Physics307L:Schedule/Week 4 agenda
Agenda for Lecture (items will be continued next week)
- Note about O-scope labs
- Note about labs this week and next
- Introduction to terminology
- Mean +/- error
- Probability density function
- Histogram (maybe use height distribution)?
- Normal Distribution
- Central Limit Theorem
- http://en.wikipedia.org/wiki/Illustration_of_the_central_limit_theorem
- downloadable simulation: http://www.vias.org/simulations/simusoft_cenlimit.html
- Java dice rolling: http://www.stat.sc.edu/~west/javahtml/CLT.html
Agenda for Labs
- Spend about 2 hours figuring out how to set up the equipment and planning out the experiments. Record your goals for the experiments in your lab notebook.
- Partnering up is not required, but is permitted this week.
Scraps
Uncertainty
This is copied directly from wikipedia: http://en.wikipedia.org/wiki/Uncertainty
Relation between uncertainty, accuracy, precision, standard deviation, standard error, and confidence interval
The uncertainty of a measurement is stated by giving a range of values which are likely to enclose the true value. This may be denoted by error bars on a graph, or as value ± uncertainty, or as decimal fraction(uncertainty). The latter "concise notation" is used for example by IUPAC in stating the atomic mass of elements. There, 1.00794(7) stands for 1.00794 ± 0.00007.
Often, the uncertainty of a measurement is found by repeating the measurement enough times to get a good estimate of the standard deviation of the values. Then, any single value has an uncertainty equal to the standard deviation. However, if the values are averaged and the mean is reported, then the averaged measurement has uncertainty equal to the standard error which is the standard deviation divided by the square root of the number of measurements.
When the uncertainty represents the standard error of the measurement, then about 68.2% of the time, the true value of the measured quantity falls within the stated uncertainty range. For example, it is likely that for 31.8% of the atomic mass values given on the list of elements by atomic mass, the true value lies outside of the stated range. If the width of the interval is doubled, then probably only 4.6% of the true values lie outside the doubled interval, and if the width is tripled, probably only 0.3% lie outside. These values follow from the properties of the normal distribution, and they apply only if the measurement process produces normally distributed errors. In that case, the quoted standard errors are easily converted to 68.2% ("one sigma"), 95.4% ("two sigma"), or 99.7% ("three sigma") confidence intervals.
Normal Distribution
Copied from: http://en.wikibooks.org/wiki/Probability:Important_Distributions
Normal Distribution
The normal or Gaussian distribution is a thing of beauty, appearing in many places in nature. This probably is a result of the normal distribution resulting from the law of large numbers, by which a sum of many random variables (with finite variance) becomes a normally distributed random variable according to the central limit theorem.
Also known as the bell curve, the normal distribution has been applied to many social situations, but it should be noted that its applicability is generally related to how well or how poorly the situation satisfies the property mentioned above, whereby many finitely varying, random inputs result in a normal output.
The formula for the density f of a normal distribution with mean [math]\displaystyle{ \mu }[/math] and standard deviation [math]\displaystyle{ \sigma }[/math] is
- [math]\displaystyle{ f(x) = \frac{1}{\sigma\sqrt{2\pi}} \, e^{ -\frac 12 \left( \frac{x- \mu}{\sigma}\right)^2 } }[/math].
A rather thorough article in Wikipedia could be summarized to extend the usefulness of this book: Normal distribution from Wikipedia.
