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

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! 2

3. Final Exam Marks for PHY131 – Summer 2012 The red curve n max = maximum value = value of m for which n(m) = n max = standard deviation 3

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. 4

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

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

7. The Standard Deviation is a Measure of the Width of the Gaussian 7 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 Deviation is a Measure of the Width of the Gaussian 8 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 the Curve Has an Area = 0.68 9 If you choose one measurement of d i 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 d i

10. Characterising Repeated Measurements as a Gaussian is Almost Always Only an Approximation A true Gaussian only approaches zero at If the number of measurements random 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 10