1. Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The Parallelogram & Related Methods 1

2. 0 5 10 15 20 051015 x y Example of Parallelogram Method to obtain errors in gradient & intercept Figure 1 :  Give the graph a title to which you can refer ! & Add a descriptive caption. Blah, blah …… (dimensionless) Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 2

3. 0 5 10 15 20 051015 x (dimensionless) y (dimensionless) Estimate ”best fit” line gradient m [ e.g. here m = 0.59 ] intercept c [ e.g. here c = 6.5 ] Example of Parallelogram Method to obtain errors in gradient & intercept Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 3

4. 0 5 10 15 20 051015 x (dimensionless) y (dimensionless) Draw 2 lines parallel to best line so as to enclose roughly 2/3 of data points Example of Parallelogram Method to obtain errors in gradient & intercept Why 2/3? - It’s because we assume the data obey a Normal Distribution in which there is a 68.3% (  66.7% = 2/3) confidence that the “true” value lies within  of the measured value 22 68.3% of area under Normal curve Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 4

5. 0 5 10 15 20 051015 x (dimensionless) Draw “extreme lines” between opposite corners of parallelogram Max gradient m H Min intercept C L Min gradient m L Max intercept C H Example of Parallelogram Method to obtain errors in gradient & intercept Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 5

6. 0 5 10 15 20 051015 x (dimensionless) Draw “extreme lines” between opposite corners of parallelogram Max gradient m H Min intercept C L Min gradient m L Max intercept C H Final Results including errors Example of Parallelogram Method to obtain errors in gradient & intercept Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 6

7. 0 5 10 15 20 051015 x (dimensionless) y (dimensionless) Add expt. “error bars” Example of Parallelogram Method to obtain errors in gradient & intercept Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 NB : these error bars are estimated from the scatter in the data. Here, they play no part in getting the errors in the gradient and intercept. 7

8. Recommended final appearance of graph for Diary or Reports, if using Parallelogram Method Figure 1 : Descriptive caption. Blah, blah …… 0 5 10 15 20 051015 x (dimensionless) y (dimensionless) Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 8

9. What if the scatter is so small that it is difficult to draw the max. and min. lines by eye? Method 2 : Modified Parallelogram Method Plot the experimental points (x, y data ) x y 9

10. What if the scatter is so small that it is difficult to draw the max. and min. lines by eye? Method 2 : Modified Parallelogram Method Plot the experimental points (x, y data ) Draw the best fit line and determine equation y fit = m 1 x + c 1 x y 10

11. Draw the best fit line and determine equation y fit = m 1 x + c 1 What if the scatter is so small that it is difficult to draw the max. and min. lines by eye? Method 2 : Modified Parallelogram Method Plot the experimental points (x, y data ) x y 0 The small difference (y data – y fit ) will be dominated by the random scatter 11

12. x ( y data – y fit ) 0 Replot (y data – y fit ) on an expanded scale. Draw the best fit line and determine equation y fit = m 1 x + c 1 What if the scatter is so small that it is difficult to draw the max. and min. lines by eye? Method 2 : Modified Parallelogram Method Plot the experimental points (x, y data ) 0 The small difference (y data – y fit ) will be dominated by the random scatter x y 12

13. Draw the best fit line and determine equation y fit = m 1 x + c 1 What if the scatter is so small that it is difficult to draw the max. and min. lines by eye? Method 2 : Modified Parallelogram Method Plot the experimental points (x, y data ) x y 0 The small difference (y data – y fit ) will be dominated by the random scatter x ( y data – y fit ) 0 Replot (y data – y fit ) on an expanded scale. Fit the best line (y data – y fit ) = m 2 x + c 2 13

14. Draw the best fit line and determine equation y fit = m 1 x + c 1 What if the scatter is so small that it is difficult to draw the max. and min. lines by eye? Method 2 : Modified Parallelogram Method Plot the experimental points (x, y data ) x y 0 The small difference (y data – y fit ) will be dominated by the random scatter x ( y data – y fit ) 0 Replot (y data – y fit ) on an expanded scale. Fit the best line (y data – y fit ) = m 2 x + c 2 Form the parallelogram enclosing 2/3 of points 14

15. Draw the best fit line and determine equation y fit = m 1 x + c 1 What if the scatter is so small that it is difficult to draw the max. and min. lines by eye? Method 2 : Modified Parallelogram Method Plot the experimental points (x, y data ) x y 0 The small difference (y data – y fit ) will be dominated by the random scatter x ( y data – y fit ) 0 Replot (y data – y fit ) on an expanded scale. Fit the best line (y data – y fit ) = m 2 x + c 2 Use the extreme lines to find the max and min values of m 2 and c 2 Form the parallelogram enclosing 2/3 of points 15

16. Replot (y data – y fit ) on an expanded scale. Fit the best line (y data – y fit ) = m 2 x + c 2 Draw the best fit line and determine equation y fit = m 1 x + c 1 What if the scatter is so small that it is difficult to draw the max. and min. lines by eye? Method 2 : Modified Parallelogram Method Plot the experimental points (x, y data ) x y 0 The small difference (y data – y fit ) will be dominated by the random scatter x ( y data – y fit ) 0 Use the extreme lines to find the max and min values of m 2 and c 2 Form the parallelogram enclosing 2/3 of points Final Results including errors 16

17. There is another method commonly used when the scatter is small Method 3 : Using Predetermined Error Bars (e.g. by multiple measurements at a single value of x) x y 0 Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 17

18. Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Add predetermined error bars to the plotted points x y 0 There is another method commonly used when the scatter is small Method 3 : Using Predetermined Error Bars (e.g. by multiple measurements at a single value of x) 18

19. Add predetermined error bars to the plotted points x y Draw best line through points (if predetermined error is consistent with the scatter in the points, the line should go through ~2/3 of error bars & miss remaining ~1/3 ) 0 There is another method commonly used when the scatter is small Method 3 : Using Predetermined Error Bars (e.g. by multiple measurements at a single value of x) Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 19

20. Add predetermined error bars to the plotted points x y Put in extreme lines so as to still pass through ~2/3 of error bars Draw best line through points (if predetermined error is consistent with the scatter in the points, the line should go through ~2/3 of error bars & miss remaining ~1/3 ) 0 There is another method commonly used when the scatter is small Method 3 : Using Predetermined Error Bars (e.g. by multiple measurements at a single value of x) Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 20

21. Add predetermined error bars to the plotted points Put in extreme lines so as to still pass through ~2/3 of error bars Draw best line through points (if predetermined error is consistent with the scatter in the points, the line should go through ~2/3 of error bars & miss remaining ~1/3 ) There is another method commonly used when the scatter is small Method 3 : Using Predetermined Error Bars (e.g. by multiple measurements at a single value of x) Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 x y 0 Final Results including errors 21

22. 1.If the “scatter” in plotted data looks different to the size of error bars (much smaller or larger), something has gone wrong! 2.Example shows all y-axis error bars of same length. This might not be true in any given case, so do not assume this unless you have confirmed it! 3.Example also shows no error bars on the horizontal x-axis : there might be errors in this direction too! Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Caution with Method 3 x y 0 22