Lessons of Statistics and Probability
§ 1. STATISTICS
1. Definition.
Statistics gathers (or data collection), organize (or process), analyze and interpret selected information using mathematical tools.
2. Population Statistics and representative sample.
the statistical population is the total set of data (information) on which we base our survey.
representative sample, however, is a small set of data that represents the population statistics. Population Statistics
Example: The Italian government, for political consultations, gives the right to vote to all adult residents, who have Italian citizenship, to elect the parliament. Example
representative sample: The Italian Parliament represents the population statistics.
In the statistical, ie the first stage of investigation or interview, you always use a representative sample (many statistical agencies as doxa, exit the pool, etc.. Use it) because you have a successful result and the while interviewing a few individuals is a saving of money. If the statistical population consists of women between 50% and 50% men, even the representative sample must be formed from 50% to 50% of men and women. If the statistical population has 5% of graduates also the representative sample size should be 5% of graduates and so on. The representative sample must be scaled down image of the statistical population.
3. Observation.
Statistical data are expressed graphically through an orthogonal Cartesian graph where the x-axis could be represented about the characters and the y-axis the frequency of the characters.
4. Statistical characters.
I caratteri statistici sono qualitativi e quantitativi.
Qualitativi se il carattere statistico è un dato non numerico come ad esempio un sostantivo come vino, macchina, auto, bicicletta.
Esempio: Svolgo un’indagine e scopro che:
40 persone usano auto; 20 persone usano scooter e 10 persone usano bicicletta.
Questo dato può essere rappresentato così:
Un classico grafico delle rilevazioni statistiche qualitative è l’ideogramma. In questo grafico, i dati sono rappresentati per mezzo di immagini simboliche. É utile quando si vuole dare un’idea immediata dell'argomento trattato. É facilmente comprensibile, perché basta un'occhiata per capire di cosa si sta parlando; per questo motivo è molto utilizzata sui giornali. Svantaggi: i dati espressi in questo modo non sono molto precisi. Esempio: Consumo di vino in Italia e in Germania.
Quantitativi se il dato statistico può essere espresso da un numero.
I caratteri quantitativi si suddividono in due categorie caratteri discreti e continui.
I caratteri quantitativi sono discreti se sono espressi da un numero (età, voto etc), sono continui se sono espressi da un certo numero di dati (infiniti sotto un profilo teorico).
Esempio dei caratteri quantitativi discreti: Se io misuro l’altezza degli alunni di una classe allora I can say that measure 1.60 m 3 children, two boys 1.62 m, 1.64 m 1 boy; 1.66 m 3 children, a boy 1.68 m, 1.70 m 4 boys; 1.72 m 1 guy, 1.74 m 3 boys.
If I carry out a quantitative representation using discrete values \u200b\u200bI get a chart like this:
Example of discrete quantitative traits. If I measure the height of the pupils in a class (using the same statistical analysis) I can say that measure 1.60 m by 1.64 am six boys, from 1.65 am 1.69 m 4 boys, 1 m, 70 am 1.74 8 boys. If
performs a continuous representation using quantitative values \u200b\u200bI get a chart like the following:
This graph is called Histogram. It may be noted that in statistical graphs of the lines are discrete frequencies (the sum of all frequencies gives rise to the total number of our representative sample of the population or in some cases) instead of continuous frequencies in statistical graphics are some areas (the sum of all frequencies gives rise to the total number of our representative sample, or in some cases to the population).
1. The Media.
If we perform a statistical representation of our data using statistical continuous and equally spaced characters almost always happens that we see a series of rectangles all born with the same amplitude but having different heights. The highest, ie the rectangle that has the highest frequency, is almost always the rectangle that is at the center.
from this experience was born the concept of "expected value" or "true value" or simply Media.
In a statistical distribution of the central value of a distribution is more representative of the entire distribution, mathematically you get by using the following formula:
1. Variance.
Another very important value in a distribution is the variance (the square root is called standard deviation or standard deviation) indicating the degree of dispersion statistics.
Example: If a student reported in classwork 4,6,8 his average is 6. If a student B reported in classwork 2,6,10 his average is always 6. What is the difference between these two students? In the first case the data are less dispersed real axis and in the second case are more dispersed in the first case then the variance will be smaller than the latter.
Mathematically, the standard deviation is indicated by the following formula:
1. Median
Fashion and Fashion is the statistical value (or range that has the highest frequency).
The median is the value that is at the center in a statistical even if, as often happens it is said that both the value with greater frequency.
2. Processing Errors and
phase where you determine the mean, variance, fashion and the median is the phase of data processing. This phase will end with the verification of the presence of possible errors. The errors are divided into accidental and systematic.
accidental errors are the sum of many different types of errors among which the error that arises from the sensitivity of the instrument and the variance.
Example: We perform a statistical measure of the length of tables. Note that we could never have the exact size of each table, because the measure could have a value between 1.64 and 1.65 meters. If we have a more precise measurement tool, we realize that the measurement could have a value between 1.6465 and 1.6466 meters. This speech can be repeated indefinitely, and then we realize that we can not ever get a value of "precise" and "correct". This kind of error is called accidental and born with the inherent concept of measurement.
systematic errors arise from using unsuitable equipment. The first born
mistakes so unintentionally, it can reduce equipment with more precise but can never be entirely clear but the second group of errors can, indeed must, set aside by using instrumentation fit for purpose and must be eliminated by using a statistical probability of the particular procedures.
3. Absolute and relative frequency
Having carried out a survey and processing of data you can begin to plot the statistical data obtained, using a custom histogram as we have seen on pg. 3.
almost always use the unit interval to continue characters statistical quantities so that the area of \u200b\u200beach rectangle, from the point number (and size), is equal to the numerical value of the height and therefore the value of the frequency . The frequency used is the absolute frequency. We can observe
the total area of \u200b\u200ball rectangles will coincide with the total value of the frequencies.
In this case, we assign a frequency range for each call that the absolute (as in the example page. 3).
If we decide to use the relative frequency we associate with each interval the value of
Absolute Relative frequency = frequency / total number of tests
Again, since you use the unit interval, each rectangle will have the same area from the point number (but not size) to relative frequency.
We can see that the total area of all rectangles will coincide with the total value of the relative frequencies and then always have the numerical value of 1.
10._Rappresentazione statistics. The Gauss curve.
The vast majority of statistical graphs using histograms.
If we use the relative frequencies and if the intervals are infinitesimal (ranges theoretical because in practical terms it would never be realized), then the graph is transformed into the Gaussian curve (bell).
This graph is also very popular for applications in various fields have (not just statistics).
All data are found distributed along the curve Gauss. The area under this curve, for the clarification made in the previous paragraph, will be the same (given that you use relative frequency) to 1.
This result also has an important application of probability because the probability is: • 1
when considered in toto the entire distribution,
is a value between 0 and 1 when it is considered a portion, or the sum of several portions of the distribution,
· 0 is not considered when no portion of the distribution.
If the Gaussian has these characteristics:
· the height is greater
· the majority of the data are near (to the right and left) the central value (which represents the true value or the average)
then we are in the presence of a distribution with very few mistakes.
If the Gaussian has, however, these features:
· the height is small
· the lower part of the data are far (to the right and left) of the central value
then we are in the presence of a distribution with many errors .
§ 2. THE LIKELIHOOD
1. Classical probability and frequenstista
There are two concepts of probability: the classical probability (or a priori) in this case the probability is given by:
god of successes / number of cases judged equally valid
· the frequentist probability (or later) in this case the probability is given by:
god of successes / number of tests carried out
These two models of probability even if they initially seem in stark contrast, however, are two sides of same coin.
We note that the frequentist probability (or rear) is closely linked to the statistical concepts.
Just as in statistics and then carry out the first survey to determine some important elaborations (such as the mean, variance, fashion and the median) at the same so the probability is frequentist determines an outcome that took place after the tests or interviews.
1.
The law of large numbers law of large numbers states:
"If we can play endless tests (or interviews) for a given object to infinity then statistically the probability of occurrence of an event will coincide with the classical probability."
From This law created a dilemma.
We see a classic example.
If we consider the various sorties in the lottery in accordance with the classical probability for each number every time extraction is likely to be "caught" (although it was raffled off several times above). The urn is in the absence of memory.
If we consider the frequentist probability for each draw of the lot, the various numbers are not as likely to be precisely drawn and the numbers "laggards" are more likely than others who have been drawn recently. The urn is in the presence of memory.
This example should not put ourselves in a crisis because if we follow the definition of the law of large numbers we can see that the probability that each number has to be raffled will be equal to infinity, and then only with the growth of draws increases, yes, but this probability we have no means of quantifying "Infinity."
The law of large numbers is useful because:
· able to unify the concepts of classical probability and frequentist probability that apparently can appear contradictory,
· can predict certain trends developing reliable statistics.
