EDEXCEL ANALYTICAL METHODS FOR ENGINEERS H1 UNIT 2 - NQF LEVEL 4 OUTCOME 4 - STATISTICS AND PROBABILITY TUTORIAL 3 LINEAR REGRESSION

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1 EDEXCEL AALYTICAL METHODS FOR EGIEERS H1 UIT - QF LEVEL 4 OUTCOME 4 - STATISTICS AD PROBABILITY TUTORIAL 3 LIEAR REGRESSIO Tabular and graphical form: data collection methods; histograms; bar charts; line diagrams; cumulative frequenc diagrams; scatter plots Central tendenc and dispersion: the concept of central tendenc and variance measurement; mean; median; mode; standard deviation; variance and interquartile range; application to engineering production Regression, linear correlation: determine linear correlation coefficients and regression lines and appl linear regression and product moment correlation to a variet of engineering situations Probabilit: interpretation of probabilit; probabilistic models; empirical variabilit; events and sets; mutuall eclusive events; independent events; conditional probabilit; sample space and probabilit; addition law; product law; Baes theorem Probabilit distributions: discrete and continuous distributions, introduction to the binomial, Poisson and normal distributions; use of the ormal distribution to estimate confidence intervals and use of these confidence intervals to estimate the reliabilit and qualit of appropriate engineering components and sstems D.J.Dunn 1

2 ITRODUCTIO This section is mainl concerned with the collection of data that is thought to have a directl proportional relationship. Suppose there are two measured variables and and it is thought that the are related b a formula m + C (the straight line law). Often it is fairl apparent from the, plot if such a relationship eists but often there is enough scatter in the points to make it doubtful and difficult to decide what the best values of m and C (slope and intercept) should be. FIDIG THE BEST STRAIGHT LIE Much statistical work has been done to tr and find a wa of fitting a mathematical model (equation) to obtain the best possible representation of data found from eperiments or sampling. Let's use as an eample, the results of an eperiment to find the coefficient of friction between materials. The law of dr friction (Coulomb's Laws) applied to sliding objects states that F µ R where F is the friction force, µ is a constant called the coefficient of friction and R is the force squeezing the two surfaces together. If we conduct an eperiment we might have small errors in the instruments for measuring the weights and forces and we might have variations in the surface teture but overall we epect the results to follow the law. We might get a graph like this. Just joining the points is not satisfactor. We can see that it looks as though it should be a straight line graph so the question is how to draw the best straight line that places the data as close to the line as possible. For linear relations, the most widel used method is the method of least squares also called the linear regression method. For an given point (R) the difference between the raw data (F) and the eact point predicted b the law is ε F µr Research has shown that the best fit occurs when the values of ε is a minimum. This is the method of least squares. The reason for using ε and not ε is based around the fact that a negative value squared is positive and produces a better result. For n points the total error is (ε 1 + ε + ε ε n ) and the object is to make ε a minimum. The problem now is to find the value of the constant µ (the gradient) that produces the minimum value of ε and this is a big problem. It would be ver laborious to guess at values and then work out all the values of ε and repeat until we have a value of µ that gives the minimum. That is basicall what regression is. D.J.Dunn

3 Fortunatel these das we have computer programs that will do the work for us and produce the best fit. You will find one such aid at this web address The following is the wa to use Microsoft Ecel to do it. The following set of instructions enables ou to create a best fit and displa the best equation for an data plot using Ecel. 1. Create our raw data plot in two columns (ideall with the values in the first column).. Highlight all the cells b clicking and dragging. If the data ou want to highlight is not in two adjacent columns highlight them separatel b holding down the Ctrl ke as ou drag first one then the other. 3. Click on Insert on the menu bar. 4. Click on Chart Under Standard Tpes, Chart tpe: click on XY (Scatter). 6. Under Chart sub-tpe: click on the chart with onl data markers and no lines. 7. Click on et>. 8. Click on et>. 9. Under Titles, Click in the tet bo under Chart title: and enter a title for the graph. 10. Click in the tet bo under Categor (X) ais: and enter a title for the -ais. Click in the tet bo under Value (Y) ais: and enter a title for the -ais. 11. Click on the Gridlines tab. Click in the checkboes to turn gridlines on or off. 1. Click on the Legend tab. Click in the checkbo net to Show legend to turn the legend on or off. Placement of the legend on the graph can also be selected here. 13. Click on the Data Labels tab. Click on the radio buttons to turn data labels on or off. 14. Click on the Data Table tab. Click in the checkboes to turn a data table on or off. 15. Click on et. 16. Under Place chart: click on the button net to As new sheet. Enter a name for the graph. 17. Click on Finish. Having created a data plot the following instructions are to create a best fit graph and print the function. 1. Move the mouse cursor to an data point and left click. All of the data points should now be highlighted. Right click and click on Add Trendline.. From within the Add Trendline window, under Tpe, click on the bo with the tpe of fit ou want (e.g., Linear). 3. Click on Options at the top of the Add Trendline window. 4. Click in the checkbo net to Displa equation on chart and the checkbo net to Displa R-squared value on chart. 5. Click on OK. D.J.Dunn 3

4 WORKED EXAMPLE o. 1 In an eperiment to find the coefficient of friction, the force F and normal load R was measured and the following results obtained. F R Using ecel, produce the best linear graph and the function produced. What is the coefficient of friction? SOLUTIO Plotting according to the above instructions we get the following. COEFFICIET OF FRICTIO R , , 8 F () 6 7, 6 4 1, 4 11, 0 0, R() Series1 Linear (Series1) The function is F R The best plot does not pass through zero but this would have been an option in the last instruction. The coefficient of friction is the gradient D.J.Dunn 4

5 UDER LAYIG THEORY OF LIEAR REGRESSIO EQUATIOS If ou don't want to understand the theor, go straight to the net page. ormall a straight line function is of the form m + C where C is the intercept and m the gradient. If we were tring to fit the data to this law, we would have two constants m and C to determine so the work becomes even more comple. ε - (m + C) where is the raw data. ε ( - m - C) m -C - m + Cm + C Σ(ε) Σ( m -C - m + Cm + C ) Σ(ε) Σ - mσ - CΣ - m Σ + Cm Σ + ΣC Σ(ε) Σ - mσ - CΣ - m Σ + Cm Σ + C is the number of data points. The problem now is to find the minimum value. Π Σ - mσ - CΣ - m Σ + Cm Σ + C dπ For the value of m that makes Π a minimum 0 -Σ -mσ + C Σ 0 dm C m...(1) dπ For the value of C that makes Π a minimum 0 - Σ + m Σ + C dc - m C...() Substitute () into (1) m - m { } m m m 1 { } m + { } { } m 1 { } { } With further work () gives ( ) C but is easier to use () once m is found. D.J.Dunn 5

6 The method of linear regression for the best straight line gives: Gradient ( ) m Intercept b C ( ) Here is a little eercise ou don't find in tet books that eplains the net simplification. WORKED EXAMPLE o. Prove that ( - )( ) SOLUTIO ( - )( ) [ ] ( - )( ) [ ] ( - )( ) [ - - ] + The mean value of the and samples are respectivel ( )( ) - ( - )( ) and Substitute these ( )( ) WORKED EXAMPLE o. 3 Prove that ( ) ( ) SOLUTIO ( ) [ + ] ( ) + ( ) + ( ) + ( ) ( ) Substitute ( ) ( ) Using the above simplification we now have m ( - )( ) ( ) D.J.Dunn 6

7 WORKED EXAMPLE o. 4 Check out the answers from worked eample o.1 using the formulae to calculate m and C. Prove that ( - )( ) m gives the same result as m ( ) F R ( ) Using ecel, produce the best linear graph and the function produced. What is the coefficient of friction? SOLUTIO n - - (- )( - ) (- ) Total Mean ( 113) ( 154)( 30) 17 m ( ) 6( 5844) ( - )( ) 36 m Both formulae give the same answer. ( ) m 30 (0.1914)(154) C Or ( ) (30)(5844) (154)(113) (6)(5844) 154 C SELF ASSESSMET EXERCISE o In an eperiment to verif Ohm's Law, the voltage and current acting on a resistor was measured and the following results obtained. V I Assuming the law should be linear, find the best fit straight line law and check the answers with using Ecel. (Calculation gives m 10.73, C Ecel gives same answers) D.J.Dunn 7

8 LIEAR CORRELATIO COEFFICIET The linear correlation coefficient is a wa to measure how good a linear relationship would be without gouing through all the preceding work. Let's suppose we are eamining a scatter plot and we are tring to work out the best straight line to use. The linear correlation coefficient (r or R) is a number between -1 and 1 which measures how close to a straight line a set of points falls. A zero value means the widest scatter with no relationship. A value of 1 is a perfect correlation with a positive slope. A value of -1 is a perfect correlation with a negative slope. Other values in between represent various degrees of scatter. ote that the R value can be produced b ecel using the option tab and this is printed on the plot in worked eample 1. The diagrams illustrate correlation. Without proof the formula for r is r This is more easil understood as r ( ( ) ) ( ) standard deviation of the values. is the product moment. ( ) ( ) ( ) where is the standard deviation of the values, is the ( )( ) This works best when and are both normall distributed. The following eplains the process. CALCULATIG THE CORRELATIO COEFFICIET: Suppose we have sets of data, ( 1, 1 ), (, )... ( n, n ) STEP 1. Calculate the average of the and vales and ( )/ ( )/ STEP. Calculate the standard deviations for and ( and ). ( ) + ( ) + ( ) +...( ) 1 ( ) + ( ) + ( ) +...( ) n n D.J.Dunn 8

9 STEP 3 STEP 4 Calculate the covariance between the two data sets: [( )( ) + ( )( ) + ( )( ) +...( )( ) ] 1 1 The correlation coefficient is then defined as 3 3 n r ote some tet give -1 for the calculations of standard deviations but since all three terms contain the same it makes no difference to the answer. n WORKED EXAMPLE o. 5 Calculate the correlation coefficient for the following and data sets. n SOLUTIO Calculate and ( )/5 3.0 ( )/5.8 Calculate and, ( 1 3) + ( 3) + ( 3 3) + ( 4 3) + ( 5 3) ( 4.8) + ( 4.8) + ( 3.8) + (.8) + ( 1.8) Calculate the covariance [( 1 3)( 4.8) + ( 3)( 4.8) + ( 3 3)( 3.8) + ( 4 3)(.8) + ( 5 3)( 1.8) ] [( )( 1.) + ( 1)( 1.) + ( 0)( 0.) + ( 1)( 0.8) + ( )( 1.8) ] [ ] Calculate r, the correlation coefficient. 1.6 r ( )( ) 5 5 D.J.Dunn 9

10 SELF ASSESSMET EXERCISE o. 1. Calculate the correlation coefficient for the following and data sets and establish the best straight line law. n (Answer r ). Calculate the correlation coefficient for the following and data sets and establish the best straight line law. n (Answer r ). Calculate the correlation coefficient for the following and data sets and establish the best straight line law. n (Answer r ) D.J.Dunn 10

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