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Level 1 Laboratories A Rough Summary of Key Error Formulae for samples of random data. For details see Physics Lab Handbook (section 4.5.2 to 4.5.4) 1 Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2008 Normal, or Gaussian, distribution – a “bell-shaped” curve Frequency Mean x Standard Deviation 68.3% of area under curve 1 n x xi n i 1 ( 4 1) 1 n lim xi x 2 n n i 1 (4 5) 2 Quantity x (e.g. rebound height) 2 Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2008 Normal, or Gaussian, distribution – a “bell-shaped” curve Frequency Mean x Standard Deviation 68.3% of area under curve Quantity x (e.g. rebound height) 1 n x xi n i 1 ( 4 1) 1 n lim xi x 2 n n i 1 (4 5) 1 n 2 x x i n 1 i 1 (4 6) 2 s 2 n 1 In reality, only a finite amount of measurements can be made. If we then plotted a histogram, it would only approximate the true Normal Distribution. Nevertheless, we can still estimate the mean and standard deviation. 3 Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2008 Normal, or Gaussian, distribution – a “bell-shaped” curve Frequency Mean x 1 n x xi n i 1 ( 4 1) 1 n lim xi x 2 n n i 1 (4 5) 1 n 2 x x i n 1 i 1 (4 6) Standard Deviation 2 68.3% of area under curve s 2 n 1 Quantity x (e.g. rebound height) Note : most simple calculators will provide x and Then one should quote the final result as : sn 1 (often shown on calc. as “xn-1” ) x sm (4 9) sn 1 (4 8) Where s m is called the “Standard Error in the Mean”, given by : sm n 4 Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2008 Example calculation of n n 1 1 s 1 2 x 8) smxi xi( 4 x , sn 1 & n i n1 1 i 1n n 1 n 1 x xi 4) 6) smn i 1 1 ( 4 (1 Suppose we have a set of 10 measurements of nominally the same thing (e.g. bounce height): 64, 66, 68, 70, 72, 68, 72, 70, 71 and 70 cm • Mean s n 1 1 n 1 10 x xi xi 69.1 cm n i 1 10 i 1 • Variance (or Mean Squared Deviation) • Standard Deviation • Standard Error (in the Mean) s 2 n 1 1 n 1 9 2 xi x xi 69.12 6.8 cm 2 n 1 i 1 9 i 1 sn1 6.8 2.6 cm (same as “xn-1” on Casio calculators) 2.6 sn1 sm 0.82 cm n 10 • Quote final result as Mean ± Standard Error : i.e. 5 x sm 69.1 0.8 cm Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2008