# Idea weigh the points with the inverse of their error

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Idea: weigh the points with the ~ inverse of their error bar 0 5 10 15 20 25 x 0 10 20 30 y(x)

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Weight-adjusted average: How do we average values with different uncertainties? Student A measured resistance 100±1 Ω (x 1 =100 Ω , σ 1 =1 Ω ) Student B measured resistance 105±5 Ω (x 2 =105 Ω , σ 2 =5 Ω ) 2 1 2 2 1 1 w w x w x w x + + = 2 1 1 1 σ = w 2 2 2 1 σ = w N N N i i i w w w x w x w x w w x w x + + + + + + = = ... ... 2 1 2 2 1 1 Or in this case calculate for i=1, 2: with “statistical” weights: BOTTOM LINE: More precise measurements get weighed more heavily!
0 5 10 15 20 25 x 0 10 20 30 y(x) How good is the agreement between theory and data? χ 2 TEST for FIT (Ch.12) ( ) ( ) = = N j j j j x f y 1 2 2 2 σ χ

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0 5 10 15 20 25 x 0 10 20 30 y(x) χ 2 TEST for FIT (Ch.12) N N y y = 2 2 σ σ d 2 2 ~ χ χ = d = N - c # of degrees of freedom # of data points # of parameters calculated from data # of constraints 1 ( ) ( ) = = N j j j j x f y 1 2 2 2 σ χ (Example: You can always draw a perfect line through 2 points)
0 5 0 10 20 30 y(x) y 3 -(A+Bx 3 ) y 4 -(A+Bx 4 ) LEAST SQUARES FITTING 1. 2. Minimize χ 2 : 0 2 = A χ 0 2 = B χ 3. à A in terms of x j y j ; B in terms of x j y j , … 4. Calculate χ 2 5. Calculate d χ χ = 2 0 ~ 6.

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