Physics 270 – Experimental Physics

Standard Deviation of the Mean (Standard Error) When we report the average value of n measurements, the uncertainty we should associate with this average value is the standard deviation of the mean, often called the standard error (SE): This reflects the fact that we expect the uncertainty of the average value to get smaller when we use a larger number of measurements n.

The Gaussian distribution works well for any random variable because of the Central Limit Theorem. A simple description of it is… When data that are influenced by many small and unrelated random effects, it will be approximately normally distributed. Let Y 1, Y 2, … Y n be an infinite sequence of independent random variables, usually from the same probability distribution function, but it could be different pdf’s.

Suppose that the mean, µ, and the variance  2 are both finite. For any two numbers a and b… CLT tells us that under a wide range of circumstances the probability distribution that describes the sum of random variables tends towards a Gaussian distribution as the number of terms in the sum → ∞.

Random number generator gives numbers distributed uniformly in the interval [0,1] µ = 1/2 and σ 2 = 1/12 Take 12 numbers, add them together, then subtract 6. You get a number that looks as if it is from a Gaussian pdf!

Take 12 numbers, add them together, then subtract 6…

 Some statements are exact…  Jessica has 6 Webkinz.  = 9  All measurements have uncertainty.  “uncertainty” versus “error”  measurement = best estimate ± uncertainty  Tennis ball example Measurement = (measured value ± standard uncertainty) unit of measurement

 Random errors are statistical fluctuations (in either direction) in the measured data due to the precision limitations of the measurement device. Random errors can be evaluated through statistical analysis and can be reduced by averaging over a large number of observations.  Systematic errors are reproducible inaccuracies that are consistently in the same direction. These errors are difficult to detect and cannot be analyzed statistically. If a systematic error is identified when calibrating against a standard, the bias can be reduced by applying a correction or correction factor to compensate for the effect. Unlike random errors, systematic errors cannot be detected or reduced by increasing the number of observations.

= 50 grams = 20 grams = 5 grams 70 g ≤ mass ≤ 80 g mass = 75 ± 5 g

 more precise  more accurate?  ± 0.1 g  m = 74.6 ± 0.1 g  m = 74.6 ± 0.2 g  two 200 g calibration masses included

 results of one experiment  results of many experiments  National Institute Standards and Technology ◦  other international organizations

 Outliers ◦ Could be significant or insignificant. ◦ Don’t just throw away data.