But the determination of this constant mark just the beginning of our work. We assume that this formula was given under certain laboratory conditions in which there were two masses separated by a distance known as accurately as possible. The exponent 10 -11
that appears in the numerical value the constant G
tells us in a way that the effect to be measured is extremely small, which is expected, because two smaller masses the size of a marble used for the experiment will be attracted to a very weak force barely detectable almost . The existence of a force of attraction between two small masses can be confirmed in an experiment of medium complexity, this does not present major challenges. But the measure G
constant and not only to find that two bodies attract represents serious difficulties. The first to evaluate the constant G
was Cavendish laboratory, who used a device that was essentially a torsion balance by implementing the next schmo:
Although at first glance one thinks of the possibility of increasing all bodies used in the experiment to make the effect more intense attraction between them, no such thing can lead to out moving masses (red) without bursting the thin thread which hangs these weights. You can, however, increase the mass of blue, and this was precisely what he did Cavendish. Note that there are only two masses being attracted but two pairs
allying mass, which increases the effect on the torsion balance. Either way, the enormous difficulties in obtaining a numerical value G
reliable under this experiment will not increase our confidence in the value of G
thus obtained.
However, a value of G
obtained under these conditions and put in the formula does not guarantee that the formula will work as predicted by Newton for other conditions that are very different from the conditions used in the laboratory, which involved masses much greater distances. This formula is not the only one who can give us an attractive force between two masses decreases with increasing distance between the geometric centers of the masses. We can formulate a law that says: "Two bodies attract because direct product of their masses and inversely as the distance that separates them. "Note the absence of the term" square
of the distance separating them. And we can make both formulas coincide numerically for some distance apart, but the variation in the strength of attraction given by the two formulas, it will become more and more evident as the masses are being brought closer or separated more and more. As both formulas, mathematically different, are not equally valid to describe the same phenomenon, one has to be discarded
with the help of experimentally obtained data . Again we have to go to the lab. The confirmation of Newton's law is valid we get if we can measure the force of attraction for various distances by a graph of the results. For a variation inversely proportional to the square of the distance, the graph should be as follows:
one way or another, are laboratory experiments that help us to confirm or rule out any theory like this. And in experiments difficult by their very nature, which introduces a statistical random error that we can introduce experimental variation in each reading we take, we are almost obliged to collect the maximum amount of data to increase the reliability of the results, in which case the problem will be trying to draw any conclusion from the mass of data collected, because you can not expect all or perhaps none of the data will "fall" smoothly and accurately on a continuous curve. This forces us to try to find somehow the mathematical expression for a smooth, continuous curve, among many other possible ones, that best fits the experimental data.
case we have spoken this is the typical case in that before carrying out measurements in the laboratory and there is a theoretical model
-a formula
- waiting to be confirmed experimentally by measurements or observations made after the formula was obtained. But many other cases in which although experimental data, despite the unavoidable sources of variation and measurement errors seem to follow any law that can be adjusted to a theoretical model, such a formula does not exist, either because they have not found or perhaps because it is too complex to be stated in a few lines. In such cases the best we can do is carry out the fit of the data obtained experimentally to a
empirical formula, a formula selected from many to be the one that best fits the data. The most notorious example of this today has to do with the alleged global warming on Earth, confirmed independently by several experimental data collected over several decades in many places around the Earth. Still no exact formula or even an empirical formula that allows us to predict the Earth's temperatures will be in later years if things continue as usual. All we have are graphs in which, broadly speaking, one can see a tendency towards a gradual increase in temperature, inferred from the data trend
(
trend data), and even some of these data are cause controversial, such as temperature data registered in Punta Arenas, Chile, between 1888 and 2001:
According to the graph of this data which is set in a straight line of red that, mathematically speaking, represents the trend Data averaged over time, the temperature in that part of the world has been rising for over a century, but instead has been declining by a cumulative average of 0.6 degrees Celsius, contrary to the measurements have been carried out in other parts of the world. We do not know exactly why that happens there is different from that observed in other parts of world. Possibly there are interactions with sea temperatures, with the climatic conditions in that region of the planet, or even up to the rotation of the Earth, which are influenced to cause a fall rather than a rise in temperature observed at Punta Arenas. Anyway, despite ups and downs of the data, with such data is possible to obtain the straight line of red-superimposed on the data, which mathematically speaking is "fit" better than other lines to the average mobile data accumulated . Trends continue, this straight line allows us to estimate, between high and low that they occur in the data in subsequent years, average temperatures will short term in Punta Arenas in the years ahead. On these data, the line of best fit ( best fit) is a completely empirical formula for which there is as yet no theoretical model to support it. And just as there are many in this formula which is a mathematical model to simplify something that is being observed or measured.
Often, when carrying out the plotting of the data (the most important step prior to the selection of the mathematical model which will try to "adjust" the data) since before attempting to carry out the adjustment data to a formula we can detect the presence of an abnormality in the same due to an unexpected source of error that has nothing to do with the error of a statistical nature, as shown in the following graph of the temperatures of lakes in Detroit:
Note
carefully on this chart that there are two points that were not linked with lines for researchers to highlight the presence of a serious anomaly in the data. These are the points representing the end of 1999 and early 2000. As we begin 2000, the data show a "jump" disproportionate to the data prior history. Although we try to force
all data are grouped under a certain trend predicted by an empirical formula, an anomaly as seen in this graph we practically cries out for explanation before being buried in such empirical formula. A review of the data revealed that, indeed, the "jump" disproportionate had to do with a phenomenon that was already expected at that time was going to happen with some computer systems are not prepared for the consequences of the change of digits in the date 1999 to 2000, then dubbed as the Y2K phenomenon
(an acronym for the phrase "
Y ear 2000" where k
symbolizes thousand). The discovery of this effect gave rise to an exchange of explanations documented in places like the following:
http://www.climateaudit.org/?p=1854
http://www.climateaudit.org/?p=1868
This exchange for clarification led to the same U.S. space agency NASA to correct their own data, taking into account the effect Y2k, and corrected data displayed on the following site:
http://data.giss.nasa.gov/gistemp/graphs/
Fig.D.txt
The examples we have seen have been instances in which the experimental data despite variations in the same setting allows them to approximate a mathematical formula or even allow the detection of some error in gathering them. But there are many other occasions in which to carry out the data to plot the presence of a trend is not at all obvious, as shown in the following sample data collected on the frequency of sunspots (which may have some effect on global warming of the Earth):
In the graph of the data has been superimposed a red line under a statistical mathematical criterion represents the line of best fit to the data. But in this case this line is clearly an obvious decline, including online almost seems to be a horizontal line. If you delete line, the data seem so scattered that have chosen a straight line to try to "bundle" the trend of data seems rather an act of faith than scientific objectivity. It may be no reason to expect a statistically significant change on the frequency of sunspots over the course of several centuries or even several thousand years, given the enormous complexity of the nuclear processes that keep the Sun in constant activity. This last example demonstrates the enormous difficulties that face any researcher trying to analyze a set of experimental data on which there is any theoretical model.
The data set formulas is of vital importance always bear in mind the law of cause and effect
. In the case of the law of universal gravitation, as set forth by an exact formula, assuming the masses of two bodies as unchanged, then a variation of the distance between the masses will have a direct effect on the strength of gravitational attraction is between them. A series of experimental data on a chart positions will confirm this. And even in cases where there is an exact model, we can (or rather, we need to) make a cause-effect relationship for a model of two variables can have some meaning. Such is the case when performing measurements of stature between students of different grades of elementary school. In this case, the average heights for each grade students will be different, it will increase as increasing the grade, for the simple fact that students at this age are growing in stature every year. In short, the higher the grade, the higher the average height of students who expect to find in a group. This is a cause-effect relationship. In contrast, if we find a direct relationship between the temperature of a city for one day of the year and the number of pets that people have in their homes, most likely will not find any relationship and come out empty handed because there is no reason to expect that the average number of pets per household (case) may have some influence on the temperature (effect), and if so, this effect would be negligible by mathematically smallness.
we have seen cases involve situations based on natural phenomena which we can carry out measurements in the laboratory or outside the laboratory using something as simple as a thermometer or as an amateur telescope. But there are many other cases in which it is not necessary to conduct measurements, because more than getting data in the lab what is needed is a mathematical model that allows us to make a projection or prediction with data already on hand, as data from a census or a survey. An example is the expected growth of Mexico's annual population. The national population census is to be in charge of obtaining figures on the population of Mexico, so that to try to make a prediction on the expected population growth in future years all we have to do is go to National Institute of Statistics, Geography and Informatics (INEGI) for the results of previous censuses. It is assumed that in these census figures are not exact, there is no reason to expect such a thing, given the enormous number of variables that have to deal with workers who must carry out the census and the changing day to day can affect the "reality" of the census. Even assuming that the census could be carried out exactly, we would be another problem. If we plot the population ratios of 5 years (for example), there would be no problem in making future predictions based on past data if the data when plotted fall all in a "straight line". The problem is that, when plotted almost never fall on a straight line, usually grouped around what appears to be a curve
. Here we try to "adjust" the data to a Various formulas and use the one that best approximates all the data you already have, for which we need a mathematical-statistical approach that is less subjective as possible. It is precisely for this so we require the principles to be discussed here.
The data adjustment formulas, there are cases in which it is not necessary to go into detailed mathematical calculations for the simple reason that for such cases have been obtained and formulas that only require the calculation of simple things such as the arithmetic mean (often designated as
μ, the Greek letter mu
equivalent of the Latin letter "m", so on average arithmetic) data and standard deviation σ
(the Greek letter sigma
, the equivalent of the Latin letter "s" so the "standard") of them. We are referring to adjust the data to a Gaussian curve
. One such adjustment is applied to situations where instead of having a dependent variable and whose values \u200b\u200bdepend on the values \u200b\u200bthat can make an independent variable X on which perhaps we can exercise some control data set in which what matters is the frequency with which data are collected are located within certain ranges. An example would be grades in certain subjects in a group number of 160 students whose scores show a distribution as follows:
Between 4.5 and 5.0: 4 students
Between 5.0 and 5.5: 7 students
Between 5.5 and 6.0: 11 students
Between 6.0 and 6.5: 16 students
Between 6.5 and 7.0: 29 students
Between 7.0 and 7.5: 34 students
Between 7.5 and 8.0: 26 students
Between 8.0 and 8.5: 15 students
Between 8.5 and 9.0: 11 students
Between 9.0 and 9.5: 5 students
Between 9.5 and 10.0: 2 students
This type of distribution, when plotted, statistically show a tendency to reach a peak in a curve that resembles a bell. The first calculation that we made on such data is the arithmetic average or arithmetic mean
defined as
The way in which data are presented, we must make a slight modification in our calculations to obtain the average arithmetic them, using as the representative value of each interval the average value between the minimum and maximum of each interval. Thus, the representative value of the interval between a score of 4.5 and 5.0 is 4.75, the representative value in the range between 5.0 and 5.5 is 5.25, and so on. Each of these values \u200b\u200brepresentative of each interval we have to give weight "fair" that belongs in the calculation of the mean multiplied by the frequency with which it occurs. Thus, the value 4.75 is multiplied by 4 since that is the frequency with which it occurs, and the value 5.25 is multiplied by 7 since that is the frequency with which it occurs, and so on. Thus, the arithmetic mean of the population of 150 students will:
X = [(4) (4.75) + (7) (5.25) + (11) (5.75) +. .. (9.75) (2)] / 160
X = 7,178
Having obtained the average arithmetic
X , the next step would be to obtain the dispersion of these data with respect to the arithmetic average, through a calculation and variance σ ² in which we average the sum of the squares of the differences
d i of each data with respect to the arithmetic mean for the
variance σ ² population data:
Σd ² = 4 ∙ (4.75-7.178) ² + 7 (5.25-7.178) ² + ... + 2 ∙ (2-7178) ²
178,923
Σd ² = σ ² =
Σd ² / N = 1,118
with which we can
obtain the standard deviation σ population data (also known as the root mean square deviation-of-range data to the arithmetic mean) with the simple operation of taking the square root of variance: σ
= √ 1118 = 1057
worth noting that the standard deviation σ
evaluated for a sample
taken at random from a population has a slightly different definition of the standard deviation
σ evaluated on all data the total
of the population. The standard deviation σ
sample of a population is obtained by replacing the term
N in the denominator
N-1, because the words of the "purists" value obtained is a better estimate of the standard deviation of the population from which the sample was taken. However, for sufficiently large sample (N
greater than 30), there is little difference between the two estimates of σ
. Anyway, when you want to get a "best estimate" can always be obtained by multiplying the standard deviation we have defined √
N/N-1 . It is important to bear in mind that σ
is a somewhat arbitrary measure of data dispersion, is something that we have defined ourselves, and we use
N-1 instead of N actually denominator is not an absolute. However, it is a universally accepted convention, perhaps among other things (besides the theoretical reasons cited by purists) and the fact that to calculate a dispersion of information is required at least two
data, which is implicitly recognized by using N-1
in the denominator since this way is not possible to give
N worth one without falling into a division by zero, the definition used
N-1 removed from the scene any possible interpretation of σ
with only one value. And another important reason has more to do with the reasons given by the purists is that the use of N-1
in the denominator has to do with something called the
degrees of freedom in the analysis of variance ANOVA known as
(Analysis of Variance
) used in the design of experiments (though that's get out a bit of subject we are discussing this document).
In descriptive statistics, which is carried out taking all values \u200b\u200bof a population of data and not taking a sample of that population, the most important of the graph of the relative frequencies of the data or histogram
is the "area under the curve" rather than the formula of the curve to be pass by "height" of each data, the curve being used to find the
mathematical probability of having a group of students from a range of scores, for example between 7.5 and 9.0, a mathematical probability value always lies between zero and unity. This is what is traditionally taught in textbooks.
However, before applying the statistical tables to carry out some probabilistic analysis of the area under the curve ", it is interesting how well the data fit a continuous curve can be drawn connecting the heights of the histogram. The formula that best describes a set of data which have been shown in the example is one that gives rise to the Gaussian curve
. Then, to the following formula "Gaussian"
have the graph of the continuous curve drawn by this formula:
As can be seen, the curve indeed has the form of a bell, which derives a name with cuales es conocida.
Se puede demostrar, recurriendo a un criterio matemático conocido como el método de los mínimos cuadrados, que una fórmula general que modela una curva Gaussiana a un conjunto dado de datos con la apariencia de una "campana" es la siguiente:
Y resulta que
µ es precisamente la media aritmética de la población de datos designada también como
X , mientras que σ² es la varianza presentada por la población de datos. Esto signica que, para modelar una curva a un conjunto de datos como el que hemos estado manejando en el ejemplo, basta con calcular the mean and variance of the data, and put this information directly to the Gaussian formula, which will curve "best fit" (under the criterion of least squares) to the data. Parameter evaluation
A no problem, since the curve must reach (but not exceeding)
to a height of 34 (the number of students who represent the greater frequency with respect to other ranks of skills) so that the formula of the curve fitted to the data example is the following:
The graph of this Gaussian curve, superimposed on the graph bar that contains discrete data from which it was generated, is as follows:
We can see that the fit is reasonably good, considering the fact that in real life or experimental data never observed exactly fit the ideal Gaussian curve.
One thing which we must deal from the start, which almost never sufficiently clarified and explained well in the classroom is the fact that the general Gaussian formula not only allows positive values \u200b\u200bof X but also even allows values negative
, which have no interpretation in the real world in cases like the one just see (on a grading system for students as we are assuming, any qualification can only vary from zero to ten minimum qualification as highest). In principle, X can range from X to X =- ∞ = + ∞. In many cases this is no problem, since the curve is rapidly approaching zero before X goes down to zero taking negative values, as in our example where the arithmetic is sufficiently far from X = 0 and dispersion of data is small enough to consider negative values \u200b\u200bof X as irrelevant, although the formula allows. But in cases in which the arithmetic mean
X is too close to X = 0 and the data show a large dispersion, there is always the possibility that a complete end of the curve dropping the "other side" in the zone for which X takes negative values . If this happens, it could even force us to abandon the Gaussian model and other alternatives that will certainly be more unpleasant to handle from a mathematical point of view.
has provided a procedure to obtain the formula of a curve to connect the height of each bar of a histogram data showing the shape of the "bell", but it is important to clarify that individual
points curve without real meaning, which is equivalent to say that a point as X = 7.8 for which the value of Y is equal to 28,595 is not something we should mean absolutely nothing, since it is
the region under curve which makes sense, since the curve was generated from
histogram bars that change from one interval to another. However, what we have done here is justified for comparative purposes because, before applying our notions of statistical data set using the Gaussian distribution about whether they want to make sure the data we analyze are shaped like a bell, because if data seem to follow a linear trend ever upward or if instead of a bell we have two bells (the latter occurs when data is accumulated from two different sources), would be wrong to try to force such data on a Gaussian distribution. It is important to add that we have seen the curve is not symmetrical distribution studied in statistical texts, as so we must normalize
formula so that not only the arithmetic mean is shifted to the left the diagram to have a zero value to both sides being symmetrical with respect to X = o, but also the area under the curve has the value unity, this in order to give a performance curve
probability as applied to non-descriptive statistics
but inferential statistics
in which a random sample trying to figure out the behavior of the data from a general population. In this normalization process derives its name from the curve as
normal curve.
Before trying to invest time and effort to set a formula empirical data before doing any arithmetic, it is important to an early graphical data, as this is the first thing that must guide us in selecting the mathematical model to be used for modeling. In the case of distributions frequency as we've seen, which are represented by histograms, if doing a graph of the data we get something like the following:
then if we can "force" the data to fall within a formula modeled on a continuous independent variable whose stroke is "fit" to the heights of the bars of the histograms, obtaining in this last example a setting like this (this adjustment is carried out simply by adding the expressions for two Gaussian curves with means other than by changing individual variances and amplitudes of each curve to the score):
this setting is a setting meaningless, since a chart like this, known as graphical
bimodal, which has two "caps" peaks, is telling us that instead of having a population of data the same source we have is
two populations of different origins data, data that came tucked into a single "package" at the hands of the analyst. This is when the analyst is almost forced to go to the "field" to see how and where data were collected. It is possible that the data represent the lengths of certain beams which were produced by two different machines. It is also possible These data have originated in an experiment in which he was testing the effect of a new type of fertilizer on yield of some crops and the fertilizer was being supplied to experimental plots for two different people in two different places, in which case
there something that is causing a significant difference in the performance of the fertilizer and the effect it may have the fertilizer itself, either that both persons have been supplying different quantities of the same fertilizer, or the characteristics of different areas have caused Gaussian distribution altered the performance of each type of fertilizer.
For the continuous curve "double hump" shown above, this curve was drawn by the following formula score obtained by adding two Gaussian curves and setting the "top" of each curve to match the approximate-manipulating the arithmetic mean μ
at each end-caps with each of the two top bars, modifying also the variance σ
in each term for "open" or "close" the width of each fitted curve will:
Then we separated individual plots (not added) to each of the Gaussian curves shown in the formula, showing a probable range of data from two distinct populations of which came from the scrambled data.
In this example, it was easy to just view the histogram, the bar graph data-the presence of two Gaussian curves instead of one, thanks to the arithmetic mean of each curve (5.7 and 10.4) are separated by a margin of almost two to one. But we will not always so lucky, and there will be cases in which the arithmetic will be so close to each other that will be somewhat difficult for the analyst to decide if it considers all data as one or try to find two different curves, as would occur with a graph whose curve joining the heights of the bars would look like this:
is in cases like these in which the analyst must draw on all her wit and all his experience to decide if he tries to find two discernible groups of data in the data set at hand, or if it is not worth for the presence of two distinct populations riots in one, opting to perform modeling based on a single Gaussian formula.
Discovering the influence of unknown factors that may affect the performance of something as a fertilizer is just one of the primary objectives the design of experiments
. In the design of experiments are not interested in carrying out a modeling of the data to a formula, that comes after it has been established unequivocally how many and what are the factors that can affect performance or response to something. Once passed this stage, we can collect data to carry out the adjustment of data to a formula. In the case of a bimodal distribution, instead of trying to set a formula to describe all data with a single distribution as we have seen, is much better to try to separate the data from the two different populations that are causing the "double hump camel ", a fact that can analyze two data sets separately with the assurance that for each data set we obtain a Gaussian distribution with a single hump. It can be seen from this that the formulas data modeling is a continuous cycle experimentation, analysis and interpretation of results, followed by a new cycle of experimentation and analysis and interpretation of new results to be improving a process or could go better describing the data being collected in a laboratory or field. The data modeling formula goes hand in hand with the procedures for the collection of the same.
PROBLEM :
experimentally in the laboratory is the boiling point for some organic compounds known as alkanes (chemical formula C n H 2n +2 \u200b\u200b ) has the following values \u200b\u200bin degrees Celsius: Methane (1 carbon atom): -161.7
ethane (2 carbons): -88.6
propane (3 carbon atoms): -42.1
butane (4 carbons): -0.5
pentane (5 carbon atoms): 36.1
hexane (6 carbon atoms): 68.7
heptane (7 carbons): 98.4
octane (8 carbon atoms): 125.7
Nonane (9 carbon atoms): 150.8
Dean (10 carbon atoms): 174.0
Make a graph of the data. Does it show any tendency the boiling point of these organic compounds according to the number of carbon atoms that have each compound? The graph of discrete data is as follows:
The graph we can see that the data seem to accommodate a super smooth continuous curve, following the cause-effect, which we suggests that behind this data is a natural law waiting to be discovered by us. Since the data do not follow a straight line, the relationship between them is not a linear relationship, is a non-linear relationship
, and do not expect that the mathematical formula that is behind this curve is that of a line straight. In the absence of a theoretical model that allows us to have the exact formula, the graph arising from this data set is an excellent example of the places where we can try to adjust the data to an empirical formula, the better the data fit , the better we will suggest the nature of natural laws operating behind this phenomenon.
