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A Review of Bell-Shaped Curves

A Review of Bell-Shaped Curves. David M. Harrison, Dept. of Physics, Univ. of Toronto, May 2014. A Perhaps Apocryphal Story. In the early 1800’s Gauss’ “graduate students” were doing astronomical measurements When they repeated the measurements, they didn’t give exactly the same values

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A Review of Bell-Shaped Curves

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  1. A Review of Bell-Shaped Curves David M. Harrison, Dept. of Physics, Univ. of Toronto, May 2014

  2. A Perhaps Apocryphal Story • In the early 1800’s Gauss’ “graduate students” were doing astronomical measurements • When they repeated the measurements, they didn’t give exactly the same values • Gauss said they were incompetent, and stormed into the observatory to show them how it should be done • Gauss’ repeated measurements didn’t give exactly the same values either!

  3. Final Exam Marks forPHY131 – Summer 2012 The red curve nmax= maximum value = value of m for which n(m) = nmax = standard deviation Fit result

  4. Another Approximately Bell-Shaped Curve: a Quincunx • Bell-shaped curve aka • Gaussian aka • Normal distribution The Gaussian describes the probability that a particular ball will land at a particular position: it is a probability distribution function.

  5. Another Approximately Bell-Shaped Curve: a Quincunx For a finite number n of balls, their distribution is only approximately Gaussian If you use balls their distribution will be: A perfect Gaussian shape Still only approximately Gaussian

  6. Repeat of an Earlier Slide:Another Bell-Shaped Curve: a Quincunx • Bell-shaped curve • Gaussian • Normal distribution approximately The Gaussian describes the probability that a particular ball will land at a particular position: it is a probability distribution function.

  7. The Standard Deviationis a Measure of the Width of the Gaussian All probability distribution functions must have a total area under them of exactly 1 These two curves are properly normalised: the area under each is = 1

  8. The Standard Deviationis a Measure of the Width of the Gaussian Physical scientists tend to characterise the width of a distribution by the standard deviation. Social scientists instead often use the variance.

  9. The Shaded Area Under theCurve Has an Area = 0.68 If you choose one measurement of di at random, the probability that it is within of the true value is: A. 0 B. 68% C. 95% D. 99% E. 100% is the standard uncertainty u in each individual measurement di

  10. Characterising Repeated Measurements as a Gaussian is Almost Always Only an Approximation • A true Gaussian only approaches zero at • If the number of measurementsrandom fluctuations mean that the measured values can be too high, or too low, or too scattered, or not scattered enough • Therefore, we may only estimate the mean and the standard deviation

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