Stadard deviatio Stadard error Histograms Regressio
3 Material Properties have a Probability Distributio Example: Cosider a property like the strai at which fracture occurs i a compoet. stress Stress vs. strai curve typical of alumium 1. Ultimate tesile stregth 2. Yield stregth 3. Proportioal limit stress 4. Fracture 5. Offset strai (typically 0.2%) strai Your Task: Determie the strai at fracture.
4 Material Properties have a Probability Distributio What if two shifts use differet heat treatig procedures resultig i compoets with two differet fracture strais? The distributio of fracture strais could look somethig like this: This true probability distributio is ukow! How ca you ivestigate it?
5 Samplig Iformatio about the distributio of a property ca be obtaied from samplig. The Quality Assurace (QA) egieer tests a umber of compoets ad records the strai at fracture. A mea or average ca be evaluated. x x x x x x i1 x i sample # measuremet mea
6 Mea Value The estimate of the mea gets better with more sample poits mea
7 Mea Value Eve a fairly accurate mea, calculated with a lot of sample poits, ca t reveal the shape of the uderlyig distributio. We might like more iformatio tha the mea provides.
8 Stadard Deviatio The stadard deviatio provides the lowest order descriptio of the distributio of the data aroud the mea. s s x x i i1 2 mea stadard mea deviatio
9 Stadard Deviatio The estimate of the stadard deviatio reaches a costat with more sample poits. stadard mea deviatio x s
10 Stadard Deviatio The stadard deviatio i this example reflects a broad distributio of possible results. We might like more iformatio tha the mea ad stadard deviatio provide.
11 Distributio of the Sample Mea The cetral limit theorem states that, give certai coditios, the arithmetic mea of a sufficietly large umber of iterates of idepedet radom variables, each with a well-defied expected value ad well-defied variace, will be approximately ormally distributed. ormal distributio
12 Stadard Error The stadard deviatio provides a descriptio of the distributio of the data aroud the mea. SE s mea mea stadard deviatio stadard error
13 Stadard Error The stadard error is a measure of ucertaity i the sample mea. The stadard error becomes smaller with more sample poits stadard mea error x SE
14 Stadard Error The stadard error represets your ucertaity i the sample mea, but does ot tell you much about the actual distributio.. We might like more iformatio tha the mea ad stadard error provide.
15 Histograms Iformatio about the distributio of a property ca be obtaied from samplig. The Quality Assurace (QA) egieer tests a umber of compoets ad records the strai at fracture. A histogram ca be created. sample # measuremet
16 Histograms become more accurate with more samplig
17 Regressio A liear regressio provides the coefficiets for a liear model relatig a depedet ad idepedet variable. y mx b Cosider the strai at fracture for a series of compoets i which the heat treatmet time is varied. f mt b If we ca fid the missig coefficiets (slope ad itercept) the we ca use them to predict the strai at fracture for a give heat treatmet time. sample treatmet time strai at fracture
18 Regressio Frequetly, the results of a regressio are preseted as a plot. The R 2 Measure of Fit is boud betwee 0 (o fit) ad 1 (perfect fit).
19 Documet Access These slides ad a sample excel file presetig examples for Mea Stadard deviatio Stadard error Histograms Regressio are located olie at
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