A Review of BellShaped Curves David M Harrison

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

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

Final Exam Marks for PHY 131 – Summer 2012 The red curve nmax = maximum value result = value of m for which n(m) = n. Fit max = standard deviation 3

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

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. A perfect Gaussian shape B. Still only approximately Gaussian 5

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. 6

The Standard Deviation is 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 7

The Standard Deviation is 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. 8

The Shaded Area Under the Curve 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% is the standard uncertainty u in each individual measurement di E. 100% 9

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