Central Limit Theorem and Introduction to Uncertainty

Physics 270 – Experimental Physics

Standard DeviationTo calculate the standard deviation for a sample of 5 (or more generally n) measurements: 1. Sum all the measurements and divide by 5 (n) to get the average or mean, . 2. Now, subtract this average from each of the 5 (n) measurements to obtain 5 "deviations." 3. Square each of these 5 (n) deviations and add them all up. 4. Divide this result by 5 (n), and take the square root.

Example: Width of sheets of paper

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.

Example: Width of sheets of paper

0.10 0.0455x

sn

31.19 0.05 cmw

Central Limit TheoremThe 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 Y1, Y2, … Yn be an infinite sequence of independent random variables, usually from the same probability distribution function, but it could be different pdf’s.

Central Limit TheoremSuppose 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 distributionas the number of terms in the sum → ∞.

Example: Computer Random NumbersRandom number generator gives numbers distributed uniformly in the interval [0,1]

µ = 1/2 and σ2 = 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!

Example: Computer Random NumbersTake 12 numbers, add them together, then subtract 6…

Some statements are exact… Jessica has 6 Webkinz. 7 + 2 = 9

All measurements have uncertainty. “uncertainty” versus “error” measurement = best estimate ± uncertainty

Tennis ball example

Uncertainty of Measurements

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.

Types of Errors

Example: Mass Determination

= 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

Example: Mass Determination

0.0074.6

results of one experiment results of many experiments National Institute Standards and Technology

◦ http://physics.nist.gov/cuu/Constants/ other international organizations

Textbook Values

Outliers

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

Anomalous Data