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## Experiment

Experiment is the step in the scientific method that arbitrates between competing models or hypotheses.[1][2] Experimentation is also used to test existing theories or new hypotheses in order to support them or disprove them.[3][4] An experiment or test can be carried out using the scientific method to answer a question or investigate a problem. First an observation is made. Then a question is asked, or a problem arises. Next, a hypothesis is formed. Then experiment is used to test that hypothesis. The results are analyzed, a conclusion is drawn, sometimes a theory is formed, and results are communicated through research papers.

A good experiment usually tests a hypothesis. However, an experiment may also test a question or test previous results.

• Replication of results is "a standard procedure in the validation of any scientific discovery." [4]
• "Science was long protected from fraud by a built-in safety mechanism: to be generally accepted, experiments must be repeatable by others." [5]

It is important that one knows all factors in an experiment. It is also important that the results are as accurate as possible. If an experiment is carefully conducted, the results usually either support or disprove the hypothesis. An experiment can never "prove" a hypothesis, it can only add support. However, one repeatable experiment that provides a counterexample can disprove a theory or hypothesis. An experiment must also control the possible confounding factors -- any factors that would mar the accuracy or repeatability of the experiment or the ability to interpret the results.

• "... the results of an experiment can never uniquely identify the explanation. They can only split the range of available models into two groups, those that are consistent with the results and those that aren't."[5]

Experiments are not the only method that scientists use to test hypotheses. An experiment usually refers to observations in which conditions are artificially controlled and manipulated by the experimenter to eliminate extraneous factors, often in a scientific laboratory. Information about nature is also gathered and hypotheses tested in observational studies and field studies, which are observations of phenomena in a natural setting, without control by the experimenter.

## Types of experiments

### Controlled experiments

To demonstrate a cause and effect hypothesis, an experiment must often show that, for example, a phenomenon occurs after a certain treatment is given to a subject, and that the phenomenon does not occur in the absence of the treatment. (See Baconian method.)

Standard curve

A controlled experiment generally compares the results obtained from an experimental sample against a control sample, which is practically identical to the experimental sample except for the one aspect whose effect is being tested (the independent variable). A good example would be a drug trial. The sample or group receiving the drug would be the experimental one; and the one receiving the placebo would be the control one. In many laboratory experiments it is good practice to have several replicate samples for the test being performed and have both a positive control and a negative control. The results from replicate samples can often be averaged, or if one of the replicates is obviously inconsistent with the results from the other samples, it can be discarded as being the result of an experimental error (some step of the test procedure may have been mistakenly omitted for that sample). Most often, tests are done in duplicate or triplicate. A positive control is a procedure that is very similar to the actual experimental test but which is known from previous experience to give a positive result. A negative control is known to give a negative result. The positive control confirms that the basic conditions of the experiment were able to produce a positive result, even if none of the actual experimental samples produce a positive result. The negative control demonstrates the base-line result obtained when a test does not produce a measurable positive result; often the value of the negative control is treated as a "background" value to be subtracted from the test sample results. Sometimes the positive control takes the quadrant of a standard curve.

An example that is often used in teaching laboratories is a controlled protein assay. Students might be given a fluid sample containing an unknown (to the student) amount of protein. It is their job to correctly perform a controlled experiment in which they determine the concentration of protein in fluid sample (usually called the "unknown sample"). The teaching lab would be equipped with a protein standard solution with a known protein concentration. Students could make several positive control samples containing various dilutions of the protein standard. Negative control samples would contain all of the reagents for the protein assay but no protein. In this example, all samples are performed in duplicate. The assay is a colorimetric assay in which a spectrophotometer can measure the amount of protein in samples by detecting a colored complex formed by the interaction of protein molecules and molecules of an added dye. In the illustration, the results for the diluted test samples can be compared to the results of the standard curve (the blue line in the illustration) in order to determine an estimate of the amount of protein in the unknown sample.

Controlled experiments can be performed when it is difficult to exactly control all the conditions in an experiment. In this case, the experiment begins by creating two or more sample groups that are probabilistically equivalent, which means that measurements of traits should be similar among the groups and that the groups should respond in the same manner if given the same treatment. This equivalency is determined by statistical methods that take into account the amount of variation between individuals and the number of individuals in each group. In fields such as microbiology and chemistry, where there is very little variation between individuals and the group size is easily in the millions, these statistical methods are often bypassed and simply splitting a solution into equal parts is assumed to produce identical sample groups.

Once equivalent groups have been formed, the experimenter tries to treat them identically except for the one variable that he or she wishes to isolate. Human experimentation requires special safeguards against outside variables such as the placebo effect. Such experiments are generally double blind, meaning that neither the volunteer nor the researcher knows which individuals are in the control group or the experimental group until after all of the data have been collected. This ensures that any effects on the volunteer are due to the treatment itself and are not a response to the knowledge that he is being treated.

In human experiments, a subject (person) may be given a stimulus to which he or she should respond. The goal of the experiment is to measure the response to a given stimulus by a test method.

Original map by John Snow showing the clusters of cholera cases in the London epidemic of 1854

### Natural experiments

A natural experiment is an observational study in which the assignment of treatments to subjects has been haphazard: That is, the assignment of treatments to subjects has not been made by experimenters (and certainly not by randomization). Natural experiments are most useful when there has been a clearly defined and large change in the treatment (or exposure) to a clearly defined subpopulation, so that changes in responses may be plausibly attributed to the change in treatments (or exposure). Natural experiments are considered for study designs whenever controlled experimentation is difficult, such as in epidemiology and economics.

One of the most famous natural experiments was the 1854 Broad Street cholera outbreak in London, England. On 31 August 1854, a major outbreak of cholera struck Soho. Over the next three days 127 people near Broad Street died. By the end of the outbreak 616 people died. The physician John Snow identified the source of the outbreak as the nearest public water pump, which he identified using a map of deaths and illness. In this example, Snow discovered a strong association between the use of the water and deaths and illnesses due to cholera. Snow found that the water company (the Southwark and Vauxhall Company) that supplied water to districts with high attack rates obtained the water from the Thames downstream from where raw sewage was discharged into the river. By contrast, districts that were supplied water by the Lambeth Company, which obtained water upstream from the points of sewage discharge, had low attack rates. The water supply in mid-Nineteenth Century London wsa not developed by scientists studying cholera, and so exposure to this well may be considered a haphazard event. [6] Therefore, this exposure has been recognized as being a natural experiment. [7][8]

### Field experiments

Field experiments are so named in order to draw a contrast with laboratory experiments. Often used in the social sciences, and especially in economic analyses of education and health interventions, field experiments have the advantage that outcomes are observed in a natural setting rather than in a contrived laboratory environment. However, like natural experiments, field experiments suffer from the possibility of contamination: experimental conditions can be controlled with more precision and certainty in the lab.

## History

### Francis Bacon

Francis Bacon was an English philosopher and scientist in the 1600s and an early and influential supporter of experimental science. He disagreed with the method of answering scientific questions by deduction and described it as follows: âHaving first determined the question according to his will, man then resorts to experience, and bending her to conformity with his placets, leads her about like a captive in a procession.â[9] Bacon wanted a method that relied on repeatable observations, or experiments. He was notably the first to order the scientific method as we understand it today.

There remains simple experience; which, if taken as it comes, is called accident, if sought for, experiment. The true method of experience first lights the candle [hypothesis], and then by means of the candle shows the way [arranges and delimits the experiment]; commencing as it does with experience duly ordered and digested, not bungling or erratic, and from it deducing axioms [theories], and from established axioms again new experiments.

â Francis Bacon. Novum Organum. 1620.[10]

When the problem or conditions do not permit a controlled experiment, such as in astronomical research, observational studies can be useful. For example, Tycho Brahe made careful observations and recorded measurements of stellar and planetary positions over time. After Brahe's death, his measurements proved useful in the development of Johannes Keplerâs laws of planetary motion, which offered a better fit than did Ptolemy's theory.

In the centuries that followed, important advances and discoveries were made by people who applied the scientific method in different areas. For example, Galileo Galilei was able to accurately measure time and experiment to make accurate measurements and conclusions about the speed of a falling body. Antoine Lavoisier was a French chemist in the late 1700s who used experiment to describe new areas such as combustion and biochemistry and to develop the theory of conservation of mass (matter).[11] During the 1800s, Louis Pasteur used the scientific method to disprove the prevailing theory of spontaneous generation and to develop the germ theory of disease.[12] Because of the importance of controlling potentially confounding variables, the use of well-designed laboratory experiments is preferred when possible.

### Galileo Galilei

Measured fall time of a small steel sphere falling from various heights. The data is in good agreement with the predicted fall time of $\sqrt{2h/g}$, where h is the height and g is the acceleration of gravity.

Galileo Galilei (1564â1642) was a scientist who performed many quantitative experiments addressing many topics. Using several different methods, Galileo was able to accurately measure time. Previously, most scientists had used distance to describe falling bodies using geometry, which had been used and trusted since Euclid.[13] Galileo himself used geometrical methods to express his results. Galileoâs successes were aided by the development of a new mathematics as well as cleverly designed experiments and equipment. At that time, another kind of mathematics was being developedâalgebra. Algebra allowed arithmetical calculations to become as sophisticated as geometric ones. Algebra also allowed the discoveries of scientists such as Galileoâas well as later scientists like Newton, Maxwell and Einsteinâto be later summarized by mathematical equations. These equations described physical relationships in a precise, self-consistent manner.

One prominent example is the âball and ramp experiment.â[14] In this experiment Galileo used an inclined plane and several steel balls of different weights. With this design, Galileo was able to slow down the falling motion and record, with reasonable accuracy, the times at which a steel ball passed certain markings on a beam.[15] Galileo disproved Aristotleâs assertion that weight affects the speed of an objectâs fall. According to Aristotleâs Theory of Falling Bodies, the heavier steel ball would reach the ground before the lighter steel ball. Galileoâs hypothesis was that the two balls would reach the ground at the same time.

Other than Galileo, not many people of his day were able to accurately measure short time periods, such as the fall time of an object. Galileo accurately measured these short periods of time by creating a pulsilogon. This was a machine created to measure time using a pendulum.[16] The pendulum was synchronized to the human pulse. He used this to measure the time at which the weighted balls passed marks that he had made on the inclined plane. He measured to find that balls of different weights reached the bottom of the inclined plane at the same time and that the distance traveled was proportional to the square of the elapsed time.[17] Later scientists summarized Galileoâs results as The Equation of Falling Bodies.[18][19]

 Distance d traveled by an object falling for time t where g is gravitational acceleration (~ 9.8 m/s2): $\ d=\frac{1}{2}gt^2$

These results supported Galileoâs hypothesis that objects of different weights, when measured at the same point in their fall, fall at the same speed because they experience the same gravitational acceleration.

### Antoine Lavoisier

Lavoisier and Laplace's ice calorimeter device

Antoine Lavoisier (1743â1794) was a French chemist regarded as the founder of modern chemistry. Lavoisier's experiments were among the first truly quantitative chemical experiments. He showed that, although matter changes its state in a chemical reaction, the quantity of matter is the same at the end as at the beginning of every chemical reaction.[citation needed] In one experiment, he burned phosphorus and sulfur in air to see whether the results further supported his previous conclusion (Law of Conservation of Mass). In this experiment, however, he determined that the products weighed more than the original phosphorus and sulfur. He decided to do the experiment again. This time he measured the mass of the air surrounding the experiment as well. He discovered that the mass gained in the product was lost from the air. These experiments provided further support for his Law of Conservation of Mass.

One of Lavoisierâs experiments connected the worlds of respiration (physiology) and combustion. Lavoisierâs hypothesis was that combustion and respiration were one and the same, and combustion occurs with every instance of respiration. Lavoisier, working with Pierre-Simon Laplace, designed an ice calorimeter apparatus for measuring the amount of heat given off during combustion or respiration. This machine consisted of three concentric compartments. The center compartment held the source of heat, in this case, the guinea pig or piece of burning charcoal. The middle compartment held a specific amount of ice for the heat source to melt. The outside compartment contained packed snow for insulation. Lavoisier then measured the quantity of carbon dioxide and the quantity of heat produced by confining a live guinea pig in this apparatus. Lavoisier also measured the heat and carbon dioxide produced when burning a piece of charcoal in the calorimeter. Using this data, he concluded that respiration was in fact a slow combustion process. He also discovered through precise measurements that these processes produced carbon dioxide and heat with the same constant of proportionality. He found that for 224 grains of âfixed airâ (CO2) produced, 13 oz. of ice was melted in the calorimeter. Converting grains to grams and using the energy required to melt 13 oz. of ice, one can compute that for each gram of CO2 produced, about 2.02 kcal of energy was produced by the combustion of carbon or by respiration in Lavoisier's calorimeter experiments. This compares well with the modern published heat of combustion for carbon of 2.13 kcal/g.[20] This continuous slow combustion, which Lavoisier and Laplace supposed took place in the lungs, enabled the living animal to maintain its body temperature above that of its surroundings, thus accounting for the puzzling phenomenon of animal heat.[21] Lavoisier concluded, âLla respiration est donc une combustion," That is, respiratory gas exchange is combustion, like that of burning a candle.

Lavoisier was the first to conclude by experiment that the Law of Conservation of Mass applied to chemical change.[22] His hypothesis was that the mass of the reactants would be the same as the mass of the products in a chemical reaction. He experimented on vinous fermentation (biochemistry). He determined the amounts of hydrogen, oxygen, and carbon in sugar. He weighed a quantity of sugar, added yeast and water in measured amounts, and allowed the mixture to ferment. Lavoisier measured the mass of the carbonic acid gas and water that were given off during fermentation and weighed the residual liquor, the components of which were then separated and analyzed to determine their elementary composition.[23] In this way he controlled a couple of potential confounding factors. He was able to capture the carbonic acid gas and water vapor that were given off during fermentation so that his final measurements would be as accurate as possible. Lavoisier then concluded that the total mass of the reactants was equal to the mass of the final product and residue.[24] Moreover, he showed that the total mass of each constituent element before and after the chemical change remained the same. Similarly, he demonstrated via experimentation that the mass of products of combustion is equal to the mass of the reacting ingredients.

### Louis Pasteur

(Above) A swannecked flask is pictured with broth and no bacteria in it after being boiled to remove any germs or microorganisms. (Below) Another swannecked flask is pictured with outside contaminants in the air allowed in. This broth grows microorganisms.

Louis Pasteur (1822â1895), regarded as the âFather of Microbiological sciences and immunology,â was a French biologist during the 1800s.[25] He discovered and supported by experimental results the idea that disease-causing agents do not spontaneously appear but are alive and need the right environment to prosper and multiply. Stemming from this discovery, he used experiment to develop vaccines for chicken cholera, anthrax and rabies, and to develop methods for reducing bacteria in some food products by heating them (pasteurization). His work also led him to advocate (along with the English physician Dr. Joseph Lister) for antiseptic surgical techniques. Most scientists of that day believed that microscopic life sprang into existence from nonliving matter. This idea was called spontaneous generation.

Pasteurâs observations of tiny organisms under the microscope caused him to doubt spontaneous generation. He designed an experiment to test it. His hypothesis was that life could not arise from where there is no life. He took care to control possible confounding factors. For example, he needed to make sure there was no life, even microscopic, in the flasks of broth he used as a test medium. He decided to kill any microscopic organisms already present by boiling the broth until he was confident that any microorganisms present were killed. Pasteur also needed to make sure that no microscopic organisms entered the broth after boiling, yet the broth needed exposure to air to properly test the theory. A colleague suggested a flask with a neck the shape of an âSâ turned sideways. Dust (which Pasteur thought contained microorganisms) would be trapped at the bottom of the first curve, but the air would flow freely through.[26]

Thus, if bacteria should really be spontaneously generated, then they should be growing in the flask after a few days. If spontaneous generation did not occur, then the contents of the flasks would remain lifeless. In the end, it was a complete success; not a single microorganism appeared in the broth. Then Pasteur allowed the dust containing the microorganisms to mix with the broth. In just a few days the broth became cloudy from millions of organisms growing in it. For two more years, he repeated the experiment in various conditions and locales to assure himself that the results were correct. In this way Pasteur supported his hypothesis that spontaneous generation does not occur.[27] Despite the experimental results supporting his hypotheses and his success curing or preventing various diseases, correcting the public misconception of spontaneous generation was a slow, difficult process.

As he worked to solve specific problems, Pasteurâs notions were sometimes corrected by the results of his experiments, such as when he was asked to find the cause of disease devastating the French silkworm industry in 1865. After a year of diligent work he correctly identified a culprit organism and gave practical advice for developing a healthy population of moths. However, when he tested his own advice, he found disease still present. It turned out he had been correct but incomplete â there were two organisms at work. It took two more years of experimenting to find the complete solution.[28]

## Observational science

Observational science is used when it is impractical to fit a system into a laboratory setting. It can also be used when confounding factors are either limited or known well enough to analyze the data in light of them. In order for an observational science to be valid, the confounding factors must be known and accounted for.

### Astronomy

An early European observational scientist was Tycho Brahe (1546â1601). Braheâs observations of stellar and planetary positions were noteworthy both for their accuracy and quantity.[29] His celestial positions were much more accurate than those of any predecessor or contemporary. In an observatory funded for him by King Frederick II of Denmark, Brahe built some of the largest observing instruments yet constructed.[30] Because of the large size of this equipment, Brahe was able to measure angles to an accuracy of better than 0.1 degree.[31] This was more accurate than any previous observations and close to the limit that the human eye can observe.[31] In this way, Brahe was able to make observations about stellar and planetary positions in a lab setup.

Brahe himself was not a Copernican but proposed a system in which the Sun and Moon orbited the Earth, while the other planets orbited the Sun. His system provided a safe position for astronomers who were dissatisfied with older models but were reluctant to accept the Earth's motion. It gained a considerable following after 1616 when Rome decided officially that the heliocentric model was contrary to both philosophy and Scripture, and could be discussed only as a computational convenience that had no connection to fact. His system also offered a major innovation: while both the geocentric model and the heliocentric model as set forth by Copernicus relied on the idea of transparent rotating crystalline spheres to carry the planets in their orbits, Brahe eliminated the spheres entirely.

Johannes Kepler (1571â1630) used the accurate observations of Brahe to discover the shape of Marsâ orbit. His first hypothesis was that the orbit was circular. After four years of research and testing 70 different combinations of circles and epicycles, he devised a shape that would fit Marsâ orbit. However, the model was accurate to only 0.13 degrees.[32] Kepler knew that Braheâs observations could be used to develop an orbit shape more accurate than this. Kepler eventually decided to try various oval shaped orbits. This implied that the speed of the planet changed as it traveled around the oval. After nine years, he found that elliptical orbits fit satisfactorily with the observed path of Mars. He found that this shape worked not only for Mars, but also for every planet that Brahe had observed.[32]

### Biology

Observational studies are not experiments. By definition, observational studies lack the manipulation required for Baconian experiments. In addition, observational studies in biological systems often involve variables that are challenges to quantify or control. Nevertheless, observational studies are used because it is sometimes too difficult (too expensive, or too much time required) or unethical to conduct longitudinal experiments with human or animal subjects. In these situations, observational studies have value because they often suggest hypotheses that can be tested with randomized experiments or by collecting fresh data.

In providing therapies for human subjects, for example in psychology or health care, it is unethical to provide a substandard treatment to patients. Therefore, ethical review boards are supposed to stop clinical trials and other experiments unless a new treatment is believed to offer benefits as good as current best practice.[33] It is also unethical and often illegal to conduct randomized experiments on the effects of substandard or harmful treatments, such as the effects of ingesting arsenic on human health. To understand the effects of such exposures, scientists use observational studies.

Observational studies are limited because they lack the statistical properties of randomized experiments. In a randomized experiment, the method of randomization specified in the experimental protocol guides the statistical analysis, which is usually specified also by the experimental protocol.[34] Without a statistical model that reflects an objective randomization, the statistical analysis relies on a subjective model.[34] Inferences from subjective models are unreliable in theory and practice.[35] In fact, there are several cases where carefully conducted observational studies consistently give wrong results, that is, where the results of the observational studies are inconsistent and also differ from the results of experiments. For example, epidemiological studies of colon cancer consistently show beneficial correlations with broccoli consumption, while experiments find no benefit.[36]

A particular problem with observational studies involving human subjects is the great difficulty attaining fair comparisons between treatments (or exposures), because such studies are prone to selection bias, and groups receiving different treatments (exposures) may differ greatly according to their covariates (age, height, weight, medications, exercise, nutritional status, ethnicity, family medical history, etc.). In contrast, randomization implies that for each covariate, the mean for each group is expected to be the same. For any randomized trial, some variation from the mean is expected, of course, but the randomization ensures that the experimental groups have mean values that are close, due to the central limit theorem and Markov's inequality. With poor randomization, the systematic variation in covariates between the treatment groups (or exposure groups) makes it difficult to separate the effect of the treatment (exposure) from the effects of the other covariates, most of which have not been measured. The mathematical models used to analyze such data must consider each differing covariate (if measured), and the results will not be meaningful if a covariate is neither randomized nor included in the model.

To avoid these conditions that render an experiment far less useful, physicians conducting medical trials (such as for Food and Drug Administration approval), will quantify and randomize the covariates that can be identified. Researchers attempt to reduce the biases of observational studies with complicated statistical methods such as propensity score matching methods, which require large populations of subjects and extensive information on covariates. Outcomes are also quantified when possible (bone density, amount of some cell or substance in the blood, physical strength or endurance, etc.) and not based on a subject's or a professional observer's opinion. In this way, the design of an observational study can render the results more objective and therefore more convincing.

## References

1. ^ Cooperstock, Fred I. General Relativistic Dynamics: Extending Einsteinâs Legacy Throughout the Universe. Page 12. World Scientific. 2009. ISBN 9789814271165
2. ^ Griffith, W. Thomas. The Physics of Everyday Phenomena: A Conceptual Introduction to Physics. Page 4. New York: McGraw-Hill Higher Education. 2001. ISBN 0072328371.
3. ^ Devine, Betsy. Fantastic realities: 49 mind journeys and a trip to Stockholm. Page 62. Wilczek, Frank. World Scientific. 2006. ISBN 9789812566492
4. ^ Griffith, W. Thomas. The Physics of Everyday Phenomena: A Conceptual Introduction to Physics. Page 3. New York: McGraw-Hill Higher Education. 2001. ISBN 0072328371.
5. ^ Kevin Brown, Reflections on Relativity
6. ^ Snow viewed the developments as "an experiment...on the grandest scale." Snow, J. (1855). On the Mode of Communication of Cholera (2nd ed.). London: Churchill. Excerpted in MacMahon, B. & Pugh, T.F. (1970). Epidemiology. Boston: Little Brown.
7. ^ The 1854 cholera outbreak is the example of a natural experiment discussed often by David A. Freedman, e.g. in Statistical Models: Theory and Practice (Cambridge University Press) [1], chapter 1.3 (pages 6-9).
8. ^ Snow's studies of the pattern of the disease were convincing enough to persuade the local council to disable the well pump by removing its handle. After the handle of the well-pump was replaced, the incidence of new cases dropped. In stopping the use of water from the well-pump, the authorities did an uncontrolled experiment (without a control group) and without randomization.
9. ^ Bacon, Francis. Novum Organum. In Durant, Will. The Story of Philosophy. Page 101. Simon & Schuster Paperbacks. 1926. ISBN 9780671695002
10. ^ Durant, Will. The Story of Philosophy. Page 101 Simon & Schuster Paperbacks. 1926. ISBN 9780671695002
11. ^ Bell, Madison Smartt. Lavoisier in the Year One. Page 57. W.W. Norton & Company, Inc. 2005. ISBN 0393051552
12. ^ Dubos, Rene J. Louis Pasteur: Free Lance of Science. Page 155. De Capo Press. 1986. ISBN 9780306802621
13. ^ Drake, Stillman; Swerdlow, Noel M.; Levere, Trevor Hardly. Essays on Galileo and the history and philosophy of science, Volume 3. Page 22. University of Toronto Press. 1999. ISBN 9780802047168.
14. ^ Solway, Andrew. Exploring forces and motion. Page 17. The Rosen Publishing Group. 2007. ISBN 9781404237476
15. ^ Stewart, James. Redlin, Lothar. Watson, Saleem. College Algebra. Page 562. Cengage Learning. 2008. ISBN 9780495565215
16. ^ Massachusetts Medical Society, New England Surgical Society. The Boston Medical and Surgical Journal, Volume 125. Page 314. Cupples, Upham & Co. 1891
17. ^ Tiner, John Hudson. Exploring the World of Physics: From Simple Machines to Nuclear Energy. New Leaf Publishing Group. 2006. ISBN 0890514666
18. ^ Longair, M.S. Theoretical concepts in physics: an alternative view of theoretical reasoning in physics. Page 37. Cambridge University Press. 2003. ISBN 9780521528788
19. ^ Schutz, Bernard F. Gravity from the ground up. Page 3. Cambridge University Press. 2003. ISBN 9780521455060
20. ^ Holmes, Frederic Lawrence. Lavoisier and the chemistry of life: an exploration of scientific creativity. Page 188 Univ. Wisconsin Press. Reprint. 1987. ISBN 9780299099848; The published value of the heat of combustion for carbon is usually expressed as 393.5 kJ/mol; unit conversion yields the figure in units for comparison of 2.13 kcal/g
21. ^ Holmes, Frederic Lawrence. Lavoisier and the chemistry of life: an exploration of scientific creativity. Page 197. Univ. Wisconsin Press. Reprint. 1987. ISBN 9780299099848.
22. ^ Bell, Madison Smartt. Lavoisier in the Year One. Page 44. W.W. Norton & Company, Inc. 2005. ISBN 0393051552
23. ^ Holmes, Frederic Lawrence. Lavoisier and the chemistry of life: an exploration of scientific creativity. Page 382 Univ. Wisconsin Press. Reprint. 1987. ISBN 9780299099848.
24. ^ Bell, Madison Smartt. Lavoisier in the Year One. Page 92W.W. Norton & Company, Inc. 2005. ISBN 0393051552
25. ^ Simmers, Louise. Simmers-Nartker, Karen. Diversified Health Occupations. Page 10. Cengage Learning 2008. ISBN 9781418030216
26. ^ Dubos, Rene J. Louis Pasteur: Free Lance of Science. Page 169. Da Capo Press. 1986. ISBN 9780306802621
27. ^ Debré, Patrice. Louis Pasteur. Page 300. JHU Press, 2000. ISBN 9780801865299
28. ^ Dubos, Rene J. Louis Pasteur: Free Lance of Science. Page 210. Da Capo Press. 1986. ISBN 9780306802621
29. ^ Noel Swerdlow, Astronomy in the Renaissance, pp. 187-230 in Christopher Walker, ed., Astronomy before the Telescope, (London: British Museum Press, 1996), pp. 207-10.
30. ^ Kupelis, Theo. Kuhn, Karl F. In Quest of the Universe. Page 55. Jones and Bartlett Publishers. 2007. ISBN 9780763743871.
31. ^ a b Kupelis, Theo. Kuhn, Karl F. In Quest of the Universe. Page 55. Jones and Bartlett Publishers. 2007. ISBN 9870763743871
32. ^ a b Kupelis, Theo. Kuhn, Karl F. In Quest of the Universe. Page 57. Jones and Bartlett Publishers. 2007. ISBN 9870763743871
33. ^   Pre-publication chapters are available on-line.
34. ^ a b *Hinkelmann, Klaus and Kempthorne, Oscar (2008). Design and Analysis of Experiments, Volume I: Introduction to Experimental Design (Second ed.). Wiley. ISBN 978-0-471-72756-9.
35. ^ David A. Freedman, R. Pisani, and R. A. Purves. Statistics, 4th edition (W.W. Norton & Company, 2007) [2] ISBN 978-0-393-92972-0
36. ^ David A. Freedman (2009) Statistical Models: Theory and Practice, Second edition, (Cambridge University Press) [3] ISBN 9780521743853


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# Proof of the Weak Mordell-Weil Theorem

Suppose is an elliptic curve over a number field , and fix a positive integer . Just as with number fields, we have an exact sequence

Then we have an exact sequence

From this we obtain a short exact sequence

 (12.1)

Now assume, in analogy with Section 12.1, that , i.e., all -torsion points are defined over . Then

and the sequence (12.2.1) induces an inclusion

 (12.2)

Explicitly, this homomorphism sends a point to the homomorphism defined as follows: Choose such that ; then send each to . Given a point , we obtain a homomorphism , whose kernel defines an abelian extension  of that has exponent . The amazing fact is that can be ramified at most at the primes of bad reduction for and the primes that divide . Thus we can apply theorem 12.1.1 to see that there are only finitely many such .

Theorem 12.2.1   If is a point, then the field  obtained by adjoining to all coordinates of all choices of is unramified outside and the primes of bad reduction for .

Proof. [Sketch of Proof] First one proves that if is a prime of good reduction for , then the natural reduction map is injective. The argument that  is injective uses formal groups'', whose development is outside the scope of this course. Next, as above, for all . Let be the inertia group at . Then by definition of interia group, acts trivially on . Thus for each we have

Since is injective, it follows that for , i.e., that is fixed under all . This means that the subfield of  generated by the coordinates of is unramified at . Repeating this argument with all choices of  implies that is unramified at  .

Theorem 12.2.2 (Weak Mordell-Weil)   Let be an elliptic curve over a number field , and let be any positive integer. Then is finitely generated.

Proof. First suppose all elements of have coordinates in . Then the homomorphism (12.2.2) provides an injection of into

By Theorem 12.2.1, the image consists of homomorphisms whose kernels cut out an abelian extension of  unramified outside and primes of bad reduction for . Since this is a finite set of primes, Theorem 12.1.1 implies that the homomorphisms all factor through a finite quotient of . Thus there can be only finitely many such homomorphisms, so the image of is finite. Thus itself is finite, which proves the theorem in this case.

Next suppose  is an elliptic curve over a number field, but do not make the hypothesis that the elements of have coordinates in . Since the group is finite and its elements are defined over  , the extension  of  got by adjoining to  all coordinates of elements of is a finite extension. It is also Galois, as we saw when constructing Galois representations attached to elliptic curves. By Proposition 11.3.1, we have an exact sequence

The kernel of the restriction map is finite, since it is isomorphic to the finite group cohomology group . By the argument of the previous paragraph, the image of in under

is finite, since it is contained in the image of . Thus is finite, since we just proved the kernel of is finite.

William Stein 2012-09-24

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# The Sands of Time

Random video! Although I feel that I’ve heard it earlier, my first conscious memory of getting linked to it is from this post. At first I thought it would be the right background music for this post, but upon further reflection I think it mainly suited me while I was writing this post. Well, it’s topical if you mentally replace “day” with “year”.

Anyway. Around this time a year ago, I paused my participation in big high-school competitions, for a variety of reasons.

Firstly, I stopped attempting to make IMO both because I wouldn’t get that much from the training and because other people ought to have the opportunity. I was concerned that I might condition myself to only be able to do math with the short-term motivation of contests. Better to focus on college math and maybe some original research, I thought. During the year, I did lots of the former and very little of the latter. Meh.

As for the IOI, my obvious next target: I was tired of training and going abroad while paranoid about whether my immune system would hold up. I didn’t feel that the IOI was worth that. To some degree, I also felt burned out about programming. Long story short, my treatment should end soon, and learning Haskell completely resolved the burnout problem.

But the most important reason, I think, was that “high school was too short”. I started math competitions ridiculously early and didn’t spend much time exploring other interests. I thought I knew myself well enough that I could say I didn’t have many more interests at all, but I was completely wrong (psych nerds will reflexively note this to be the Dunning-Kruger effect). I coded lots in weird languages — Haskell, as mentioned previously, plus Scala, plus all manner of other magical command line tools. I wrote my first math problem and submitted it officially, picked up a new instrument, went to a debate competition, served as an unimportant tech guy for MUN, discovered and became hooked on Pentatonix, participated in three puzzle hunts in Australia and one in Massachusetts, figured out my rough political stance, rode a boat, got retweeted by @eevee and @Kyrgyzstan_News, increased my Neopets™ fortune by over 3400%, and lurked on FurAffinity a little too much.

But now, dear competition world, I’m back.

Not ready to declare IOI contestant status, but it’s very probable. To a prosperous 2014! raises glass

The rest of this post will be an increasingly technical rambling rundown of programming competitions in an attempt to preserve the competitions in the period of over a year since my last competitions post. If you aren’t familiar with programming or at least math, you can probably stop reading.

The last programming competition I entered last year was the national round of the competition whose name roughly translates as “Academic Ability Contest,” which I’ll call the AAC for the sake of convenience below.

The hardest problem on the regional AAC, which I briefly mentioned in that last competitions post, involved combining math operators (as in the four 4s game). However, it only required you to find solutions with at most two operators, so an ugly, unstructured brute-force sufficed, and I got full score. Oops.

The hardest problem on the national AAC was a barely-disguised assignment problem, which apparently ought to be solved with the Hungarian algorithm, neither of which I had heard of before. (The backstory of that problem was easily the most pathetic I’ve ever seen. “Bob entered a math competition where one of the problems went like this. Help Bob out!”) I also very stupidly misread another problem and got a confusing “Second prize”, which actually meant somewhere between fourth to tenth place.

2013 rolled around. I took two quick math competitions, the silly APMO (on which three countries got ten perfect scores each) and the Tournament of Towns, since they didn’t require any extra training.

As for programming competitions, I competed in IPSC 2013 with Yoshi(ap), something I actually forgot about until I wrote this post. IPSC had a highly diverse and interesting selection of problems — some algorithmic, some mathematical, some puzzle-like (think rankk; some skills I learned there came in handy for Problem H.) Other small things later in the year: I qualified for round 1 of the Facebook Hacker Cup but was too lazy and/or forgetful to compete further; I survived Brilliant.org’s Hunger Games thing with a far-too-simple program; and, of course, I got a few miscellaneous victories on CodeForces and TopCoder (very few because time zones). All from the comfort of my home.

Back to classical algorithmic competition programming. During the summer, I attended a NTHU camp, learned a few algorithms, and solved many problems, most notably Problem E of the 2006 ACM-ICPC Shanghai Regionals, which is Dijkstra on a complicatedly transformed graph.

By “solved” I mean “spent hours on this problem and submitted fifteen wrong solutions, even while reading another person’s carefully documented solution at the same time.” That’s how hard it was. Chekhov’s Skill in 3… 2… 1…

Fast-forward to the new school year, 2013’s local AAC: the hardest problem was… Dijkstra on a not-so-complicatedly transformed graph. Guess how I did.

In-between, I participated in the National Problem-Solving Contest (NPSC) with a classmate. This contest is structured like ACM-ICPC, except that it’s for high-school students. We came alarmingly close to solving all the problems; the only one we didn’t get was the last one, which was essentially “write an interpreter of this primitive dialect of Lisp”. I say “primitive”, but just having to implement define and lambda sent the complexity out the window. To drive this point home, the organizers wrote a solution to Problem A in the Lisp dialect. Problem A, by the way, was a troll shortest-path problem that tricked me into thinking it was a maximum-flow problem for troublingly long.

Speaking of troll problems, this gem appeared in the previously mentioned summer camp, as a diversion of sorts during a weird game session (translated):

Problem B

?

Input Specification: Each line contains an integer n with 1 ≤ n ≤ 26.

Output Specification: Output the answer to each line.

Sample Input:
5
6
9
Sample Output:
B
C
F

Also, around this time, I finally remembered to do USACO, and (as expected) got promoted to gold. It would be nice if they sent reminder emails, although I should blame myself for not writing it down. The time limit was so wide that I spent half the time eating dinner, washing dishes, and coding a dumb client-tester in Perl to simplify redirecting input and output.

Later in December 2013, National AAC — the hardest problem was a geometric DP thing that I still don’t know how to solve. However, I managed to get four out of five test cases through repeatedly submitting unproved modifications of my brute-force solution with a pruned search space. There was instant grading with no submission penalty for this one because the powers that be were trying out our IOI system. The judging part went well.

On the other hand, compared to the posted previous competition environments, the work environment was much less convenient. The I/O spec was more annoying than USACO: all the problems had to read to and write from the same filenames, input.txt and output.txt (though of course you could do separate problems in separate subdirectories). The computers ran Windows XP with only basic Cygwin installations. But luckily, they had a reasonable version of Vim and a sufferable high-level language — Perl.

So I dashed out a client-side tester program in Perl.

Chekhov’s Skill is really an overused trope in my life.

Also came in handy: perl -de 0 is a REPL (thanks, HyperPolyglot). Or maybe a REL (read-eval-loop) because it doesn’t print anything unless you tell it to. But it’s still helpful for quick tests.

In hindsight, my Perl was redundant and would still be longer than a shell script if I fixed it, so that was pretty silly.

#!/bin/bash
cat < /dev/stdin > input.txt
./\$1
cat output.txt

That’s my final lesson of the year: pick the right language for the job.

This is something I occasionally wish more fellow programming contestants realized. Yes, C++ is fast, but Java has many more portable goodies like its GUI and image-manipulation systems, Python has nice features for mathematical exploration like built-in arbitrary-size integers and itertools, and Perl has unreadable but quick magic variables and file tests. Learning extra languages is not very hard and lots of fun.

It is also something I am learning again for an ongoing competition that will end in just over a day — dear future self, no matter how much you love Haskell, please don’t expect normal people to be able to collaborate with you if you use it.

Until next year!

(note: the commenting setup here is experimental and I may not check my comments often; if you want to tell me something instead of the world, email me!)


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# Calculate the area of circle having diameter as 10 cm?

Feb 19, 2018

$\text{Area} = 25 \pi c {m}^{2} \approx 78.5 c {m}^{2}$

#### Explanation:

$\text{Area of a circle} = \pi {r}^{2}$

$r = \frac{d}{2} = \frac{10}{2} = 5 c m$

$\text{Area} = \pi \cdot {5}^{2} = 25 \pi c {m}^{2} \approx 78.5 c {m}^{2}$

Feb 19, 2018

See below...

#### Explanation:

Area of a circle = $\pi \cdot {r}^{2}$

Where $r$ is the radius.

The radius is half of the diameter.

$\therefore r = \frac{10}{2} = 5 c m$

Area = $\pi \cdot {5}^{2} = 25 \pi c {m}^{2}$

I'd just leave the answer as an exact value unless specified in the question to a certain accuracy.

Feb 19, 2018

$\text{area "=25pi~~78.54" square cm}$

#### Explanation:

$\text{the area of a circle is calculated using}$

• " area of circle "=pir^2

$\text{where r is the radius of the circle}$

$\text{here } d = 10 \Rightarrow r = \frac{10}{2} = 5$

$\Rightarrow \text{area "=pixx5^2=25pi~~78.54" square cm}$


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# Pedagogical introduction to Stückelberg renormalization (group)?

+ 3 like - 0 dislike
1193 views

Das anybody know a nice pedagogical introduction (something that is shorter than a heavy textbook, maybe a lecture note) to the Stückelberg renormalization (group)? Conversely to the renormalization methods often applied in (high-energy) theoretical physics, in this approach the renormalization group transformation is reversible which makes the renormalization group a true group, and it is (for me personally a bit surprisingly at the first glance) often applied in contexts that have nothing to do with relating effective theories valid at different energy scales to each other or integrating out high-energy effects.

While I am generally interested in applications of any kind, I would enjoy most reading about how the Stückelberg renormalization is applied in a QFT context.

Brunetti, Romeo, Michael Dütsch, and Klaus Fredenhagen. "Perturbative algebraic quantum field theory and the renormalization groups."Advances in Theoretical and Mathematical Physics 13.5 (2009): 1541-1599.

We discuss the connection between the Stückelberg–Petermann renormalization group which describes the freedom in the perturbative construction with the Wilsonian idea of theories at different scales. In particular, we relate the approach to renormalization in terms of Polchinski’s Flow Equation to the Epstein–Glaser method.

On the simplest, classical level (applied to ordinary differential equations), the similarities and differences between the Stückelberg–Petermann renormalization group and the Wilsonian renormalization group is well described in

G.C. Paquette, Renormalization group analysis of differential equations subject to slowly modulated perturbations, Physica A 276 (2000), 122-163.

See especially p. 9-10 and Section 7.2.

+ 3 like - 0 dislike

The Stueckelberg renormalization group is what removes the freedom left in the prescription of the renormalization conditions after the limit where the regularization (whether by a cutoff or another regularization recipe) is already taken. Because the physics must be unique, the theory cannot depend on this freedom, as with the gauge freedom in a gauge theory. The Stueckelberg renormalization group is the analogue of the gauge group.

I don't know of a good introduction to Stueckelberg renormalization. But (for example) the following articles show the use of the renormalization group in contexts very different from quantum field theory. You need to judge for yourself how pedagogical the articles are written.

A renormalization group treatment of the classical van der Pol oscillator is given in https://arxiv.org/abs/1305.2372

classical singular perturbation theory: https://arxiv.org/abs/hep-th/9506161https://arxiv.org/abs/cond-mat/9407024

quantum anharmonic oscillator: https://arxiv.org/abs/hep-th/9710087

similarity renormalization group in nuclear physics: https://arxiv.org/pdf/nucl-th/0611045

nonlinear optics; https://arxiv.org/abs/hep-th/0001210 ; this might come closest to your request; there it is named after Bogoliubov rather than Stueckelberg.

For the history, see https://arxiv.org/abs/hep-th/9602024

For relativistic QFT, see, e.g., https://arxiv.org/abs/hep-th/0501228

a resummation technique is required. There are several RG formulations that may be used to achieve this. We believe the most powerful is still the original RG of Stuckelberg–Peterman, Gell–Mann Low and Bogoliubov–Shirkov [68–70], which we refer to as the reparametrization RG. Although discovered in perturbative studies of quantum electrodynamics in the process of removing the ultraviolet (UV) divergences from measured quantities, it was pointed out very early in the development of the subject, by Blank Bonch–Bruervich and Shirkov [71], that the RG is not dependent on the existence of such UV divergences and that it could be a useful tool in a variety of fields (they mention condensed matter physics). [...] From a modern perspective, the original field theoretic renormalization can now be seen to be nothing more than a coordinate change from original bare parameters to renormalized parameters. A coordinate transformation, of itself, does not change the physics, but, as we shall see and demonstrate, one coordinate system may be vastly superior to another when doing perturbative calculations, especially when combined with the notion of a “sliding scale” for the renormalization point.

In contrast, Wilson-style renormalization is not just a reparameterization but simplifies the model by changing the short-distance, high-energy physics, leaving only the long-distance, low-energy physics intact. The relationship between the two is discussed in Section 5.2 of

Brunetti, Romeo, Michael Dütsch, and Klaus Fredenhagen. "Perturbative algebraic quantum field theory and the renormalization groups."Advances in Theoretical and Mathematical Physics 13.5 (2009): 1541-1599.

We discuss the connection between the Stückelberg–Petermann renormalization group which describes the freedom in the perturbative construction with the Wilsonian idea of theories at different scales. In particular, we relate the approach to renormalization in terms of Polchinski’s Flow Equation to the Epstein–Glaser method.

answered Oct 23, 2016 by (15,458 points)
edited Jan 13, 2017

A lengthy comment discussion involving issues of integrating out degrees of freedom (Stückelberg Renormalization has nothing to do with integrating out degrees of freedom) has been moved to chat.Please continue the discussion about such issues there if you like, any further off-topic comments here will be moved there too.

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# Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.$\displaystyle \lim_{x\to \infty} \frac{x + x^2}{1 - 2x^2}$

## $-\frac{1}{2}$

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### Video Transcript

So for this problem here we're looking at X plus X squared. Provided by one minus two X squared. That's our function. And we see that if we plug in infinity we would get infinity over negative infinity. And that's considered an indeterminate form. So instead what we can do is take the derivative of the numerator and we would get to acts Then we take the derivative of the denominator. We get a -4 X. And we plug in infinity. We're going to get infinity over negative infinity. So this is going to be yet another and determinant form. That's no problem because we can use the hotel's rule again, taking the derivative of the numerator this time we'll get to and then the derivative of the denominator here will get negative four. That would just be a negative one half. So when we take the limit of a constant value, it's just going to be that constant value. So our limit is negative .5

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This is the course website for Applied Statistical Methods 1 course taught by Dr. Robin Donatello in Fall 2019. This landing page is used for posting of regular announcements and information for students of the class.

### 12-16-2019 Week 16: Finals Week

• Poster Session: Monday 4-6pm Tehama 116
• Poster peer review Google Form: https://forms.gle/QWu3YMZtiAbjzxj29
• Some paper copies will be available.
• You will be expected to review 10 posters during the final session.

Final take home exam Due Wednesday 12/18 - midnight.

Final assignments (Due Friday 5/20)

OH for Finals Week: All hours will be held in Holt 202
- Mon: 9-10am, 2-3pm
- Tue: 10:30-11:30, 8-9pm (Zoom link in Slack) - Wed: 9:30-11am

### 12-09-2019 Week 15: Poster building!

• Poster building week! Due dates are FIRM
• Take Home Final exam
• Distributed Wed 12/11 via Slack.
• Due Wednesday 12/18 EOD.
• All links to files and for submission in Slack post.
• Makeup Midterm - In class on Friday. Otherwise, open work day.

### 12-02-2019 Week 14: Model building / Model fit

• You’ve been exploring relationships on variables related to your individual research question all semester. Now is time to bring all this information together and build a ‘final’ model.
• Build a final multivariable model. y ~ x1 + x2 + …
• The goal is to explain the variabilty in your response variable, using multiple predictor variables.
• Model building is cyclical, we’ll go over some criteria this week on how to know when to stop and how to compare different models.
• Poster prep stage IV is due this week - this is the final stage before you build your posters. Pay close attention to the requirement details on the project page.
• describe the limitations to your study
• propose further work: What can someone DO with the information you’ve found?

### 11-25-2019 Fall Break

Relax, get caught up on sleep.

### 11-18-2019 Week 13: Categorical Variables & Logistic Regression

• Swapping ordering of topics - Categorical variables first on Monday, Logistic regression starts Wednesday
• Quiz 13 (on both categorical variables and logistic regression) due Tuesday night.
• Homework 10 due Saturday before you go off on fall Break.
• My Math Seminar talk on Missing Data video has been posted on the notes page. - Any EC is due by 12/6

Other schedule updates: No more quizzes after break. Gradebook total has been adjusted to dropping the lowest 2 quiz scores.

### 11-11-2019 Week 12: Multivariable Regression Modeling

• Writing and interpreting multivariable models
• How else can tertiary variable muck up an observed bivariate relationship? Confounding.

TWO THINGS TO WORK ON THIS WEEK

• Homework 10
• Poster prep stage III - Take your bivariate relationship and add an appropriate analysis.

### 11-11-2019 Week 11: Moderation

We’re starting to talk about multivariable models this week. This involves understanding how to think about simultaneous relationships between multiple variables.

Life isn’t bivariate. We “know” there is more to how big a caterpillar is than simply the species. What about the temperature? was it a drought year? what type of tree is it on? How heavy is the bird predation in that area? What other species are on the same tree that could be competing for resources? To understand the singular effect of bird predation on caterpillar size, we must consider how all other factors affect caterpillar size, and how those other factors might be related to each other as well.

The course notes is a “cliff notes” version of the lecture material for this coures. If you’re not watching the videos (even on 1.5x speed) you are missing critical information that I will not lecture on in class, but that you will be responsible for on the homework and exam. You will do poorly on the quizzes if you have not watched the videos and taken your own set of notes.

### 10-28-2019 Week 10: Regression Analysis

We’re finishing up Bivariate Inference this week.

• Simple Linear Regression - similar to correlation, but now we care about how much the response variable changes as the explanatory varible changes.
• HW 8 due this Saturday
• Think about which inferential tools you are going to want to use on your personal research questions.

### 10-21-2019 Week 9: Bivariate Inference

• See below for quiz schedule.
• Covering 3 core statistical tests
- ANVOA: Q ~ C
- Chi-Squared: C ~ C
- Correlation: Q~Q
• Questions come from the PDS video AND the course packet.
• To be on track, you should have the first question in Hw8 (2-sample T-Test) complete before class on Monday.
• I highly encourage you to get the 1:1 extra R help available. Link below.

### 10-14-2019 Week 8: Statistical Inference via Hypothesis Testing

The rest of the semester is all about Inference! This is what you’ve been training for, and waiting for. This week we’ll start out with setting the foundations of the process on 1 variable, then quickly move into testing relationships between two variables.

#### Midterm handed back Friday.

• Change in exam wrapper. Not going to do error assessment.
• [Exam reflection handout].
• Hard copies handed out in class.
• Hard copy due next Friday 10/25

#### Temporary change in quiz protocol

Next week we are covering ANOVA, $$\chi^{2}$$, and Correlation. Quiz 09 covered topics on all three.

• Quiz09 will be split into three smaller quizzes.
• Due the night before the topic is covered in class.
• ANOVA due Sun 10/20
• $$\chi^{2}$$ due Tue 10/22
• Correlation due Thu 10/24
• I will pick the 1 lowest scoring question to do as a group quiz that next day.
• Scores will be re-combined into a Quiz09 score.

### 10-07-2019 Week 7: Simulation, sampling, probability distributions

Now that you’ve had a chance to learn what your data looks like, exploring relationships and learning more about the variables that you are interested in researching, let’s take a step back and look at the big picture. You will use lecture notes on writing empirical research (found on the materials page of this website) to guide the writing of a research proposal.

Applied statistics is used to support research. Numbers and measures don’t stand on their own. It is important that you understand how to use data to support scienctific research.

So far you have been conducting an exploratory data analysis to learn the distributions of your measures, and to assess for any possible relationships in your research. You cannot make any statistical claims about whether or not a relationship is significant at this point. That is under the realm of inferential statistics and will be the topic of the class after Spring break.

To understand why inferential statistics can be used to make valid claims, we have to learn a little bit about Probability first. This topic is only briefly covered in the PDS videos. You will have to spend more time in the course packet, AND in the OpenIntro textbook if you have never taken a statistics course before.

This is a much more mathematical section of the course, so be sure to plan your time accordingly. Over the next two weeks you should be writing your research proposal, working on your poster preparation slides, and filling out the notes and examples in the course packet. We will be working heavily out of the course packet, so it is essential that you read the material ahead of time so the examples we work through in class are meaningful.

Week overview

• Two in class exercises (mon, fri)
• HW07 due in google drive & worksheet – Part of this will be due Wednesday
• Quiz 7 - Quantifying Uncertainty. Due Thursday. Group quiz on Friday.
• Relies heavily on PDS video and reading the course notes section 4.5-4.6
• Must have a fundamental understanding of how to calculate a confidence interval

### 09-30-2019 Week 6: Reflection, Midterm, and new stuff.

We’re going to switch gears here and talk about probability, but before we do that let’s take a step back and think about what we’ve learned so far and consider how our thoughts about Statistics has changed due to this new information. It’s also a time for self reflection. How are you doing in this class? Are you happy with your grade? What can you do differently?

This week is mostly about prepping for the exam. We will do a probabilty activity on wednesday to get us ready for the next unit.

Updates I have cxl’d HW 6 (research proposal). Quiz 6 is still required but will stay open until Monday night.

### 09-23-2019 Week 5: Describing Relationships

It’s ok to struggle, it’s not OK to not do anything. If you are having trouble with coding, you need to ask for help. This class builds on each assignment. You can’t do HW05 well if you didn’t do HW04. You won’t be able to do HW06 if you can’t do HW05. Don’t risk falling behind, you will struggle to catch up and may not pass. There are five of us available to help you during office hours and Community Coding. You can make appointments with any of us outside of our scheduled hours as well.

• This week we will discuss how to create visualizations to assess the relationship between two variables.
• The homework assignments are training grounds for you to practice how to create and interpret graphics to explore your research question. The graphs you make for the homework may or may not be directly related to your research question.
• You may have to go back to your raw data and pull in new measures. Or adjust your variables (like collapsing levels) based on what you observed in last week’s assignment.

• Monday 2pm OH cancelled this week. Extra OH Monday at 9, and Wed at 9am.

### 09-16-2019 Week 4: Describing Distributions of Data

I’m alive!

OH Cancelled Monday. Sorry, lot of Drs appts in the upcoming weeks that were scheduled months ago. Additional OH Wednesday at 9am

If there’s one take home message for this week, is to describe the distribution of variables you need:

1. a picture
2. summary statistics
3. a complete sentence interpretation

While your goal should be to make figures stand alone, you won’t be there quite yet. Your coding skills will develop as you go on throughout the semester.

• You will also start the work to build your research poster. See the project instructions page for details. This means you have two assignments due this Friday. Procrastinate at your own peril.

Re: Homework 03. I will adjusted the grading rubric to downweight the data management portion. This is a wakeup call for all of you who tried to start the HW on Saturday evening. You won’t be able to do HW for this class in an evening. There is a reason we work on it all week in class. I advise going and reading the [advice from last year’s students].

HW03 is a critical assignment You can’t complete any remaining homework until you get this file working. If you were unsuccessful this week, your goal is to work with your resources (each other, tutors, peer mentors, office hours, community coding) until you get it working.

### 09-09-2019 Week 3: Hello R! - Preparing Data for Analysis.

I may be late to my OH on Tuesday. I have to take my dog 🐶 to the vet :(

• We are starting to code in R this week. You should follow along with the coding that is done in the PDS video. Practicing typing commands and mimicking the output that is in the video is very important and will contribute to your learning.

• In class on Monday you will work with your partner to take a closer look at your codebook to identify what types of data management you will have to do.

• I have created a [video] to walk you through how to get your data into R Studio.

• The [Help with R] page has more info such as useful functions and how to navigate data.

#### Entering a challenging time

You may feel like this in the next couple weeks:

But if you use the references in the course notes, the peer mentors, videos and engage in the most effective way of learning a new programming language,

then you will learn how to dominate debugging and come out a stronger and more capable of being able to answer research questions using data.

### 09-02-2019 Week 2: Generating research questions

Hope everyone had a relaxing 3day weekend. Office hours have been set. Thank you for contributing!

• Monday 2-3pm Holt 202
• Tuesday 10-11am Holt 202
• Wednesday 2-4pm PHSC 213 (Community Coding)
• Thursday 2-4pm MLIB 442 (Community Coding)

Due to a glitch in the system, Quiz 1 is going to be considered a “trial run” and not count towards your grade.

This week we’re going to cover the basic data types, how to formulate a testable research hypotheses on your selected data, and setting up a reproducible research pipeline to prepare data for analysis.

Haven’t selected a [research data set] yet? Look at the [sign up sheet] to see what’s left and sign up to analyze a data set. If you don’t have a partner yet, put your name down by a data set you are interested in analyzing, or post in the #assignments channel in slack that you’re looking for a partner.

Quiz 2 due Tuesday Don’t forget this week we start having Quizzes on the upcoming week material. You must watch the videos and look through the course notes to be prepared for these quizzes!

### 08-26-2019 Week 1: Welcome to Fall 19!

• This course website contains all materials except the textbook for this class. Be sure to familiarize yourself with the organization. You will be here a lot. Bookmark this page.
• I have created a handy [Materials Overview] that walks you through the components of this class.
• Take this survey to help choose when I should hold Office Hours: https://forms.gle/6ZTSfrJ2WbyKMfk39


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## How do I get the path to the current script with Node.js?

How would I get the path to the script in Node.js?

I know there's process.cwd, but that only refers to the directory where the script was called, not of the script itself. For instance, say I'm in /home/kyle/ and I run the following command:

node /home/kyle/some/dir/file.js

If I call process.cwd(), I get /home/kyle/, not /home/kyle/some/dir/. Is there a way to get that directory?

How to get the current folder in Node, Using __dirname in a Node script will return the path of the folder where the current JavaScript file resides. Using ./ will give you the current  How do I get the path to the current script with Node.js? How would I get the path to the script in Node.js? I know there's process.cwd , but that only refers to the directory where the script was called, not of the script itself.

So basically you can do this:

fs.readFile(path.resolve(__dirname, 'settings.json'), 'UTF-8', callback);

Use resolve() instead of concatenating with '/' or '\' else you will run into cross-platform issues.

Note: __dirname is the local path of the module or included script. If you are writing a plugin which needs to know the path of the main script it is:

require.main.filename

or, to just get the folder name:

require('path').dirname(require.main.filename)

How to get the path of current script using Node.js ?, We can get the path of the present script in node. js by using __dirname and __filename module scope variables. __dirname: It returns the directory name of the current module in which the current script is located. __filename: It returns the file name of the current module. Then to get the path to the current ES module one has to use: const __filename = new URL(import.meta.url).pathname; And for the directory containing the current module: import path from 'path'; const __dirname = path.dirname(new URL(import.meta.url).pathname);

This command returns the current directory:

var currentPath = process.cwd();

For example, to use the path to read the file:

var fs = require('fs');
{
if(err)
console.log(err)
else
console.log(data.toString());
});

Get the Path of the Current Working Directory in NodeJS, How do I get the current working directory in node JS? Using __dirname in a Node script will return the path of the folder where the current JavaScript file resides. Using ./ will give you the current working directory. It will return the same result as calling process.cwd (). Initially the current working directory is the path of the folder where you ran the node command,

Use __dirname!!

__dirname

The directory name of the current module. This the same as the path.dirname() of the __filename.

Example: running node example.js from /Users/mjr

console.log(__dirname);
// Prints: /Users/mjr
console.log(path.dirname(__filename));
// Prints: /Users/mjr

https://nodejs.org/api/modules.html#modules_dirname

For ESModules you would want to use: import.meta.url

Introduction to Node.js, How do I navigate to a folder in node JS? Add scripts-prepend-node-path=true into .npmrc file in Node.js Application Root folder. Go to Domains > example.com > Node.js. Click 'open' next to Application Root. Using File Manager add scripts-prepend-node-path=true into .npmrc file. Create the file if it does not exist.

When it comes to the main script it's as simple as:

process.argv[1]

From the Node.js documentation:

##### process.argv

An array containing the command line arguments. The first element will be 'node', the second element will be the path to the JavaScript file. The next elements will be any additional command line arguments.

If you need to know the path of a module file then use __filename.

Difference between __dirname and ./ in Node.js, then you simply write in console D: to change drive from c to d. Initiate the Node.js File. The file you have just created must be initiated by Node.js before any action can take place. Start your command line interface, write node myfirst.js and hit enter:

Node.js, are used to get the filename and directory name of the currently executing file. The ./ gives the current working directory. It works similar to process. Get working directory name and file name in Node. Node provides two variables named “__dirname” and “__filename” who returns the directory path of the currently executing file and the exact location of the file respectively. Both of them return string. To know in depth go through the following.

Node get path of current script, The first method to get the path of the current directory is the __dirname method. This is a Node. js core module that gets the current path of whatever directory the JavaScript file or module is running in. And you can also get the same result by using the path. Node.js® is a JavaScript runtime built on Chrome's V8 JavaScript engine.

Node.js File Paths, Using __dirname in a Node script will return the path of the folder where the current JavaScript file resides. js does. __filename is the file name  sys.path[0] is the directory that your calling script is in (the first place it looks for modules to be used by that script). We can take the name of the file itself off the end of sys.argv[0] (which is what I did with os.path.basename). os.path.join just sticks them together in a cross-platform way.

• Or simply __dirname.split(path.sep).pop()
• Or require('path').basename(__dirname);
• If your goal is just to parse and interact with the json file, you can often do this more easily via var settings = require('./settings.json'). Of course, it's synchronous fs IO, so don't do it at run-time, but at startup time it's fine, and once it's loaded, it'll be cached.


text:
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# A fish has 5000 catfish in his pond. The number of catfish by $8 \%$ per month and the farmer harvests 300 catfish per month.(a) Show that the catfish population $P_n$ after $n$ months is given recursively by $P_n = 1.08 P_{n-1} - 300$$P_0 = 5000$(b) How many catfish are in the pond after six months?

## a. $P_{n}=1.08 P_{n-1}-300$b. 5734

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so initially we have 5000 catfish in the pond. However, that will increase by 8%. But then we also lose an additional 300 fish per month due to harvest. So here we have P N for part A. So what's the amount? After n months? Well, we take the previous amount from last month, but then we'll also add 8% of that. How and because of harvest were also subtracting off 300. And then if we just simplify this, this is just 1.8 p n minus one minus 300. This verifies the formula in part A for part B. Let's just use this formula repeatedly to find B six p six. Excuse me so I'll need to go to the next page for this one p one That's just 1.8 p zero minus 300 So 5100 p. Two. 1.8 p. One minus 300 52 Oh eight. Let's keep going in this fashion P three 1.8 52 oh eight minus 300 this time for P three 53 to 4 0.64 Let's not worry about the decimals right now. Let's just keep them until we get to the very end. Then we'll figure out what to do with the decimal. Now, I'm just going to carry this decimal over minus 300 so I'm just using the formula over and over again. This time I'm on P four, 5450 6112 two more to go minus 300 and pf five. That's five, 586 66 Oh, 96 And finally P six. And then we have P five here minus 300. And then take this one back to the calculator. And this one is 5733 with some change left over. That's about 0.59 and we should round this up to be set 5734 And the units were catfish. So after six months, we have 5734 catfish after six months, and that's our final answer

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# Tag Info

110

You will find a lot of information in this answer. I will add a few personal notes. Module Use Module when you want to localize variables inside your functions body, and those variables will potentially acquire and/or change their values during the computation. Basic use For example: f[x_]:=Module[{y=x^2},y=y+x;{x,y}] Here, a local mutable variable ...

29

The differences between Module, Block and With are nicely summarized by the results of the following expressions: x = "global"; f[] := x Module[{x = "local"}, {x, f[], Hold[x]}] Block[{x = "local"}, {x, f[], Hold[x]}] With[{x = "local"}, {x, f[], Hold[x]}] which returns: {"local", "global", Hold[x$123]} (* Module *) {"local", "local", ... 29 An ugly hack, look at all things in Global context, keep in table if Dimensions didn't return {} Grid[Select[{#, Dimensions[ToExpression@#]} & /@ Names["Global*"], #[[2]] != {} &], Alignment -> Left] For this to be helpful it needs to be updated dynamically and preferably be in a palette to avoid scrolling up all the time. Instead of ... 29 Rather than answering your question as posed, let me instead save you the effort of writing such a function and at the same time demonstrate how it can be done by posting some code that I've already written for this purpose: BeginPackage["CovariancePropagation"]; Unprotect[var, cov]; ClearAll[var, cov]; SetAttributes[var, HoldAll]; SetAttributes[cov, ... 27 I'll cover a few typical uses of Block, neither of which is possible using Module or With. Temporarily removing definitions When you do Block[ {a = x}, ... ] the original definition of a is effectively replaced by whatever new definition is given in the first argument of Block, for the duration of the evaluation of Block only. If we give no ... 24 Here is the almost obligatory timing response, it probably doesn't generalise very broadly but perhaps is indicative in some respects: (* no variables *) f1[x_] := (x^2; x^3;) f2[x_] := Module[{}, x^2; x^3;] f3[x_] := Block[{}, x^2; x^3;] f4[x_] := With[{}, x^2; x^3;] (* With variable definition *) f2[x_] := Module[{y = 0}, x^2; x^3;] f3[x_] := Block[{y = ... 24 Maybe this ? ClearAll["Global*"] 23 Basic proposal There are a number of options and their attractiveness will depend on the scenario for their use, therefore it is difficult to make any broad recommendations of best practice. I will say that generally it is not recommended to rely on global assignments as in your first example, because this method scales poorly and because it is easy to ... 21 Your code reveals exactly why Clear complains: Subscript[x, r] is not a Symbol nor a String. When you assign a value to it, you're setting a DownValue not an OwnValue; in other words, you're setting the value of a function not a variable. To use$x_r$as a symbol, use the Notation package's function, Symbolize. I'd recommend using it from the palette ... 20 As the error message indicates Clear does not work that way. There are several assignment forms that automatically create a definition to something other than a raw symbol: x[5] = 1; Subscript[x, 1] = 2; x /: Subscript[x, 2] = 3; N[x] = 3.14159; DownValues[x] DownValues[Subscript] UpValues[x] NValues[x] {HoldPattern[x[5]] :> 1} ... 20 You can explicitly define variables in the global context by prefixing their name with Global, for example, Globali = 3 or Globalf[x_]:=x^2. However if you have set the notebook to have private context, you don't have Global in your$ContextPath (in order to prevent interference from other notebooks with non-private context). Therefore in your notebooks ...

19

An alternative that doesn't require protecting or using private contexts: Clear @@ DeleteCases[Names@"*", "b"];

19

You could use capital Nu, \[CapitalNu], from the Greek alphabet. It is visually almost identical to capital N from the Roman alphabet. But it has no predetermined assignment. \[CapitalNu] = 5 2 \[CapitalNu] The following shows how the input is displayed on screen.

19

Here is the simplest answer: sum[n_] := Sum[i x[i], {i, 1, n}] x /: D[x[i_], x[j_], NonConstants -> {x}] := KroneckerDelta[i, j] D[sum[n], x[2], NonConstants -> x] $\begin{cases} 2 & n>1 \\ 1-n & \text{True} \end{cases}$ The trick here is the use of the NonConstants option of the derivative operator. This then has to be ...

18

For a single code statement, this is probably an overkill. If you have two or more of them, you have to group them in any case. CompoundExpression is one obvious choice, such as f[x_]:= ( Print[x]; x^2 ) Instead, you could also do f[x_]:= Module[{}, Print[x]; x^2 ] which is what I personally often prefer. Apart from some ...

18

In the days when computers were slower, and the kernel took a long time to start up (in wall time), a little package was made to help with cleaning up without having to restart the kernel. This package is still included with Mathematica, and is found in AddOns\ExtraPackages\Utilities\CleanSlate.m (within the Mathematica installation directory). It is more ...

17

I will leave the explanation of the difference between lexical and dynamic to those more qualified than myself (and to the documentation). One practical difference between Block and Module is this: Module[{x}, x] Block[{x}, x] (* -> x$1979 x *) That is, if you construct something inside a Block with a blocked variable and then return it, you may use ... 17 There will no doubt be plenty of answers for this one. However the short answer is: Use With for local constants that you don't have to change subsequently. Use Module for local variables that are local to that piece of code. Use Block for local variables that are local to that sequence of evaluation. This tutorial in the Mathematica documentation ... 16 You can't define an unassigned symbolic variable throught itself. You are trying to do something like that: x = F[x] This is not right for symbolic computations, because x evaluates to itself as a pure symbolic value. Your code in FullFrom is: Equal[v, List[Subscript[v, 1], Subscript[v, 2], Subscript[v, 3]]] So, you get the recursion. Try different ... 16 Automatic way based on Names Here is a simple modification of the recent answer of @R.M, which is based on the definiton of a variable as a symbol which has an OwnValue defined: Clear @@ Select[ Names["Global`*"], ToExpression[#, StandardForm, Function[sym, OwnValues[sym] =!= {}, HoldAll] ] & ] If your code is in ... 16 In V10, another option is to use Association. par=<|"mu"->1,"sigma"->1,"lb"->0,"ub"->10|>; f[x_, p_Association:par] := PDF[LogNormalDistribution[p["mu"], p["sigma"]], x] Plot[f[x, ##], {x, #lb, #ub}] &@par Another form for Plot is: Plot[f[x, par], {x, par@"lb", par@"ub"}] And as @Mr.Wizard commented, you can use the default ... 15 For reading you can just use SetDelayed (:=). For writing you can define an UpValue for Set thus: mydata = {3, 14, 15, 92, 65} alice := mydata[[1]] alice /: Set[alice, value_] := mydata[[1]] = value Then for most purposes this will work as I expect you want it to: alice = alice*2; mydata {6, 14, 15, 92, 65} Even though it's already rather ... 15$ is probably the only non-alphanumeric ascii character without a special meaning in Mathematica and thus the only one you could use as a delimiter for various parts within a variable name. A warning is due: Because it is so unique, it is also used internally for the same purpose, e.g. Module and Unique will generate variable names ending in $+ an ... 14 You cannot make definitions with patterns on the left-hand side in the first argument of a scoping construct (such as Module). You need do that in the body of the Module. You should also use a different symbol for the internal function parameter. norm[x_] := Module[{fun1, fun2}, fun1[p_] := p^2 + p - 1; fun2[p_] := p^3 - p^2 + p + 1; ... 14 You can give the function one of the Hold Attributes. SetAttributes[fun, HoldFirst] Then as Leonid suggested fun[var_] := SymbolName[Unevaluated@var] Without the hold attribute, this will not work. 14 General considerations To my mind, the only robust way to do this is to build some custom object model in Mathematica, and in particular to restrict the way values can be changed to some well-defined route you can control. Because, as it follows from one of the discussions you linked to, there seems to be no reliable way to intercept arbitrary value changes ... 13 I will suggest a solution for DownValues - based definitions, but it may be generalized to other types of definitions as well. I will only consider a case of a single symbol, but again, the generalization is rather straightforward. You will also have to execute your code in a special dynamic environment. A first ingredient of my suggestion is a (slightly ... 13 I use a shortcut key Ctrl+Q for Quit[], allowing rapid clearing of all sessions variables. Here is how you can add this to Mathematica: You will be editing KeyEventTranslations.tr. This is an important system file so be careful. Start by copying the file you are going to edit from the$InstallationDirectory to $UserBaseDirectory in the same tree. This ... 13 As it is often voiced here, modifying built-in variables is not a good idea most of the times, especially in case of such fundamental symbols as N. It is used heavily through millions of underlying code lines, and you will never know where your change can (and will) cause any mischief (or catastrophe). On the other hand, the name of the variable and the way ... 13 Perhaps what you're looking for is something like this: Module[{x, a, b}, x[1] = 1; x[2] = 10; a + b/x[2] + x[1] ]$\text{a$\$$3026}+\frac{\text{b\$$3026}}{10}+1$ Here I defined a single additional local variable x but then refer to "indexed" variables sharing the same name and differing only in the index x[1] and x[2] etc. These indices are ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
## anonymous one year ago Okay, so I've got another log question, except the log bases are different. That difference is making me pretty confused. 2log4(9) - log2(3)

1. CGGURUMANJUNATH

|dw:1440359012266:dw|

2. CGGURUMANJUNATH

|dw:1440359062943:dw|

3. IrishBoy123

$$2log_4(9) - log_2(3)$$ yep?

4. anonymous

Yep :)

5. IrishBoy123

so you are now base switching?

6. anonymous

I believe so...

7. anonymous

(I'm finally on a computer so I can actually use proper equations yay)

8. IrishBoy123

$$\huge \log_b a = {\log_d a \over \log_d b}$$

9. IrishBoy123

so you have options here

10. anonymous

so what would be the common base? log 2?

11. IrishBoy123

dunno for sure, haven't done this in ages, but 2 sounds like a good way to start

12. IrishBoy123

you get a simple answer with bas 2 am pretty sure that @CGGURUMANJUNATH 's drawings are really helpful too but that scrawl does my head in

13. anonymous

So the final answer is |dw:1440360586587:dw|

14. IrishBoy123

yep

15. IrishBoy123

@CGGURUMANJUNATH is doing some really cool stuff with squaring the base itself. that is definitely worth checking out

16. anonymous

Yeah :D

17. anonymous

hanks for helping me out everyone :D

18. anonymous

Thanks*


text:
Home > Standard Error > Calculate Standard Deviation From Standard Error Of Mean

# Calculate Standard Deviation From Standard Error Of Mean

## Contents

doi:10.2307/2682923. Because the age of the runners have a larger standard deviation (9.27 years) than does the age at first marriage (4.72 years), the standard error of the mean is larger for It will be shown that the standard deviation of all possible sample means of size n=16 is equal to the population standard deviation, σ, divided by the square root of the asked 5 years ago viewed 23370 times active 4 years ago Blog Stack Overflow Podcast #89 - The Decline of Stack Overflow Has Been Greatly… 11 votes · comment · stats have a peek at these guys

Generated Thu, 06 Oct 2016 00:57:50 GMT by s_hv1002 (squid/3.5.20) Follow @ExplorableMind . . . The standard error estimated using the sample standard deviation is 2.56. R+H2O for marketing campaign modeling Watch: Highlights of the Microsoft Data Science Summit A simple workflow for deep learning gcbd 0.2.6 RcppCNPy 0.2.6 Using R to detect fraud at 1 million http://handbook.cochrane.org/chapter_7/7_7_3_2_obtaining_standard_deviations_from_standard_errors_and.htm

## Calculate Standard Error From Standard Deviation In Excel

Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. The means of samples of size n, randomly drawn from a normally distributed source population, belong to a normally distributed sampling distribution whose overall mean is equal to the mean of Greek letters indicate that these are population values.

Learn More . Search over 500 articles on psychology, science, and experiments. Thank you to... Standard Error Formula Statistics Next, consider all possible samples of 16 runners from the population of 9,732 runners.

Standard Error In the theory of statistics and probability for data analysis, Standard Error is the term used in statistics to estimate the sample mean dispersion from the population mean. Convert Standard Deviation To Standard Error In Excel The standard deviation of the sample mean is $\sigma/\sqrt{n}$ where $\sigma$ is the (population) standard deviation of the data and $n$ is the sample size - this may be what you're Boost Your Self-Esteem Self-Esteem Course Deal With Too Much Worry Worry Course How To Handle Social Anxiety Social Anxiety Course Handling Break-ups Separation Course Struggling With Arachnophobia? http://stats.stackexchange.com/questions/15505/converting-standard-error-to-standard-deviation ISBN 0-8493-2479-3 p. 626 ^ a b Dietz, David; Barr, Christopher; Çetinkaya-Rundel, Mine (2012), OpenIntro Statistics (Second ed.), openintro.org ^ T.P.

All the R Ladies One Way Analysis of Variance Exercises GoodReads: Machine Learning (Part 3) Danger, Caution H2O steam is very hot!! Standard Error Of The Mean Definition The standard deviation of all possible sample means of size 16 is the standard error. And if so, is this formula appropriate? $$SE = \frac{SD}{\sqrt{N}}$$ standard-deviation standard-error share|improve this question edited Jul 16 '12 at 11:34 Macro 24.2k496130 asked Sep 13 '11 at 13:54 Bern 86113 Standard error of the mean This section will focus on the standard error of the mean.

## Convert Standard Deviation To Standard Error In Excel

If people are interested in managing an existing finite population that will not change over time, then it is necessary to adjust for the population size; this is called an enumerative https://explorable.com/standard-error-of-the-mean When the true underlying distribution is known to be Gaussian, although with unknown σ, then the resulting estimated distribution follows the Student t-distribution. Calculate Standard Error From Standard Deviation In Excel Ecology 76(2): 628 – 639. ^ Klein, RJ. "Healthy People 2010 criteria for data suppression" (PDF). Convert Standard Error To Variance If the sample size is large (say bigger than 100 in each group), the 95% confidence interval is 3.92 standard errors wide (3.92 = 2 × 1.96).

They may be used to calculate confidence intervals. More about the author How can I kill a specific X window Why does Ago become agit, agitis, agis, etc? [conjugate with an *i*?] 2048-like array shift Zero Emission Tanks How do I debug an However, the sample standard deviation, s, is an estimate of σ. This formula may be derived from what we know about the variance of a sum of independent random variables.[5] If X 1 , X 2 , … , X n {\displaystyle Estimated Standard Error Formula

As an example, consider an experiment that measures the speed of sound in a material along the three directions (along x, y and z coordinates). Relevant details of the t distribution are available as appendices of many statistical textbooks, or using standard computer spreadsheet packages. Hyattsville, MD: U.S. check my blog Jobs for R usersFinance Manager @ Seattle, U.S.Data Scientist – AnalyticsTransportation Market Research Analyst @ Arlington, U.S.Data AnalystData Scientist for Madlan @ Tel Aviv, IsraelBioinformatics Specialist @ San Francisco, U.S.Postdoctoral Scholar


text:
# Transmission Lines and Impedance Matching: what is considered electrically 'short'?

by satchmo05
Tags: considered, electrically, hort, impedance, lines, matching, transmission
 P: 114 Hey all, Between two devices, I will need to have a coaxial line (with SMA connections on either end) connecting them. The operating frequency of the signal I am propagating is approximately 125 MHz. With this said, the wavelength (assuming ideal) is 2.4 meters. I have heard from numerous sources that if electrically 'long,' it is greater than 0.25λ, which is any length larger than 0.6 meters. Any transmission line length smaller than 0.6 meters is considered "electrically short." My question is in regards to impedance matching: if my transmission line is "electrically short," do I need to worry about impedance matching? Obviously, everything would be happy if I matched at the load, but I'm interested to know if I can skip a step. My thought is that since I'm at such a high frequency, attenuation and loss effects will be near negligible. Thoughts? Thanks for your help! - Satchmo05
 P: 2,451 More info please. If the input and output impedances are the same then the length of the line is not significant. Of course they should match the impedance of the line that connects them unless the line is considerably shorter than a wavelength. Like 0.1λ. - Why do you equate high frequency to negligible attenuation?
 P: 114 I guess what I should've asked is when do I not have to worry about impedance matching, in terms of transmission line length? When less than 0.25 lambda? When electrically small (< 0.1 lambda)? My reasoning there was more of a length issue: attenuation constants, line capacitance, line impedance, etc. are all measured in meters. If my coax line is less than 0.1 lambda < 0.24 m, then minimal attenuation can be expected.
P: 3,843

## Transmission Lines and Impedance Matching: what is considered electrically 'short'?

 Quote by satchmo05 Hey all, Between two devices, I will need to have a coaxial line (with SMA connections on either end) connecting them. The operating frequency of the signal I am propagating is approximately 125 MHz. With this said, the wavelength (assuming ideal) is 2.4 meters. First, assume the $\epsilon_r=4\;$ velocity is about half of speed of light which is 1.5EE8 m/s. At 125MHz, λ= 1.2m, what you have is free air velocity. I have heard from numerous sources that if electrically 'long,' it is greater than 0.25λ, which is any length larger than 0.6 meters. Any transmission line length smaller than 0.6 meters is considered "electrically short." The is not correct at all. A short to the source is only refer to an open end coax 0.25λ long from the source. Any length longer or shorter will not be a short anymore. My question is in regards to impedance matching: if my transmission line is "electrically short," do I need to worry about impedance matching? Obviously, everything would be happy if I matched at the load, but I'm interested to know if I can skip a step. My thought is that since I'm at such a high frequency, attenuation and loss effects will be near negligible. Thoughts? Thanks for your help! - Satchmo05
Transmission line matching has to is a lot more complicated. The easiest way is to plot the source and load impedance onto the Smith Chart, then you need to design a network using either tx line or other elements to connect the two points, then you match the circuit. If you need the transmission line in between, then you plot the two points, then draw the circle from the load with center at the center of the graph. Find the length and add element to move to the source point. It is a lot more complicated. Very few cases you can just use a section of coax and get perfect match.
 P: 1,784 You also need to remember that the wavelength in free space is not the same as the wavelength in coax. You need to look up the propagation velocity for the coax you're using and divide your wavelength by that percentage. You do have to worry about impedance matching even with short transmission lines. RF engineers match impedances between stages even though the electrical length is essentially zero. When the impedances are not matched, some of the energy is reflected from the load reducing the power out. Also that reflected energy can cause the final stage to heat up and even go into oscillation. A quarter wavelength transmission line can be your worst case. If you have a quarter wavelength attached to your source and forget to connect a load, the open at the end of the transmission line will appear to be a short at the last stage and may destroy the transistor. If you can give us the complex impedances of your source and load, we can help you match them.

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