TI-83 series
The TI-83 series is a series of graphing calculators manufactured by Texas Instruments. The original TI-83 is itself an upgraded version of the TI-82. Released in 1996, it was one of the most popular graphing calculators for students. In addition to the functions present on normal scientific calculators, the TI-83 includes many features, including function graphing, polar/parametric/sequence graphing modes, statistics and algebraic functions, along with many useful applications. Although it does not include as many calculus functions and programs can be downloaded from certain websites, or written on the calculator. TI replaced the TI-83 with the TI-83 Plus calculator in 1999, which included Flash memory, enabling the device's operating system to be updated if needed, or for large new Flash Applications to be stored, accessible through a new Apps key; the Flash memory can be used to store user programs and data. In 2001, the TI-83 Plus Silver Edition was released, which featured nine times the available flash memory, over twice the processing speed of a standard TI-83 Plus, all in a translucent grey case inlaid with small "sparkles."
The TI-83 was redesigned twice, first in 1999 and again in 2001. The 1999 redesign introduced a design similar to the TI-73 and TI-83 Plus, eliminating the sloped screen, common on TI graphing calculators since the TI-81; the 2001 redesign introduced a different shape to the calculator itself, eliminated the glossy grey screen border, reduced cost by streamlining the printed circuit board to four units. The TI-83 was the first calculator in the TI series to have built in assembly language support; the TI-92, TI-85, TI-82 were capable of running assembly language programs, but only after sending a specially constructed memory backup. The support on the TI-83 could be accessed through a hidden feature of the calculator. Users would write their assembly program on their computer, assemble it, send it to their calculator as a program; the user would execute the command "Send (9prgmXXX", it would execute the program. Successors of the TI-83 replaced the Send backdoor with a less-hidden Asm command.
The TI-83 Plus is a graphing calculator made by Texas Instruments, designed in 1999 as an upgrade to the TI-83. The TI-83 Plus is one of TI's most popular calculators, it uses a Zilog Z80 microprocessor running at 6 MHz, a 96×64 monochrome LCD screen, 4 AAA batteries as well as backup CR1616 or CR1620 battery. A link port is built into the calculator in the form of a 2.5mm jack. The main improvement over the TI-83, however, is the addition of 512 kB of Flash ROM, which allows for operating system upgrades and applications to be installed. Most of the Flash memory is used by the operating system, with 160 kB available for user files and applications. Another development is the ability to install Flash Applications, which allows the user to add functionality to the calculator; such applications have been made for math and science, text editing and day planners, editing spread sheets and many other uses. Designed for use by high school and college students, though now used by middle school students in some public school systems, it contains all the features of a scientific calculator as well as function, parametric and sequential graphing capabilities.
Symbolic manipulation is not built into the TI-83 Plus. It can be programmed using a language called TI-BASIC, similar to the BASIC computer language. Programming may be done in TI Assembly, made up of Z80 assembly and a collection of TI provided system calls. Assembly programs are more difficult to write. Thus, the writing of Assembly programs is done on a computer; the TI-83 Plus Silver Edition is a newer version of the TI-83 Plus calculator, released in 2001. Its enhancements are 1.5 MB of Flash memory, a dual-speed 6/15 MHz processor, 96 kB of additional RAM, an improved link transfer hardware, a translucent silver case, more applications preinstalled. This substantial Flash memory increase is significant. Whereas the TI-83+ can only hold a maximum of 10 apps, the Silver Edition can hold up to 94 apps, it includes a USB link cable in the box. It is completely compatible with the TI-83 Plus; the key layout is the same. A second version of the TI-83 Plus Silver Edition exists, the ViewScreen™ version.
It is identical, but has an additional port at the screen end of the rear of the unit, enabling displays on overhead projectors via a cable and panel. This feature can be useful, it looks similar to the standard TI-83 Plus, but has a silver-colored frame, identical to the standard Silver Edition, around the screen. The TI-83 Plus Silver Edition is listed on the Texas Instruments website as "discontinued." In April 2004, the TI-83 Plus Silver Edition was replaced by the TI-84 Plus Silver Edition. They feature the same processor and the same amount of Flash memory, but the TI-84 Plus Silver Edition features a built in USB port and changeable faceplates. CPU: Zilog Z80 CPU, 6 MHz, or 15 MHz, or Inventec 6S1837 ROM 24 kB ROM (TI-83
Statistics
Statistics is a branch of mathematics dealing with data collection, analysis and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics; when census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements.
In contrast, an observational study does not involve experimental manipulation. Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, inferential statistics, which draw conclusions from data that are subject to random variation. Descriptive statistics are most concerned with two sets of properties of a distribution: central tendency seeks to characterize the distribution's central or typical value, while dispersion characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets.
Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors and Type II errors. Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis. Measurement processes that generate statistical data are subject to error. Many of these errors are classified as random or systematic, but other types of errors can be important; the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics can be said to have begun in ancient civilization, going back at least to the 5th century BC, but it was not until the 18th century that it started to draw more from calculus and probability theory. In more recent years statistics has relied more on statistical software to produce tests such as descriptive analysis.
Some definitions are: Merriam-Webster dictionary defines statistics as "a branch of mathematics dealing with the collection, analysis and presentation of masses of numerical data." Statistician Arthur Lyon Bowley defines statistics as "Numerical statements of facts in any department of inquiry placed in relation to each other."Statistics is a mathematical body of science that pertains to the collection, interpretation or explanation, presentation of data, or as a branch of mathematics. Some consider statistics to be a distinct mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty and decision making in the face of uncertainty. Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, measure-theoretic probability theory.
In applying statistics to a problem, it is common practice to start with a population or process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Ideally, statisticians compile data about the entire population; this may be organized by governmental statistical institutes. Descriptive statistics can be used to summarize the population data. Numerical descriptors include mean and standard deviation for continuous data types, while frequency and percentage are more useful in terms of describing categorical data; when a census is not feasible, a chosen subset of the population called. Once a sample, representative of the population is determined, data is collected for the sample members in an observational or experimental setting. Again, descriptive statistics can be used to summarize the sample data. However, the drawing of the sample has been subject to an element of randomness, hence the established numerical descriptors from the sample are due to uncertainty.
To still draw meaningful conclusions about the entire population, in
Median
The median is the value separating the higher half from the lower half of a data sample. For a data set, it may be thought of as the "middle" value. For example, in the data set, the median is 6, the fourth largest, the fifth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is likely to fall above or below it; the median is a used measure of the properties of a data set in statistics and probability theory. The basic advantage of the median in describing data compared to the mean is that it is not skewed so much by large or small values, so it may give a better idea of a "typical" value. For example, in understanding statistics like household income or assets which vary a mean may be skewed by a small number of high or low values. Median income, for example, may be a better way to suggest; because of this, the median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median will not give an arbitrarily large or small result.
The median of a finite list of numbers can be found by arranging all the numbers from smallest to greatest. If there is an odd number of numbers, the middle one is picked. For example, consider the list of numbers 1, 3, 3, 6, 7, 8, 9This list contains seven numbers; the median is the fourth of them, 6. If there is an number of observations there is no single middle value. For example, in the data set 1, 2, 3, 4, 5, 6, 8, 9the median is the mean of the middle two numbers: this is / 2, 4.5.. The formula used to find the index of the middle number of a data set of n numerically ordered numbers is / 2; this either gives the halfway point between the two middle values. For example, with 14 values, the formula will give an index of 7.5, the median will be taken by averaging the seventh and eighth values. So the median can be represented by the following formula: m e d i a n = a ⌈ # x ÷ 2 ⌉ + a ⌈ # x ÷ 2 + 1 ⌉ 2 One can find the median using the Stem-and-Leaf Plot. There is no accepted standard notation for the median, but some authors represent the median of a variable x either as x͂ or as μ1/2 sometimes M.
In any of these cases, the use of these or other symbols for the median needs to be explicitly defined when they are introduced. The median is used for skewed distributions, which it summarizes differently from the arithmetic mean. Consider the multiset; the median is 2 in this case, it might be seen as a better indication of central tendency than the arithmetic mean of 4. The median is a popular summary statistic used in descriptive statistics, since it is simple to understand and easy to calculate, while giving a measure, more robust in the presence of outlier values than is the mean; the cited empirical relationship between the relative locations of the mean and the median for skewed distributions is, not true. There are, various relationships for the absolute difference between them. With an number of observations no value need be at the value of the median. Nonetheless, the value of the median is uniquely determined with the usual definition. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid.
In a population, at most half have values less than the median and at most half have values greater than it. If each group contains less than half the population some of the population is equal to the median. For example, if a < b < c the median of the list is b, and, if a < b < c < d the median of the list is the mean of b and c. Indeed, as it is based on the middle data in a group, it is not necessary to know the value of extreme results in order to calculate a median. For example, in a psychology test investigating the time needed to solve a problem, if a small number of people failed to solve the problem at all in the given time a median can still be calculated; the median can be used as a measure of location when a distribution is skewed, when end-values are not known, or when one requires reduced importance to be attached to outliers, e.g. because they may be measurement errors. A median is only defined on ordered one-dimensional data, is independent of any distance metric. A geometric median, on the other hand, is defined in any number of dimensions.
The median is one of a number of ways
John Tukey
John Wilder Tukey was an American mathematician best known for development of the FFT algorithm and box plot. The Tukey range test, the Tukey lambda distribution, the Tukey test of additivity, the Teichmüller–Tukey lemma all bear his name, he is credited with coining the term'bit'. Tukey was born in New Bedford, Massachusetts, in 1915, obtained a B. A. in 1936 and M. Sc. in 1937, in chemistry, from Brown University, before moving to Princeton University where he received a Ph. D. in mathematics. During World War II, Tukey worked at the Fire Control Research Office and collaborated with Samuel Wilks and William Cochran. After the war, he returned to Princeton, dividing his time between the university and AT&T Bell Laboratories, he became a full professor at 35 and founding chairman of the Princeton statistics department in 1965. Among many contributions to civil society, Tukey served on a committee of the American Statistical Association that produced a report challenging the conclusions of the Kinsey Report, Statistical Problems of the Kinsey Report on Sexual Behavior in the Human Male.
He was awarded the National Medal of Science by President Nixon in 1973. He was awarded the IEEE Medal of Honor in 1982 "For his contributions to the spectral analysis of random processes and the fast Fourier transform algorithm." Tukey retired in 1985. He died in New Brunswick, New Jersey, on July 26, 2000. Early in his career Tukey worked on developing statistical methods for computers at Bell Labs where he invented the term "bit", his statistical interests were many and varied. He is remembered for his development with James Cooley of the Cooley–Tukey FFT algorithm. In 1970, he contributed to what is today known as the jackknife estimation—also termed Quenouille–Tukey jackknife, he introduced the box plot in his 1977 book, "Exploratory Data Analysis." Tukey's range test, the Tukey lambda distribution, Tukey's test of additivity, Tukey's lemma, the Tukey window all bear his name. He is the creator of several little-known methods such as the trimean and median-median line, an easier alternative to linear regression.
In 1974, he developed, with the concept of the projection pursuit. He contributed to statistical practice and articulated the important distinction between exploratory data analysis and confirmatory data analysis, believing that much statistical methodology placed too great an emphasis on the latter. Though he believed in the utility of separating the two types of analysis, he pointed out that sometimes in natural science, this was problematic and termed such situations uncomfortable science. A. D. Gordon offered the following summary of Tukey's principles for statistical practice:... the usefulness and limitation of mathematical statistics. Tukey coined many statistical terms that have become part of common usage, but the two most famous coinages attributed to him were related to computer science. While working with John von Neumann on early computer designs, Tukey introduced the word "bit" as a contraction of "binary digit"; the term "bit" was first used in an article by Claude Shannon in 1948.
In 2000, Fred Shapiro, a librarian at the Yale Law School, published a letter revealing that Tukey's 1958 paper "The Teaching of Concrete Mathematics" contained the earliest known usage of the term "software" found in a search of JSTOR's electronic archives, predating the OED's citation by two years. This led many to credit Tukey with coining the term in obituaries published that same year, although Tukey never claimed credit for any such coinage. In 1995, Paul Niquette claimed he had coined the term in October 1953, although he could not find any documents supporting his claim; the earliest known publication of the term "software" in an engineering context was in August 1953 by Richard R. Carhart, in a Rand Corporation Research Memorandum. List of pioneers in computer science Andrews, David F. Robust estimates of location: survey and advances. Princeton University Press. ISBN 978-0-691-08113-7. OCLC 369963. Basford, Kaye E. Graphical analysis of multiresponse data. Chapman & Hall/CRC. ISBN 978-0-8493-0384-5.
OCLC 154674707. Blackman, R B; the measurement of power spectra from the point of view of communications engineering. Dover Publications. ISBN 978-0-486-60507-4. Cochran, William G. Statistical problems of the Kinsey report on sexual behavior in the human male. Journal of the American Statistical Association. Hoaglin, David C. Understanding Robust and Exploratory Data Analysis. Wiley. ISBN 978-0-471-09777-8. OCLC 8495063. CS1 maint: Multiple names: authors list CS1 maint: Extra text: authors list Hoaglin, David C. Exploring Data Tables and Shapes. Wiley. ISBN 978-0-471-09776-1. OCLC 11550398. CS1 maint: Multiple names: authors list CS1 maint: Extra text: authors list Hoagl
Outlier
In statistics, an outlier is an observation point, distant from other observations. An outlier may be due to variability in the measurement or it may indicate experimental error. An outlier can cause serious problems in statistical analyses. Outliers can occur by chance in any distribution, but they indicate either measurement error or that the population has a heavy-tailed distribution. In the former case one wishes to discard them or use statistics that are robust to outliers, while in the latter case they indicate that the distribution has high skewness and that one should be cautious in using tools or intuitions that assume a normal distribution. A frequent cause of outliers is a mixture of two distributions, which may be two distinct sub-populations, or may indicate'correct trial' versus'measurement error'. In most larger samplings of data, some data points will be further away from the sample mean than what is deemed reasonable; this can be due to incidental systematic error or flaws in the theory that generated an assumed family of probability distributions, or it may be that some observations are far from the center of the data.
Outlier points can therefore indicate faulty data, erroneous procedures, or areas where a certain theory might not be valid. However, in large samples, a small number of outliers is to be expected. Outliers, being the most extreme observations, may include the sample maximum or sample minimum, or both, depending on whether they are high or low. However, the sample maximum and minimum are not always outliers because they may not be unusually far from other observations. Naive interpretation of statistics derived from data sets. For example, if one is calculating the average temperature of 10 objects in a room, nine of them are between 20 and 25 degrees Celsius, but an oven is at 175 °C, the median of the data will be between 20 and 25 °C but the mean temperature will be between 35.5 and 40 °C. In this case, the median better reflects the temperature of a randomly sampled object than the mean; as illustrated in this case, outliers may indicate data points that belong to a different population than the rest of the sample set.
Estimators capable of coping with outliers are said to be robust: the median is a robust statistic of central tendency, while the mean is not. However, the mean is a more precise estimator. In the case of distributed data, the three sigma rule means that 1 in 22 observations will differ by twice the standard deviation or more from the mean, 1 in 370 will deviate by three times the standard deviation. In a sample of 1000 observations, the presence of up to five observations deviating from the mean by more than three times the standard deviation is within the range of what can be expected, being less than twice the expected number and hence within 1 standard deviation of the expected number – see Poisson distribution – and not indicate an anomaly. If the sample size is only 100, just three such outliers are reason for concern, being more than 11 times the expected number. In general, if the nature of the population distribution is known a priori, it is possible to test if the number of outliers deviate from what can be expected: for a given cutoff of a given distribution, the number of outliers will follow a binomial distribution with parameter p, which can be well-approximated by the Poisson distribution with λ = pn.
Thus if one takes a normal distribution with cutoff 3 standard deviations from the mean, p is 0.3%, thus for 1000 trials one can approximate the number of samples whose deviation exceeds 3 sigmas by a Poisson distribution with λ = 3. Outliers can have many anomalous causes. A physical apparatus for taking measurements may have suffered a transient malfunction. There may have been an error in data transcription. Outliers arise due to changes in system behaviour, fraudulent behaviour, human error, instrument error or through natural deviations in populations. A sample may have been contaminated with elements from outside the population being examined. Alternatively, an outlier could be the result of a flaw in the assumed theory, calling for further investigation by the researcher. Additionally, the pathological appearance of outliers of a certain form appears in a variety of datasets, indicating that the causative mechanism for the data might differ at the extreme end. There is no rigid mathematical definition of.
There are various methods of outlier detection. Some are graphical such as normal probability plots. Others are model-based. Box plots are a hybrid. Model-based methods which are used for identification assume that the data are from a normal distribution, identify observations which are deemed "unlikely" based on mean and standard deviation: Chauvenet's criterion Grubbs's test for outliers Dixon's Q test ASTM E178 Standard Practice for Dealing With Outlying Observations Mahalanobis distance and leverage are used to detect outliers in the development of linear regression models. Subspace and correlation based techniques for high-dimensional numerical data It is proposed to determine in a series of m observations the limit of error, beyond which all observations involving so great an error may be rejected, provided there are